April 24, 2005 Contents

Preface VII

I 1 and 2-dimensional results 13

Introduction to Part I 15

1 History 17 1.1 Basic intuitions ...... 18 1.2 The fundamental group and homology ...... 19 1.3 The search for higher dimensional versions of the fundamental group ...... 21 1.4 The origin of the concept of abstract groupoid ...... 22 1.5 The van Kampen Theorem ...... 23 1.6 Proof of the van Kampen theorem for the fundamental groupoid ...... 25 1.7 The fundamental group of the circle ...... 31 1.8 Higher order groupoids ...... 33

2 Homotopy theory and crossed modules 35 2.1 Homotopy groups and relative homotopy groups ...... 36 2.2 Whitehead’s work on crossed modules ...... 41 2.3 The 2-dimensional van Kampen Theorem ...... 44 2.4 The classifying spaces of a group and of a crossed module ...... 47 2.5 Cat1-groups...... 50 2.6 The fundamental crossed module of a ﬁbration ...... 52 2.7 The category of categories internal to groups ...... 56

3 Basic algebra of crossed modules 61 3.1 Presentation of groups and identities among relations...... 63 3.2 van Kampen diagrams ...... 67

I II April 24, 2005

3.3 Precrossed and crossed modules ...... 70 3.4 Free precrossed and crossed modules ...... 73 3.5 Pre Cat1-groups and the existence of colimits ...... 76 3.6 Implementation of crossed modules in GAP ...... 77

4 Coproducts of crossed P-modules 79 4.1 The coproduct of crossed P -modules ...... 80 4.2 The coproduct of two crossed P -modules ...... 81 4.3 The coproduct and the van Kampen theorem ...... 85 4.4 Some special cases of the coproduct ...... 89

5 Induced crossed modules 95 5.1 Pullbacks of precrossed and crossed modules...... 96 5.2 Induced precrossed and crossed modules ...... 99 5.3 Induced crossed modules: Construction in general...... 102 5.4 Induced crossed modules and the van Kampen Theorem ...... 103 5.5 Calculation of induced crossed modules: the epimorphism case...... 106 5.6 The monomorphism case. Inducing from crossed modules over a subgroup ...... 108 5.7 On the ﬁniteness of some induced crossed modules ...... 113 5.8 Inducing crossed modules by a normal inclusion ...... 114 5.9 Computational issues for induced crossed modules ...... 122

6 Double groupoids and the 2-dimensional van Kampen theorem. 127 6.1 Double categories ...... 129 6.2 The category XMod of crossed modules of groupoids...... 134 6.3 Fundamental double groupoid ...... 138 6.4 Thin structures on a double category. The category DGpds of double groupoids...... 145 6.5 Connections in a double category: equivalence with thin structure...... 151 6.6 Equivalence between XMod and DGpds: folding ...... 156 6.7 Homotopy commutativity lemma ...... 162 6.8 Proof of the 2-dimensional van Kampen Theorem ...... 169

II Crossed complexes 177

Introduction to Part II 179

7 Crossed complexes: Basic properties 185 7.1 Topological prerequisites ...... 187 April 24, 2005 III

7.2 Crossed complexes and related categories ...... 188 7.2.1 Crossed complexes ...... 189 7.2.2 n-truncated crossed complexes and relation to Crs ...... 191 7.2.3 Modules over a groupoid ...... 192 7.2.4 Homotopy and homology groups of crossed complexes ...... 195 7.3 Colimits of crossed complexes ...... 196 7.3.1 Reduction of the computation of colimits to groupoids, modules and crossed modules . . 196 7.3.2 Induced modules and crossed modules ...... 197 7.3.3 Coproducts...... 200 7.3.4 Free modules and free crossed modules...... 202 7.4 GvKT for ﬁltered spaces and crossed complexes ...... 204 7.5 Induced modules and relative homotopy groups ...... 210 7.6 Crossed complexes of free type ...... 214

8 Crossed complexes: tensor product and homotopies 219 8.1 Modules over groupoids: Monoidal closed structure...... 223 8.2 The monoidal closed category Crs of crossed complexes ...... 226 8.2.1 The internal hom structure of Crs(C,D)...... 227 8.2.2 The bimorphisms as an intermediate step ...... 231 8.2.3 The tensor product of two crossed complexes ...... 232 8.3 Analysis of the tensor product C ⊗ D ...... 236

8.3.1 The groupoid (C ⊗ D)1...... 236

8.3.2 The crossed module (C ⊗ D)2 → (C ⊗ D)1...... 238

8.3.3 The tensor product (C ⊗ D)m,n for m, n ≥ 2 ...... 241

8.3.4 The tensor product (C ⊗ D)m,n for n = 0, 1...... 241 8.4 The tensor product of free crossed complexes is free ...... 242 8.5 The monoidal closed category FTop of ﬁltered spaces ...... 244 8.6 Tensor products and the fundamental crossed complex ...... 245 8.7 The homotopy addition lemma for a simplex...... 247

9 The simplicial classifying space 251 9.1 Simplicial sets and the nerve of a crossed complex ...... 252 9.1.1 Simplicial sets ...... 252 9.1.2 Homotopy of simplicial sets ...... 255 9.1.3 Realisation of simplicial sets...... 257 9.1.4 Simplicial sets as a closed category ...... 258 9.1.5 Simplicial sets and ﬁltered topological spaces ...... 260 IV April 24, 2005

9.2 The classifying space of a crossed complex ...... 260 9.2.1 The nerve of a crossed complex ...... 260 9.2.2 The classifying space of a crossed complex ...... 263 9.2.3 The pointed case ...... 264 9.3 Applications ...... 265 9.4 Fibrations of crossed complexes ...... 269

10 Resolutions. 275 10.1 The notion of syzygy and of resolution ...... 275 10.2 Free crossed resolutions of groups and groupoids ...... 277 10.3 Free products with amalgamation and HNN-extensions ...... 282 10.4 Acyclic models ...... 285

11 Construction of free crossed resolutions of groupoids 291 11.1 Covering morphisms of groupoids and crossed complexes ...... 292 11.2 Coverings of free crossed complexes ...... 295 11.3 A computational procedure ...... 297 11.4 Examples ...... 302 11.4.1 The standard crossed resolution of a group ...... 302 11.4.2 A small crossed resolution of ﬁnite cyclic groups ...... 303

12 Crossed complexes and chain complexes with operators 307 12.1 Modules and chain complexes over groupoids ...... 308 12.2 Augmentation modules ...... 312 12.3 Derived modules ...... 315 12.4 The functor ∆0 : Crs → Chn ...... 316 12.5 The functor ∆ : Crs → Chn ...... 318 12.6 Properties of ∆ : Crs → Chn ...... 320 12.6.1 ∆ and colimits ...... 320 12.6.2 ∆ and the closed category structure ...... 321 12.6.3 ∆ and the free standard resolution of a group ...... 323 12.7 The chain complex of a ﬁltered space and of a CW -complex...... 324

13 Cohomology of groups 327 13.1 Cohomology of a group ...... 330 13.2 Cohomology of groups as classes of crossed extensions ...... 331 13.3 An interpretation of H3(G, A) and some examples ...... 334 April 24, 2005 V

13.4 Dimension 2 cohomology with coeﬃcients in a crossed module and extension theory ...... 338 13.5 Obstructions to existence and classiﬁcation of extensions ...... 342 13.6 Local systems ...... 343

III ω-groupoids 347

Introduction to Part III 349

14 The algebra of cubes 351 14.1 Connections and compositions in cubical complexes ...... 351 14.2 ω-groupoids ...... 355 14.3 Crossed complexes ...... 356 14.4 Folding operations ...... 360 14.5 Skeleton and coskeleton ...... 366 14.6 The equivalence of ω-groupoids and crossed complexes ...... 370 14.7 Properties of thin elements ...... 373

15 Colimit theorems 377 15.1 Filter-homotopies ...... 378

15.2 %X∗ is an ω-groupoid ...... 381 15.3 The ﬁbration and deformation theorems ...... 383 15.4 The union theorem for ω-groupoids ...... 386 15.5 The colimit theorem for crossed complexes ...... 390 15.6 Homotopy and homology ...... 391 15.7 The free ω-groupoid on one generator ...... 393

16 Tensor products and homotopies 399 16.1 The monoidal closed structure on cubical sets ...... 400 16.2 The monoidal closed structure on ω-groupoids ...... 403 16.3 The monoidal closed structure on Crossed complexes ...... 408 16.4 The symmetry of tensor products ...... 418 16.5 The pointed case ...... 420 16.6 Computations ...... 421

16.7 The natural transformation ρ(X∗) ⊗ ρ(Y∗) → ρ(X∗ ⊗ Y∗)...... 427 VI April 24, 2005

Appendices 431

A A resum´eof category theory. 431 A.1 Categories, functors, natural transformations...... 431 A.2 Graphs. Free category associated to a graph ...... 433 A.3 Directed graphs. Free groupoid associated to a directed graph ...... 433 A.4 Limits and colimits...... 434 A.5 Adjoints...... 436 A.6 Some properties...... 437

B Closed categories. 439 B.1 Products of categories ...... 439 B.2 Closed categories ...... 441 B.3 The internal hom for categories and groupoids ...... 441 B.4 The monoid of endomorphisms in the case of Gpds ...... 443 B.5 The symmetry groupoid and the actor of a groupoid ...... 445 B.6 The case of a group ...... 447 B.7 Crossed modules and quotients of groups ...... 448 Part II

Crossed complexes

177

Introduction to Part II

The aim of Part II is to present a powerful tool in algebraic topology and the cohomology of groups, that of crossed complexes, and to show some major results and calculations that can be obtained using this tool. A major point is that crossed complexes can be used for some explicit calculations involving nonabelian information coming from the fundamental group. The fact that a van Kampen Theorem holds shows that this algebra gives a tool for some nonabelian local to global problems in dimensions > 1. Thus our aim is not just solve some speciﬁc problems in homotopy theory or algebraic topology, but also to suggest the kind of tools that might be required for a wider investigation of local to global problems, in view of the important part such problems play in mathematics and its applications. Recall that Part I has given a transition from 1-dimensional to 2-dimensional homotopy theory. Dimension 1 involves in a natural way the fundamental groupoid and the theory of groupoids. Dimension 2 homotopy has been expressed principally using the fundamental crossed module

∂ Π(X, A, x) = (π2(X, A, x) −→ π1(A, x)) of a pointed pair of spaces x ∈ A ⊆ X, and the theory of crossed modules. A Generalised van Kampen Theorem (GvKT) in dimension 2 allowed for many nonabelian computations of this crossed module, and so of these second relative homotopy groups. However for the proof of the GvKT we had to move to the category of double groupoids, and in this respect it was convenient to use also the fundamental crossed module of groupoids

∂ Π(X, A, C) = (π2(X, A, C) −→ π1(A, C)) for a triple of spaces C ⊆ A ⊆ X, so that C is thought of as a set of base points. This groupoid viewpoint will be essential in this Part II. It is also an important feature of crossed modules over groupoids that they model all weak homotopy 2-types. This modelling is done via the classifying space functor on crossed modules of groupoids

B : XMod → Top.

However the deﬁnition of this functor and the proof of its properties requires the work of this Part II, since it involves the generalisation from crossed modules to crossed complexes (see Chapter 9). Some aspects of the situation in dimensions > 3 are analogous to that in lower dimensions. In particular, a convenient generalisation of a triple of spaces is a ﬁltered space

X∗ : X0 ⊆ X1 ⊆ · · · ⊆ Xn ⊆ · · · ⊆ X, so that the case of a triple is where Xn = Xn+1 = X for n > 2. The above fundamental crossed module of a triple generalises to the fundamental crossed complex of a ﬁltered space

∂ ∂ ∂ ∂ ∂ Π(X∗): ··· −→ πn(Xn,Xn−1,X0) −→ πn−1(Xn−1,Xn−2,X0) −→ · · · −→ π2(X2,X1,X0) −→ π1(X1,X0).

179 180

Our ﬁrst task is to explain the algebraic notion of crossed complex and the structure and properties of this

Π(X∗). The ﬁrst major result we state and apply in Chapter 7, is that this functor

Π: FTop → Crs satisﬁes a Generalised van Kampen Theorem (GvKT): Π preserves certain colimits. Part of Chapter 7 thus discusses colimits of crossed complexes and in particular the notion of free crossed complex. Consequence of the GvKT are some results which are commonly regarded as basic in algebraic topology and homotopy theory, for example:

1. the Brouwer degree theorem: homotopy classes of maps Sn → Sn are classiﬁed by their degree while any map Si → Sn is inessential if i < n (see Corollary 7.5.9);

2. the relative Hurewicz theorem: if πi(X,A) = 0 for 1 < i < n and X,A are connected, then the Hurewicz map

πn(X, A, x) → Hn(X,A)

determines Hn(X,A) as πn(X, A, x) factored by the action of π1(A, x) (see Theorem 7.5.10);

n 3. if n > 2, then πn(A ∪ {eλ}, A, x) is a free π1(A, x)-module (crossed π1(A, x)-module if n = 2) on the characteristic maps of the n-cells (see Theorem 7.6.1);

4. if X∗ is the skeletal ﬁltration of a CW -complex, then Π(X∗) is a free crossed complex.

Put in another way, these results lend some support to the idea that crossed complexes give a convenient algebraic formalism for the theory and applications of relative homotopy theory. This was first recognised in a 1946 paper by Blakers ([15]) using the name “group system” for what is now called crossed complex (of groups). The applications were next developed by J.H.C.Whitehead in his 1949 paper ([175]) This paper has been considerably neglected in sharp contrast with the first part ([174]) where definition of CW -complexes has contributed a basic tool for algebraic topology (See more history on [43])

A second major set of results, stated in Chapter 8 and applied there and in later chapters, deals with homotopies for ﬁltered spaces and for crossed complexes. We introduce them in terms of more general structures like monoidal closed categories and the cylinder construction.

The category of filtered spaces has a tensor product X∗ ⊗ Y∗ defined by [ (X∗ ⊗ Y∗)n = Xp × Yq p+q=n for filtered spaces X∗,Y∗. This models the skeletal filtration of a product X × Y of CW -complexes, where the n-cells of the product are of the form ep × eq for all p + q = n and p-cells ep of X, q-cells eq of Y .

This tensor product allows one to define homotopies for filtered spaces as maps I∗ ⊗ X∗ → Y∗ where I∗ is the unit interval with its usual cell structure. There is also an internal hom filtered space FTOP(Y∗,Z∗), with total space Top(X,Y ), and an exponential law

∼ FTop(X∗ ⊗ Y∗,Z∗) = FTop(X∗, FTOP(Y∗,Z∗)).

We require analogous structures for crossed complexes, and indeed it is these which give crossed complexes extra power. 181

In Chapter 8 we define for crossed complexes D,E an internal hom crossed complex CRS(D,E), whose elements in dimension 0 are morphisms D → E of crossed complexes, in dimension 1 are homotopies of morphisms, and in higher dimensions are forms of ‘higher homotopies’. The definition is completely explicit, except that the verification of the axioms for a crossed complex is left till Part III. Then a tensor product C ⊗ D of crossed complexes is defined precisely so that the exponential law holds, i.e. there is a natural bijection

Crs(C ⊗ D,E) =∼ Crs(C, CRS(D,E)) for all crossed complexes C,D,E. This adjoint relation enables us to prove that a tensor product of free crossed complexes is free.

Here C ⊗ D is generated in dimension n by all elements c ⊗ d for c ∈ Cp, d ∈ Dq and with p + q = n. The complication of the rules required to hold reﬂects the complications of the geometry which is being modelled, namely that of the product Ep × Eq where the cell Ep has a cell subdivision

0 0 1 0 1 p 0 p−1 p E = e ,E = e± ∪ e ,E = e ∪ e ∪ e , p > 2.

Thus in general the product has 9 cells, and a ‘cylinder’ E1 × Ep has cells in dimensions 0, 1, p − 1, p, p + 1. The capacity to contain information on what happens in a range of dimensions is what gives crossed complex the extra power over chain complexes, even with a group of operators. A precise comparison with the latter and crossed complexes is given in Chapter 12. We have to assume some properties of the tensor product and of the functor Π whose proofs are deferred to Part III. There we give a precise algebraic relation between crossed complexes and a cubical algebraic theory, and this relation is the basis of the power of the crossed complexes theory and applications. One of the results we are able to prove on these assumptions is that the functor Π preserves homotopies. This is based on an important natural transformation

η : Π(X∗) ⊗ Π(Y∗) → Π(X∗ ⊗ Y∗) which enables us to translate homotopies from ﬁltered spaces to crossed complexes. Further, η is an isomorphism if Π(X∗), Π(Y∗) are free crossed complexes, for example if X∗,Y∗ are CW -ﬁltered spaces. One consequence of these results is the Homotopy Addition Lemma for a simplex: intuitively, this says that the boundary ∂σ of an n-simplex σ is the ‘sum’ of all its faces. How to express this intuition in precise terms was one of the problems that led to the foundation of algebraic topology. One longstanding description of this in chain complexes and homology is that for all n > 0

Xn i ∂σ = (−1) ∂iσ i=0 where for any n-simplex σ, ∂iσ is the face opposite to the i-th vertex, so that the vertices are given the structure of a totally ordered set. However, for homotopical purposes we need to involve not the abelian homology groups, but instead the fundamental groupoid and its action on relative homotopy groups. This leads to diﬀering formulae for n = 1, 2, 3 and n > 4. We shall show that we can model in crossed complexes the inductive construction of an n-simplex as a cone on the (n − 1)-simplex, and hence give an algebraic deduction of these formulae, based on the algebraic formulae in the tensor product of crossed complexes (see Theorem 8.7.4). These results are applied in Chapter 9 to develop using simplicial theory the classifying space functor

B : Crs → Top. 182

The main result is a homotopy classiﬁcation theorem stating that if X∗ is a CW -ﬁltered space, and C is a crossed complex, then there is natural bijection of homotopy classes ∼ [X,BC] = [Π(X∗),C].

We show how this leads to some specific calculations of homotopy classes of maps. We also develop the notion of fibration of crossed complexes, and its relation with fibration of spaces. It is at this point that the simplicial classifying space theory works more smoothly than the cubical one (to be developed in Part III) because it is a standard result that the realisation of a Kan fibration of simplicial sets gives a fibration of spaces. Another important result is the family of exact sequences derived from such a fibration of crossed complexes. It has a number of uses, for example in the theory of group extensions. This is exploited in Chapter 13.

It is time to remember from Part I that the sources for crossed modules are twofold. From Topology the fundamental crossed module of a triple of spaces, and, from Algebra, combinatorial group theory, i.e. the presentation of groups. Until now, we have presented the generalization to crossed complexes of the ﬁrst source. It is time to turn to Algebra. In Chapter 10 we move in this direction by considering the notion of free resolution of a group or groupoid. This develops the work on Identities among Relations in Chapter 3 of Part I. We prove that two of these resolutions are homotopy equivalent, thus making the homotopy computations independent of the resolution chosen. In particular, we give some examples of resolutions having few generators that make computations feasible. Also related to the concept of resolution, we introduce the theory of acyclic models for crossed complexes, and give several important applications. Chapter 11 gives a procedure for generating resolutions associated to a given presentation of a group G. This works by constructing for the universal covering groupoid Ge of G, obtained from the adjoint action of G on itself, a free crossed resolution F∗(Ge) together with a contracting homotopy of this resolution. This connects these methods with classical procedures involving paths in Cayley graphs. Here we use also Cayley graphs with relations, and higher syzygies, for which covering morphisms of crossed complexes gives a useful algebraic model and one appropriate for calculations. We end this Chapter by applying this procedure to get some free resolutions. In particular, for each group st G, we get the free standard crossed resolution F∗ (G), a crossed complex version of the bar construction so much used in homological algebra. Also, we get the resolutions with few generators that have been used in Chapter 10 Chapter 12 gives the relation between crossed complexes and chain complexes with a groupoid of operators. This builds on a number of constructions in homological and homotopical algebra which are classical in the case of a group of operators, but for which the extension to a groupoid of operators gives extra clarity and power. Thus we deﬁne the category Chn of chain complexes with a groupoid of operators and study its closed category structure. The relation to crossed complexes is given by a pair of adjoint functors

∆ : Crs → Chn :Θ that satisfy several properties with respect to both closed categories structures. As a consequence of the general theory of this Chapter, we get a relation between the free standard crossed resolution and the bar resolution of a group. In particular, for crossed complexes B, C and chain complexes L there are natural isomorphisms CRS(C, ΘL) =∼ ΘCHN(∆C,L), 183 giving isomorphism between the corresponding homotopy classes. Since the chain complexes associated by ∆ to the free standard crossed resolution of a group G is the bar resolution, we can redefine the cohomology of a group G with coefficients in a G-module using free crossed resolutions. This is carried out at the beginning of Chapter 13. This gives a natural theory which also generalises in a number of ways, for example giving cohomology of a group or space with coefficients in a crossed complex. This somewhat nonabelian theory allows for more calculations than are easily possible in the abelian theory, for example of the k-invariant of a crossed module. This ends our account of the theory and application of crossed complexes, apart from the proofs for crossed complexes and associated functors of a number of major properties which have been used extensively in Part II. 184 Chapter 7

Crossed complexes: Basic properties

This ﬁrst Chapter of Part II is thought of as a quick introduction to the themes we are going to study in this Part. It revolves around the Generalised van Kampen Theorem (GvKT) for ﬁltered topological spaces in its crossed complexes form. Recall that, as said in the preface: “Many of the main aims of the book can be summarised by stating that we construct a diagram, which we call the Main Equivalence (ME) :

(ﬁltered spaces) nn MMM nnn MMM Π nnn MMρ nn MMM (ME) nnn MM nnn MMM wn λ & (crossed complexes) o / (ω-groupoids) γ such that

(A) γ, λ give an equivalence of categories;

(B) γρ is naturally equivalent to Π;

The ﬁnal statement we call a Generalised van Kampen Theorem (GvKT)” We deal in this Chapter with the Π part of the diagram (ME), giving the deﬁnition of Π, the statement of the GvKT, some classical homotopical results that are got as consequences and some new ones. So, this Chapter can be seen as a generalisation to all dimensions of concepts introduced in Chapter 2 of Part I, i.e. the fundamental crossed module of a pair of spaces, and of some results of Chapters 4 and 5. Some particular case of the algebraic objects we are going to introduce were already studied in the pioneering years of homotopy theory (for instance, Blakers in[15] and Whitehead in [175]). They worked with the sequence of homotopy groups

∂n+1 ∂n ∂n−1 ∂3 ∂2 ··· / πn(Xn,Xn−1, x) / πn−1(Xn−1,Xn−2, x) / ······ / π2(X2,X1, x) / π1(X1, x)

185 186 associated to a ﬁltration

X∗ = X0 ⊆ X1 ⊆ X2 ⊆ · · · ⊆ Xn ⊆ · · · ⊆ X in a space X and to a base point x ∈ X0. This sequence included in what we call the fundamental crossed complex Π(X∗) of a filtered space X∗. To generalise this sequence, they introduced the concept of homotopy systems. They are a particular kind of what we call crossed complexes (in particular, they are of free type) We devote the first Section to an easy introduction of the category FTop of filtered topological spaces paying some attention to the standard example, the skeletal filtration of a CW -complex.

The next few Sections revolve around the algebraic counterpart. We are going to use groupoids because, among other advantages, they allow us to consider several base points and also take into account the full action of the fundamental groupoid. Thus we have to generalise some well known categories and introduce some new ones. In the second Section we introduce the category Crs of crossed complexes (over groupoids), generalising the idea of homotopy systems. This is our basic category in this Part II, so we spend quite a few pages relating the crossed complexes to some easier structures. First we consider n-truncated crossed modules, concentrating in the ﬁrst n-dimensions. It is particularly interesting the 2-truncation that gives a crossed module over a groupoid. They form the category Crs2 and include the crossed modules over groups studied in Chapters 2-5. We could also concentrate in the phenomena arising just in dimension n (for n ≥ 3), getting then a module over a groupoid. They form the category Mod and generalise the well known structure of module over a group.

Previous to the statement of the GvKT and the driving of some consequences, we devote Section 7.3 to the study of colimits in the category Crs. It is an easy consequence of the appropriate functors having right adjoints (and hence preserving colimits) that the colimits in Crs can be computed in a three step process. First a colimit in groupoids, then a colimit of crossed modules and, last, colimits of modules in dimensions n ≥ 3. We develop further this result, studying how to decompose the computation of colimits in Mod (and XMod) in two steps. First a change of base groupoid (via the induced module construction), and, then, a colimit in the category of modules over a ﬁxed groupoid. We ﬁnish this quite algebraic Section by studying some examples, like coproducts, pushouts and free modules.

The following Section 7.4 gives the statement of the GvKT and some direct consequences. In essence, the GvKT indicates under which conditions, the functor Π preserves some colimits. The conditions are that we have an open cover of X so that all induced ﬁltrations are connected. Then X is the coequaliser of the open cover and we conclude that its fundamental crossed complex is the coequaliser of the fundamental crossed complexes of the elements of the open cover. This already implies some results, for instance in the case of a CW -complex X, but since coequalisers are not widely known, we state the GvKT in the particular case of pushouts that is easier to use. We also generalise the Theorem getting a pushout form for the case were X is an adjunction space. We end the Section by applying this to a quotient of spaces. Now it is time to reap some classical results. We apply the GvKT in Section 7.5 to recover the Brouwer degree and the Relative Hurewicz Theorems, proving some of the power of our approach. This is done translating the general results to the situation when the ﬁltration has only two stages, i.e. it is 187 the associated to a pair of spaces (X,A). Then both the conditions and the consequences of the GvKT translate in terms of connectedness of pairs.

The last Section 7.6, gives an algebraic kind of crossed complexes that are interesting because, from them, it is easy to construct morphism of crossed complexes. They generalise the structure of the fundamental crossed complex of a CW -complex and are called crossed complexes of a free type.

7.1 Topological prerequisites

By a space is meant a compactly generated topological space X, i.e. one which has the ﬁnal topology with respect to all continuous functions C → X for all compact Hausdorﬀ spaces C. The category of spaces and continuous maps will be written Top. More about the basic properties of these spaces and this category can be found in [28].

Deﬁnition 7.1.1 A ﬁltered space X∗ consists of a space X and a sequence of nested subspaces

X∗ = X0 ⊆ X1 ⊆ X2 ⊆ · · · ⊆ Xn ⊆ · · · ⊆ X which we call a ﬁltration of X. A ﬁltration preserving map

f : X∗ → Y∗ is a continuous map f : X → Y such that f(Xn) ⊆ Yn for all n ≥ 0. These objects and morphisms give the category FTop of ﬁltered spaces and ﬁltered maps. 2

Example 7.1.2 Let us name some of the most common ﬁltered spaces.

n n 1. We denote the standard n-simplex by ∆ , and the same space with its skeletal ﬁltration by ∆∗ . In 1 1 particular, we write I for ∆ , and I∗ for the corresponding ﬁltered space ∆∗.

n 2. We write I∗ for the filtered space of the skeletal filtration of the standard n-cube. 3. To define the filtered n-ball, we fix some notation. The standard n-ball and (n − 1)-sphere are the usual subsets of the Euclidean space of dimension n:

En = {x ∈ Rn | ||x|| 6 1}, Sn−1 = {x ∈ Rn | ||x|| = 1}

where ||x|| is the standard Euclidean norm. n We write E∗ for the ﬁltered space of the ﬁltration of the n-ball given by the base point up to dimension n − 2, Sn−1 in dimension n − 1 and En in dimensions ≥ n. Thus

1 1 E∗ = I∗ = ∆∗,

and for n ≥ 2 n n−1 n E∗ = {1} = ··· = {1} ⊆ S ⊆ E .

The category FTop has a set of standard examples that include all the previous ones, namely the CW - complexes. These are spaces built up in inductive fashion by attaching cells. We recall their construction, which also gives a preparation for Section 7.6 where we work analogously with crossed complexes. 188

We begin by explaining the process of attaching cells to a space. Let A be a space, Λ a set of indexes, m −1 {fλ}λ∈Λ a family of continuous maps fλ : S λ → A, and we consider the adjunction space

mλ X = A ∪{fλ} {e }λ∈Λ, given by the pushout diagram,

F (fλ) mλ−1 / λ∈Λ S A

j F (hλ) mλ / λ∈Λ E X. Then we say that the space X is obtained from A by attaching cells. By standard properties of adjunction spaces (see [28]), the map j is a closed injection, and so we assume it is an inclusion. As examples, we have

E1 = {0, 1} ∪ e1 and En = e0 ∪ en−1 ∪ en for n ≥ 2.

m The maps hλ : E λ → X are called the characteristic maps of the cells. It is a standard fact that they m m m are homeomorphisms on the interior of each E λ . The images e λ = hλ(E λ ) in X are called the cells of X m relative to A. We say that X is obtained from A by attaching the cells {e λ }λ∈Λ. It is important to notice that a map f : X → Y is continuous if and only f|A is continuous and each composite fhλ is continuous.

We construct a CW -complex by attaching cells in an inductive process. We can form a sequence of spaces n −1 n n−1 X by starting with X = A and forming X by ‘attaching’ to X a family of n-cells indexed by a set Λn. n−1 n−1 That is for each n ≥ 0 we choose a family of maps fλ : S → X , λ ∈ Λn, and deﬁne

n n−1 n n X = X ∪{fλ} {eλ}λ∈Λn and X = colim X .

The canonical map j : A → X is called a relative CW -complex. Clearly, the Xn (called the relative n-skeleton) form a filtration of X that we denote X∗. If A = ∅, we say that X is a CW -complex. The cells, characteristic maps, etc., of a relative CW -complex are defined as before and we regard them as part of the structure. The advantage of this structure is that it allows proofs by induction on n. For instance, n a map f : X → Y is continuous if and only each restriction fn : X → Y is continuous and this happens if and only if each composite fhλ is continuous, for all λ ∈ Λn and all n ≥ 0. Thus we may construct a map f : X → Y by induction on skeleta starting with X0, which is just the disjoint union of A and Λ0. We can conveniently write n X = A ∪ {eλ}λ∈Λn,n≥0, and may abbreviate this in some cases, for example to X = A ∪ en ∪ em. For A = ∅ we get that X with its characteristic maps is simply a CW -complex. n n−1 0 Let E∗ and S∗ denote the skeletal filtrations of the standard n-ball and (n − 1)-sphere, where E = {0}, S−1 = ∅, E1 = I = {0, 1} ∪ e1, S0 = {0, 1} , and for n ≥ 2, En = {1} ∪ en−1 ∪ en, Sn−1 = {1} ∪ en−1. More detail of this, including the characteristic maps, is given in [28].

7.2 Crossed complexes and related categories

The structure of crossed complex is suggested by the canonical example, the fundamental crossed complex Π(X∗) of the ﬁltered space X∗, in particular that of a CW -complex with its skeletal ﬁltration. 189

For any ﬁltered space X∗ and any x ∈ X0, there is a sequence of groups and homomorphisms (abelian for n ≥ 3):

∂n+1 ∂n ∂n−1 ∂3 ∂2 ··· / πn(Xn,Xn−1, x) / πn−1(Xn−1,Xn−2, x) / ······ / π2(X2,X1, x) / π1(X1, x) which was extensively used by J.H.C.Whitehead in [175]. In this sequence, πn(Xn,Xn−1, x) are the relative homotopy groups, ∂n are the boundary maps and the composition and the actions are those studied in Section 2.1 of Part I.

When we vary the base point in X0 we get groupoids πn(Xn,Xn−1,X0) all having the same set of objects X0. As usual, we shall write πn(Xn,Xn−1) for πn(Xn,Xn−1,Xn−1). This sequence of groupoids satisfies some properties that could be proved directly, but for which there are a lot of details to be worked out. We shall deduce these properties from the full definition and construction of the homotopy ω-groupoid ρ(X∗). It turns out that Π(X∗) can be considered as a substructure γρ(X∗) of ρ(X∗), and in this way these properties are verified in Chapter 15. In this Section we define the category of crossed complexes, then some related categories and, in particular, we study the fundamental crossed complex of a filtered space with emphasis in the skeletal filtration of a CW -complex.

7.2.1 Crossed complexes

We now write a full deﬁnition of the category of crossed complexes and give some examples.

Deﬁnition 7.2.1 Let C1 be a groupoid. We write C0 for its objects, C1(p, q) for the set of morphisms from p to q (p, q ∈ C0) and C1(p) for the group C1(p, p).

A crossed complex C over C1 is written as a sequence

/ δn / δn−1 / δ3 / δ2 / ··· Cn Cn−1 ······ C2 C1 and it is given by the following three sets of data:

1.- For n > 2, Cn is a family of groups {Cn(p)}p∈C0 and for n > 3, the groups Cn(p) are Abelian. This is equivalent to say that Cn is a totally disconnected groupoid with the same set of objects as C1, namely C0.

We generally shall use additive notation for all groups Cn(p), n > 3, and multiplicative notation for n = 1, 2, and we shall use the symbol 0 or 1 for their respective identity elements.

2.- For (n > 2) an action of the groupoid C1 on the right on each Cn,

Cn × C1 → Cn

a a denoted (x, a) 7→ x , such that if x ∈ Cn(p) and a ∈ C1(p, q) then x ∈ Cn(q).

Later on, we shall see that this property is equivalent to say that Cn is a C1-module (see Deﬁnition 7.2.7). a −1 We shall always consider C1 acting on C1 by conjugation, i.e. x = a xa for all x ∈ C1(p) and a ∈ C1(p, q). As an example of our use of notation, two of the conditions for an action are written xab = (xa)b, x1 = x in all dimensions, but the third condition is expressed as (xy)a = xaya for n = 1, 2, and (x + y)a = xa + ya for n > 3. ∼ A consequence of the existence of this action is that Cn(p) = Cn(q) if p and q lie in the same component of the groupoid C1, i.e. when there is a morphism in C1, from p to q. 190

3.- For n > 2, δn : Cn → Cn−1 is a morphism of groupoids over C0 and preserves the action of C1. These three set of data satisfy the properties

CX1) C is analogous to a chain complex, i.e. δn−1δn = 0 : Cn → Cn−2 for n > 3.

CX2) Im δ2 acts trivially on Cn for n > 3 and as conjugation on C2.

Notice that CX2) has two parts. The second part just says that C2 is a crossed module over the groupoid δ2c −1 C1, i.e. for x, c ∈ C2, x = c xc. The ﬁrst part of CX2) indicates that C1 acts through Coker δ2. We shall come back later to this point. These two axioms give a good reason for the name ‘crossed complex’, it has a ‘head’ that is a crossed module and a ‘tail’ that is a (kind of) chain complex.

A morphism of crossed complexes f : C → D is a family of morphisms of groupoids fn : Cn → Dn (n > 1) all inducing the same map of vertices f0 : C0 → D0, and compatible with the boundary maps and the actions a f1(a) of C1 and D1. This means that δnfn(x) = fn−1δn(x) and fn(x ) = fn(x) for all x ∈ Cn and a ∈ C1. We represent the morphisms of crossed complexes by a commutative diagram

/ δn / δn−1 / δ3 / δ2 / ··· Cn Cn−1 ······ C2 C1

fn fn−1 f2 f1 / δn / δn−1 / δ3 / δ2 / ··· Dn Dn−1 ······ D2 D1

We denote by Crs the resulting category of crossed complexes.

In the case when C0 is a single point we call C a reduced crossed complex, or a crossed complex over a group. Those crossed complexes give a full subcategory of Crs.

We can also ﬁx the groupoid C1 = G and restrict the morphisms to those inducing the identity on G, getting then the category CrsG of crossed complexes over G. 2

Example 7.2.2 Let us consider some examples.

1. The ﬁrst set of examples of a crossed complex comes from topology. The fundamental crossed complex of

a ﬁltered space Π(X∗) is given by the sequence

∂n+1 ∂n ∂n−1 ∂3 ∂2 ··· / πn(Xn,Xn−1,X0) / πn−1(Xn−1,Xn−2,X0) / ······ / π2(X2,X1,X0) / π1(X1,X0)

all of them being groupoids with X0 as set of objects.

2. There are many algebraic examples of crossed complexes to be described in Subsection 7.2.3. As some immediate examples that will be generalised later in Deﬁnition 7.2.9, we have: - Associated to a positive integer n > 1 and a group G, abelian if n > 2, we have the crossed complex

En(G) deﬁned by G in dimension n and trivial elsewhere;

En(G) = ··· / 0 / G / 0 / ······ / 0 / 0

- Associated to any G-crossed module M = (µ : M → G), we have the crossed complex E(M) deﬁned by

µ E(M) = ··· / 0 / ······ / 0 / M / G. 191

7.2.2 n-truncated crossed complexes and relation to Crs

We will use ﬁnite-dimensional versions of crossed complexes.

Deﬁnition 7.2.3 An n-truncated crossed complex is a ﬁnite sequence

δn / / δ3 / δ2 / Cn Cn−1 ······ C2 C1 satisfying all the axioms in so far as they make sense. We denote by Crsn the category of n-truncated crossed complexes. Notice that a 1-truncated crossed complex is simply a groupoid, and a 2-truncated crossed complex is a crossed module over a groupoid. Thus we can consider Crs1 = Gpds and Crs2 = XMod.

An n-truncated crossed complex could be described also as a crossed complex with Ck = 0 for k > n. Thus the n-skeleton functor n sk : Crsn → Crs,

δn / δn−1 / / δ2 / maps an n-truncated crossed complex C = (Cn Cn−1 ······ C2 C1) to

n / / δn / δn−1 / / δ2 / sk (C) = ··· 0 Cn Cn−1 ······ C2 C1 .

This n-skeleton functor allows us to consider Crsn as a full subcategory of Crs. On the other direction, there are n-truncation functors

trn : Crs → Crsn given by considering only the ﬁrst n groupoids in a crossed complex. 2

Proposition 7.2.4 The n-skeleton functor is left adjoint to the n-truncation functor.

Proof For any crossed complex D and n-truncated crossed complex C there is an obvious bijection

n Crs(sk (C),D) → Crsn(C, trn(D)) because a morphism of crossed complexes skn(C) → D is just given by the ﬁrst n maps since all others are the 0 maps as in the diagram

/ / δn / δn−1 / δ3 / δ2 / ··· 0 Cn Cn−1 ······ C2 C1

0 fn fn−1 f2 f1 / / δn / δn−1 / δ3 / δ2 / ··· Dn+1 Dn Dn−1 ······ D2 D1

The n-truncation functor has also a right adjoint very close to the n-skeleton functor.

Deﬁnition 7.2.5 We deﬁne the n-coskeleton functor

n cosk : Crsn → Crs, 192

δn / δn−1 / / δ2 / on an n-truncated crossed complex C = (Cn Cn−1 ······ C2 C1) by

n / / δn / δn−1 / / cosk (C) = ··· 0 Ker δn Cn Cn−1 ······ C1 for n ≥ 2 and by 1 / / / cosk (C) = ··· 0 1C1 C1

where 1C1 is the totally disconnected groupoid formed by the vertex groups of C1. 2

Notice that the only diﬀerence with the skeleton functor is the existence of elements of dimension n + 1. The importance of this is realised when proving adjointness.

Proposition 7.2.6 The n-coskeleton functor is the right adjoint of the n-truncation functor.

Proof For any crossed complex C and n-truncated crossed complex C0 there is an obvious bijection

n Crs(C, cosk (D)) → Crsn(trn(C),D) because a morphism f from C to coskn(D) is just given by the ﬁrst n maps since the (n + 1)st has to be fn+1 = fnδn+1 and all others have to be the 0 maps as in the diagram

/ / δn+1 / δn / δn−1 / δ3 / δ2 / ··· Cn+2 Cn+1 Cn Cn−1 ······ C2 C1

0 fnδn+1 fn fn−1 f2 f1 δ0 δ0 δ0 δ0 / / 0 / n / n−1 / 3 / 2 / ··· 0 Ker δn Dn Dn−1 ······ D2 D1

n 0 Notice that we need the (n + 1)st dimensional part of cosk (D) to be Ker δn, in order to be able to deﬁne fn+1 0 by fnδn+1 because δnfnδn+1 = fn−1δnδn+1 = 0. 2

7.2.3 Modules over a groupoid

The truncation functor allows us to study phenomena that happen up to some dimension. We are going to deﬁne a functor that allows to study one dimension at a time. We need a functor to a category appropriate to such cases and the category varies with the dimension. For n = 2 the category XMod of crossed modules and the 2-truncation functor does the job. For n ≥ 3 we have to deﬁne the category of modules over groupoids which generalises that of G-modules for a group G and functors from the category of crossed complexes restricting to dimension n.

Deﬁnition 7.2.7 A module over a groupoid is a pair (M,G), where G is a groupoid with set of vertices G0, M is a totally disconnected abelian groupoid with the same set of vertices, and there is given an action of G on M.

Thus M is given by a family of abelian groups {Mp}p∈G0 and the G action is given by a family of maps

Mp × G(p, q) → Mq

a b (ab) for all p, q ∈ G0. These maps are denoted by (x, a) 7→ xa and satisfy the usual properties, i.e. (x ) = x and a a a a (x + y) = x + y . In particular, any a ∈ G(p, q) induces an isomorphism x 7→ x from Mp to Mq. 193

A morphism of modules is a pair (θ, f):(M,G) → (M 0,G0), where f : G → G0 and θ : M → M 0 are morphisms of groupoids and preserve the action. Thus θ is given by a family of group morphisms

0 θp : Mp → Mf(p)

a f(a) for all p ∈ G0 satisfying θq(x ) = (θpx) , for all a ∈ G(p, q), x ∈ Mp. We can deﬁne the category Mod having modules over groupoids as objects and the morphisms of modules as morphisms. As usual, we can restrict to modules over groups getting the more common category of modules over groups, or to modules over the same groupoid G and morphisms such that f is the identity, getting the category ModG of modules over G. 2

With this category we can restrict a crossed complex to any given dimension.

Deﬁnition 7.2.8 The functor restriction to dimension n

0 (−)n : Crs → Mod

0 is given by (−)nC = (Cn,C1)

0 We use the notation (−)n because later we are going to deﬁne later another restriction taking notice of the condition CX2) that is more interesting to us. We shall keep the notation (−)n for that restriction (see Deﬁnition 7.2.12). Clearly a module over a groupoid can be seen as a crossed complex that is G in dimension 1, M in dimension n and zero elsewhere.

Deﬁnition 7.2.9 For each n > 3, we have a functor

En : Mod → Crs deﬁned on objects so that the image of a module (M,G) is the crossed complex

En(M,G) = ··· / 0 / M / 0 / ······ / 0 / G.

It is easy to deﬁne the value on morphisms.

0 It is clear that the functor En is the right inverse of (−)n the restriction to dimension n functor for any n ≥ 3. Thus Mod can be embedded as a (full) subcategory of Crs for any dimension n ≥ 3. 0 Trying to check if En is the adjoint for either side of (−)n, we have to define a morphism of crossed complexes from C to En associated to a morphism of modules from Cn to M. There is no problem to define the morphism in dimension n, but we run into difficulties to define it in dimension n + 1 commuting the appropriate diagram.

We need to modify En on the same line that we modiﬁed the skeleton functor to get the coskeleton functor.

Deﬁnition 7.2.10 For each n > 3, we have a functor

En : Mod → Crs 194 deﬁned on objects so that the image of a module (M,G) is the crossed complex

Id En(M,G) = ··· / 0 / M / M / 0 / ······ / 0 / G. where the M are in dimensions n and n − 1 and the map between them is the identity. It is easy to deﬁne the value on morphisms.

0 Proposition 7.2.11 The functor (−)n is right adjoint to En.

Proof We need to study Crs(En(M,G),C), i. e. morphisms of crossed complexes

0 Id 0 0 0 ··· / 0 / M / M / ··· / 0 / G

0 fn fn−1 0 f1 / / / / / / ··· Cn+1 Cn Cn−1 ··· C2 C1 δn+1 δn δn−1 δ3 δ2

It is clear that these are determined by fn : M → Cn and f1 : G → C1 such that fn preserves the action via f1, because fn−1 = δnfn. Thus ∼ Mod((M,G), (Cn,C1)) = Crs(En(M,G),C) giving the adjointness. 2

To get the left adjoint is a bit more difficult because we run already into difficulties to define the morphism of crossed complexes in dimension 2. This is solved once we realize that the first part of condition CX2) in the definition of a crossed complex just means that all Cn for n ≥ 3 are Coker δ2 modules. Thus we can give another choice for the restriction functor.

Deﬁnition 7.2.12 We deﬁne the restriction to dimension n functor

(−)n : Crs → Mod given on objects by (−)n(C) = (Cn, Coker δ2).

Proposition 7.2.13 The functor (−)n is left adjoint to En+1.

Proof We need to study Crs(C,En+1(M,G)), i. e. morphisms of crossed complexes

/ δn+2 / δn+1 / δn / δ3 / δ2 / ··· Cn+2 Cn+1 Cn ··· C2 C1

0 fn+1 fn 0 f1 ··· / 0 / M / M / ··· / 0 / G 0 Id 0 0 0

It is clear that this diagram produces a morphism of modules

(θ, f):(Cn, Coker δ2) → (M,G) where f : G → Coker δ2 is induced by f1 (exists because f1δ2 = 0) and θ = fn. 195

On the other hand, given a morphism of modules

(θ, f):(Cn, Coker δ2) → (M,G) we get the morphism of crossed module putting f1 = fπ (π being the projection π : C1 → Coker δ2), fn = θ and fn+1 = fnδn+1. These correspondences give the adjointness. 2

7.2.4 Homotopy and homology groups of crossed complexes

Let us deﬁne some functors giving direct algebraic and set theoretic invariants of crossed complexes. The ﬁrst one expresses the connectivity of the basic groupoid G1.

Deﬁnition 7.2.14 The set of components π0(C) of the crossed complex C is just the set of components of the groupoid C1. This deﬁnition gives a functor

π0 : Crs → Sets.

Example 7.2.15 It is easy to see that for the skeletal ﬁltration of a CW -complex, π0Π(X∗) is bijective to π0(X).

The next invariant of a crossed complex we are going to introduce is π1C the cokernel of the crossed module part of the crossed complex.

Deﬁnition 7.2.16 The fundamental groupoid of a crossed complex C is the groupoid π1C given by the cokernel of δ2, C1 π1C = Coker δ2 = . Im δ2

A morphism f : C → D of crossed complexes induces morphisms f∗ : π1(C) → π1(D), giving a functor

π1 : Crs → Gpds.

Example 7.2.17 Using the homotopy long exact sequence, it is easy to see that for any ﬁltration X∗ satisfying ∼ π1(X2,X1,X0) = 0, we have π1Π(X∗) = π1(X2,X0). In particular for the skeletal ﬁltration of a CW -complex, ∼ we have π1Π(X∗) = π1(X,X0).

Let us consider some other functors associating π1(C)-modules to crossed complexes.

Deﬁnition 7.2.18 For any crossed complex C and for n > 2 there is a totally disconnected groupoid given by a family of abelian groups, ½ ¾ Ker δn(p) Hn(C) = {Hn(C, p) | p ∈ C0} = | p ∈ C0 . Im δn+1(p) They are called the n-homology groups of the crossed complex C.

A morphism f : C → D of crossed complexes induces morphisms f∗ : Hn(C) → Hn(D), for all n > 2. 2 196

Exercise 7.2.19 Prove that for a crossed complex C there is an induced action of π1C on the family Hn(C) of homology groups, for n > 2 making Hn(C) a π1C-module. Thus each homology group gives a functor

Hn : Crs → Mod.

Deﬁnition 7.2.20 A morphism f : C → D is a weak equivalence if it induces a bijection π0C → π0D and isomorphisms π1(C, p) → π1(D, fp), Hn(C, p) → Hn(D, fp) for all p ∈ C0 and n ≥ 2. 2

Example 7.2.21 We shall see in Section 12.7 that Hn(ΠX∗, p) is isomorphic to Hn(Xep), the nth homology group of the universal cover of X based at p.

7.3 Colimits of crossed complexes

There is one contrast between the 2-dimensional case and that in higher dimensions. Homotopy 2-types are well modelled by crossed modules, but, for n > 3, crossed complexes give only a partial model of homotopy n-types. Thus crossed complexes give only a limited amount of information on many problems we would like to study. Further, the GvKT itself has strong connectivity assumptions which again limit its utility. However the theorem does yield applications which are not obtainable by other means. The utility of crossed modules for certain nonabelian homotopical calculations in dimension 2 has been shown in Part I, and this in itself is a great advance. In Part II, we obtain homotopical calculations using crossed complexes, as applications of a GvKT, Theorem 7.4.3, stated in the next section. For this we need to know how to compute colimits of crossed complexes in more familiar terms. We ﬁrst show that the determination of colimits in Crs can be reduced to the determination of colimits in (i) the category Gpds of groupoids (ii) the category XMod of crossed modules (over groupoids), and (iii) the category Mod of modules (over groupoids). Then, we prove that getting colimits in Mod (or in XMod) may be done in two steps. First we change the base groupoid so that it is the same for all modules and then we get the colimit in ModG (resp. XMod). We end the Section by studying some particular cases of colimits: coproducts, pushouts, etc. and the concept of free module.

7.3.1 Reduction of the computation of colimits to groupoids, modules and crossed modules

We prove that the determination of colimits in Crs can be reduced to the determination of colimits in three categories.

Proposition 7.3.1 Let C = colim Cλ be a colimit in the category Crs of crossed complexes. Then

1 1 λ λ (i) for n = 1, the groupoid tr C = C1 is the colimit in Gpds of the groupoids tr C = C1 , i.e.

λ C1 = colimGpds C1 ;

(ii) for n = 2, the crossed complex tr2C is the colimit in XMod of the crossed modules tr2Cλ, i.e.

λ λ (C2 → C1) = colimXMod (C2 → C1 ); 197

(iii) for each n ≥ 3, the groupoid Cn as a module over π1(C) is the colimit in the category Mod of the groupoids λ λ Cn as modules over π1(Cn ), i.e. λ λ (Cn, π1(C)) = colimMod (Cn , π1(C )).

Proof All these facts hold because the functors appropriate to each case have right adjoints and, consequently, they preserve colimits: (i) and (ii) follow because we have proved in Proposition 7.2.6 that the coskeleton functors cosk1 : Gpds → Crs and cosk2 : XMod → Crs are the right adjoint of the truncation functors tr1 : Crs → Gpds and tr2 : Crs → XMod.

(iii) follows because we have proved in Proposition 7.2.13 that the functor En+1 : Mod → Crs is right adjoint to the restriction functor (−)n : Crs → Mod. 2

2 Note that, from this description of tr C and Cn for n ≥ 3 as colimits, the boundary maps δ : Cn → Cn−1 λ λ λ can be recovered as induced by the maps δ : Cn → Cn−1, for all λ. In the special cases of modules or crossed modules over groups, these colimits are more common; and even in the very special cases of induced modules or induced crossed modules over groups they have applications which give some classical theorems of algebraic topology, as we see in Section 7.5. Colimits of groupoids can be described by generators and relations and are as readily computed as colimits of groups (see [88, 104, 105]). This is expanded in the Appendix. Colimits in Mod and XMod are less transparent and we shall analyse their structure further in the next subsection. It is worth noticing that for the applications in Sections 7.4 and 7.5 we mainly need only colimits colim Cλ λ λ as above in the case when C0 is a singleton (i.e. each G is a group), and the colimit is taken over a connected λ diagram. Then C1 = colimGpds C1 is also a group and this colimit may be taken in the category of groups.

7.3.2 Induced modules and crossed modules

First we see how to decompose the computation of colimits of modules (over groupoids) as a two step process.

Deﬁnition 7.3.2 Given a module (M,G) and a morphism of groupoids f : G → H , the induced H-module f∗M is deﬁned by the pushout diagram

(0,f) (0,G) / (0,H)

(*) (0,idG) (0,idH ) (ψ,f) (M,G) / (f∗M,H) in Mod. 2

The existence of this construction can be obtained following the pattern for the existence and properties of induced crossed modules given in Chapter 5, i.e. as a left adjoint of the appropriate pullback functor. We get the following:

Proposition 7.3.3 For each f : G → H morphism of groupoids, we obtain a functor

f∗ : ModG → ModH which preserves colimits. 198

Exercise 7.3.4 Prove the existence of the pullback functor and that it has an adjoint.

Remark 7.3.5 Notice that the universal property of the induced module in diagram (*) can be stated as follows: For each morphism of modules

(θ, g):(M,G) → (N,K) such that g factors through H, i.e. there is a map h : H → K so that g = hf, there is a unique morphism of modules 0 (θ , h):(f∗(M),H) → (N,K) such that (θ, g) = (θ0, h)(ψ, f). This is seen in the diagram

(0,f) (0,G) / (0,H)

(0,id) (0,h) (ψ,f) (M,G) / (f∗M,H) KK KKK(θ0,h) KKK KK%% (θ,g) , (N,K)

These induced modules over groupoids constructions can be used to get colimits in a two step procedure thanks to the following result.

Proposition 7.3.6 Let (M,G) = colim (M λ,Gλ) be a colimit in Mod with canonical morphisms (θλ, αλ): λ λ λ λ λ λ (M ,G ) → (M,G). For each λ, let N = α∗ M be the induced G-module. Then M = colim N , a colimit in ModG.

Proof We have only to check that both constructions satisfy the same universal property. We leave this as an exercise. 2

Induced modules over groupoids aﬀord some interesting constructions and we will discuss them in detail in a later section. Similarly, there is a deﬁnition of crossed module induced by a morphism of groupoids generalising the induced crossed modules of Chapter 5.

Deﬁnition 7.3.7 Let (M,G) be a crossed module over G (where now M is nonabelian and we omit mention of the boundary map µ : M → G as well as the action of G). For any morphism of groupoids f : G → H we deﬁne the induced crossed module f∗M over H by the pushout diagram (*), but now a pushout in XMod . 2

As before, the existence of this construction can be obtained following the pattern for the existence and properties of induced crossed modules given in Chapter 5, i.e. as a left adjoint of the appropriate pullback functor.

Proposition 7.3.8 For each f : G → H morphism of groupoids, we obtain a functor f∗ : XModG → XModH which also preserves colimits.

Exercise 7.3.9 1. Prove the existence of the pullback functor and that it has an adjoint. 2. Write the diagram corresponding to the universal property for the induced crossed module. 199

These induced crossed modules over groupoids construction can be used to get colimits in a two step procedure thanks to the following result.

Proposition 7.3.10 Let (M,G) = colim (M λ,Gλ) be a colimit in XMod with canonical morphisms (θλ, αλ): λ λ λ λ λ λ (M ,G ) → (M,G). For each λ, let N = α∗ M be the induced crossed module over G. Then M = colim N , a colimit in XModG.

Proof We have only to check that both constructions satisfy the same universal property. We leave this as an exercise. 2

Propositions 7.3.1 and 7.3.6 give a recipe for computing C = colim Cλ, a colimit of crossed complexes:

λ (i) compute the groupoid C1 as colim C1 in Gpds; i.e.

λ C1 = colimGpds C1 ,

λ λ we call the canonical morphisms α : C1 → C1;

λ (ii) compute the C1-crossed module C2 as colim D2 in XModC1 ; i.e.

C = colim Dλ, 2 XModC1 2

λ λ λ λ λ where the C1-modules D2 = α∗ C2 are induced from C2 by the canonical morphisms α .

λ (iii) for n ≥ 3 compute the π1C-modules as colim Dn in Modπ1C ; i.e.

C = colim Dλ, n Modπ1C n

λ λ λ λ λ where the π1C-modules Dn = β∗α∗ Cn are induced from Cn by the canonical morphisms α and the quotient morphism β : C1 → π1C.

λ λ Alternatively, in this case Cn can be obtained from colim α∗ Cn in ModC1 by killing the action of δC2, i.e.

λ colimMod D C = C1 n , n N

δv where N is the sub C1-module generated by all elements x − x for all v ∈ C2.

For the applications in Sections 7.4 and 7.5 we mainly need only colimits colim Cλ as above in the case λ λ when C0 is a singleton (i.e. each G is a group), and the colimit is taken over a connected diagram. Then λ C1 = colimGpds C1 is also a group and this colimit may be taken in the category of groups. Thus C itself is the colimit colim Cλ in the category of crossed complexes over groups and therefore C can be completely described in terms of (a) colimits of groups, induced modules over groups, and colimits of modules over a ﬁxed group, all of which are familiar construction; and (b) induced crossed modules over groups, and colimits of crossed modules over a ﬁxed group, which are not so familiar. Induced crossed modules over groups have been studied in Chapter 5. 200

7.3.3 Coproducts.

We are going to consider some particular cases of colimits. Let us start by coproducts.

λ Deﬁnition 7.3.11 Let us consider a family of objects in a category {A }λ∈Λ ⊆ C0. The coproduct of this family is given by an object A and a family of morphisms f λ : Aλ → A for all λ ∈ Λ, such that for every other object C and family of morphisms gλ : Aλ → C for all λ ∈ Λ, there is a unique morphism f : A → C commuting the triangles, i.e. ff λ = gλ for λ ∈ Λ.

Example 7.3.12 Suppose we work in the category of groups. Then the coproduct of A, B is known also as the free product A ∗ B of the groups A, B. It is a non trivial theorem for groups that the canonical morphisms of the free product i : A → A ∗ B, j : B → A ∗ B are injections, but this fact is not essential for what follows. 2

It is quite easy the coproduct in Gpds the category of groupoids

Example 7.3.13 The coproduct of two groupoids G, H is the disjoint union G t H. 2

The same applies to Mod, the category of modules over groupoids

Example 7.3.14 The coproduct of two modules over groupoids (M,G) and (N,H) is the disjoint union (M t N,G t H). 2

λ Note also that in this easy way we obtain a description of the coproduct C = ∗λC in the category of crossed complexes over groups; we call C the free product of crossed complexes over groups.

A bit more involved is the coproduct in the category ModG of modules over a ﬁxed groupoid G.

Example 7.3.15 Consider two modules M,N over the same groupoid G. They can be seen as families of groups {Mp}p∈G0 and {Np}p∈G0 with some action of G. The coproduct M ∗ N is given by {Mp ⊕ Np}p∈G0 with the diagonal action.

Deﬁnition 7.3.16 A square of morphisms in a category C as follows:

a (7.3.1) C / A

b u / B v P is called a pushout square if:

(i) the square is commutative, i.e. ua = vb;

(ii) given any other commutative square with the same initial data:

a (7.3.2) C / A

b u0 B / P 0 v0

there is a unique morphism φ : P → P 0 such that φu = u0, φv = v0. 2 201

This combined situation is often represented by the diagram

a / (7.3.3) C A0 00 b u 00 00 v / 0u0 B PPP P 0 PP 00 PP φ 0 PPP 0 0 PP 0 v PP' P 0

The ‘usual universal argument’ shows that the object P of the pushout square is uniquely determined up to isomorphism by the other data, i.e. by the two morphisms a : C → A, b : C → B. However we are not at all claiming that given any two such morphisms, then we can complete the situation to a pushout square. The existence of all pushouts is a special, and often desirable, property of a category.

Example 7.3.17 A presentation for pushouts of crossed modules over a ﬁxed group G is given in Proposition λ 11 of [37]. The extension of the latter to colimits in XModG, is easy: let M = colim M and B be the colimit of the M λ in the category of groups, equipped with the induced morphism ∂ : B → G and the induced action of G; then M = B/S where S is the normal closure in B of the elements b−1c−1bc∂b for b, c ∈ B, and the boundary map M → G is induced by ∂.

In order to see how pushouts may be constructed, we need another concept, that of coequaliser. This notion is used at many places in this book.

Deﬁnition 7.3.18 A diagram in a category

a / c (7.3.4) A / B / C b is called a coequaliser diagram if:

(i) ca = cb;

(ii) given any other diagram

a / c0 (7.3.5) A / B / C0 b

in which c0a = c0b there is a unique morphism f : C → C0 such that fc = c0. 2

Example 7.3.19 Let U be an open cover of the space X. Then we have a coequaliser diagram

F a F // c / (7.3.6) (U,V )∈U 2 U ∩ V U∈U U X b where a, b are determined by the inclusions U ∩ V → U, U ∩ V → V and c is determined by the inclusions U → X. The coequaliser property then says simply that a continuous function X → Y is completely determined by continuous functions fU : U → Y,U ∈ U which coincide on the intersections U ∩ V for all U, V ∈ U. 2

Proposition 7.3.20 If a category C admits coequalisers and coproducts of two objects, then it has pushouts. 202

Proof Suppose given the pair of morphisms a : C → A, b : C → B. These deﬁne two maps a0 : C → A t B, b0 : C → A t B by composition with the canonical morphisms i : A → A t B, j : B t B of the coproduct. It is easy to verify that we can construct the pushout from the coequaliser c : A t B → P of a0, b0. More precisely, we set u = ci, v = cj and verify the universal property as required. 2

Example 7.3.21 Suppose we work in the category of groups. The coequaliser of the morphisms a0, b0 : C → A ∗ B is simply the quotient of A ∗ B by the normal closure of the subset of all a0(c)b0(c)−1 for all c ∈ C. 2

The construction of the coequaliser is more complicated in the case of the category of groupoids, because the notion of quotient by a normal subgroupoid is more complicated than in the case of groups. Let G be a groupoid. A subgroupoid N of G is called normal in G if N is wide in G (i.e. Ob(N) = Ob(G)) and for all x, y ∈ Ob(G) and a ∈ G(x, y), n ∈ G(x) we have ana−1 ∈ N(y). It follows from this that aN(x) = N(y)a. Let us consider ﬁrst an easy case

Example 7.3.22 The quotient G/N of G by a totally disconnected normal subgroupoid N of G is then easily described by Ob(G/N) = Ob(G) and that for x, y ∈ Ob(G) we have G/N(x, y) is the equivalence classes of G(x, y) under the relation a ∼ b if and only if there is an n ∈ N(y) such that na = b. We leave to the reader to verify the expected properties. 2

Now something more complicated.

Example 7.3.23 Now if R is a set of parallel pairs in G, i.e. a set of pairs (a, b) of arrows of G such that a, b have the same source and target points, then we can deﬁne N to be the normal subgroupoid of G generated by the elements (all loops) ab−1 for (a, b) ∈ R and form the quotient G/N with projection p : G → G/N. This has the universal property that of f : G → H is a morphism of groupoids such that fa = fb for all (a, b) ∈ R, then there is a unique morphism of groupoids f 0 : G/N → H such that f 0p = f. We write also G/R for G/N.

We can form G/R also as a coequaliser of morphisms i, j : CR → G where CR is the disjoint union of copies I(a,b) of the groupoid I indexed by the elements of R and i(a,b)(ι) = a, j(a,b)(ι) = b. 2

The construction of the coequaliser of i, j : CR → G without the assumption that R consists of parallel pairs in G, is more complicated.

Example 7.3.24 We ﬁrst have to take the equivalence relation ER on Ob(G) generated by the pairs (sa, sb), (ta, tb) for all (a, b) ∈ R and form the quotient function u : Ob(G) → X = Ob(G)/ER. This allows us to form the universal groupoid u∗(G) over X and under the morphism G → u∗(G) the pairs (a, b) ∈ R are sent to parallel pairs in u∗(G). We can then form the quotient u∗(G)/N as in the previous example.The composition G → u∗(G) → u∗(G)/N then gives the required coequaliser. 2

7.3.4 Free modules and free crossed modules.

In Section 3.4 we have studied free crossed modules over groups and free modules over a group are common knowledge. Now we are trying to generalise both deﬁnitions to the groupoid case. Let us try the “ad hoc” deﬁnition 203

Deﬁnition 7.3.25 Let M be a module over the groupoid G, R a set that we also consider as a discrete graph and ω : R → M a map (equivalently, let {mr | r ∈ R} be an indexed family of elements of M). We say that M is the free G-module on ω (also, that ω is a basis of M) if it satisﬁes the following universal property: for any module (N,H), morphism of groupoids f : G → H and map ω0 : R → N such that for each 0 r ∈ R, if ω(r) belongs to Mp, ω (r) lies in Nf(p), there exists a unique morphism of groupoids h : M → N such that hω = ω0 and (h, f) is a morphism of modules.

Let us try and make some changes in the universal property to make it a little bit less “ad hoc”. Another way of considering the map ω : R → M is considering the set R as a discrete groupoid, that we call 1R and using the 1R-module having Z on each r ∈ R. It is clear that a map ω : R → M is equivalent to a morphism of modules

ω :(R, 1R) → (M,G) where the morphism of groupoids 1R → G is also associated to ω. In this guise, the universal property says that for any module (N,H), morphism of groupoids f : G → H 0 and morphism of modules ω :(R, 1R) → (N,H) inducing the same map on the base groupoids, there exists a unique morphism of G modules h : M → N of G-crossed such that hω = ω0. This can be expressed in the diagram (0,ω) (0, 1R) / (0,G)

(0,id) (0,f) ω (R, 1R) / (M,G) JJ JJ(h,f) JJ JJ 0 J$ ω , (N,H)

Thus

Proposition 7.3.26 M is a free G-module on ω : R → M if the diagram

(0,ω) (0, 1R) / (0,G)

(0,id) (ψ,f) (R, 1R) / (M,G) is a push out.

This is a particular case of the induced module universal property, so we can say

Proposition 7.3.27 M is a free G-module on ω : R → M if it is isomorphic to ω∗(R).

To end this Subsection, we are going to give a construction of the free G-module associated to a map ω : R → G.

We consider a groupoid G and an injective map ω : R → G, or equivalently, an indexed family {pr | r ∈ R} of elements of G. First we create the free P -group with basis R.

For each p ∈ G0, we deﬁne the free abelian group on the formal elements {(r, g) | r ∈ R, g ∈ G1(s(r), p)}. p We think of (r, p) as r . Then, any element of Ep can be seen as a formal sum

(r1, g1) ± · · · ± (rn, gn) 204

with n ∈ N, ri ∈ R and gi ∈ G1(s(r(i)), p). This representation makes clear the deﬁnition of the G-action on generators, since to be an action it has to 0 0 satisfy (rg)g = rgg . Thus, we deﬁne a PG-action on E by

0 (r, g)g = (r, gg0) on generators and we extend it in the only possible way.

Proposition 7.3.28 (E,G) is the free G-module on ω : R → E where ω(r) = (r, 1).

Proof It is clear that G acts on E in such a way that (M,G) is a module. To prove the universal property, consider (N,H) a module, f : G → H a morphism of groupoids and a map ω0 : R → N that is fω on G. We can deﬁne the map R × G → N (r, g) 7→ (ω0r)f(g) that extends to a unique homomorphism h : E → N that is a morphism of modules over f. Actually, this is the only possible deﬁnition of h to make it a morphisms of modules. 2

There has to be some way to describe free modules over G as left adjoints of some forgetful functor. Perhaps our free construction is associated not to a function ω : S → P but to a graph morphism ω : X → G where X is a discrete graph. We deﬁne Graphs/G to be the category whose objects are graph morphisms ν : S → G, and whose morphisms are graph morphisms α : S → S0 making commutative the diagram

α / 0 S ? S ?? ~~ ?? ~~ ν ? ~ 0 ? ~~ ν G .

7.4 GvKT for ﬁltered spaces and crossed complexes

As we have seen in Chapter 1, the groupoid version of the van Kampen theorem, gives useful results for non connected spaces, but still requires a ‘representativity’ condition in dimension 0. The corresponding theorem for crossed modules, which computes certain second relative homotopy groups, as discussed in Part I, also needs a “1-connected” condition. It is thus not surprising that our general theorem requires conditions in all dimensions. The Generalised van Kampen Theorem we are going to prove states that the fundamental crossed module functor commutes with colimits in some cases. We need to be more precise about which type of colimit are preserved and in which case. We start by studying a special kind of ﬁltrations.

Definition 7.4.1 A filtered space X∗ is said to be connected if the following conditions hold for each n ≥ 0 : - φ(X∗, 0) : If r > 0, the map π0X0 → π0Xr, induced by inclusion, is surjective; i.e. X0 meets all path components of all stages of the filtration Xr. - φ(X∗, n)(for n ≥ 1): If r > n and x ∈ X0, then the map

r in : πn(Xn,Xn−1, x) → πn(Xr,Xn−1, x) 205 induced by inclusion, is surjective.

Let us state a condition that is equivalent to φ(X∗, n)(for n ≥ 1).

Proposition 7.4.2 A ﬁltered space X∗ is connected if and only if satisﬁes the following conditions: φ(X∗, 0) : If r > 0, the map π0X0 → π0Xr, induced by inclusion, is surjective. 0 φ (X∗, n)(for n ≥ 1): If r > n and x ∈ X0, πn(Xr,Xn, x) = 0.

0 Proof We only need to check the equivalence of φ(X∗, n) and φ (X∗, n) (both for n ≥ 1) in presence of φ(X∗, 0) and this is a standard application of the homotopy exact sequence of the triple (Xr,Xn,Xn−1) based at x ∈ X0. Let us give the details for those not used to this way of reasoning.

Let r > n ≥ 1. The part of the homotopy exact sequence of the triple (Xr,Xn,Xn−1) based at x ∈ X0 we are going to use is

r r in jn (*) ··· / πn(Xn,Xn−1, x) / πn(Xr,Xn−1, x) / πn(Xr,Xn, x) / ···

(where for n = 1 this is an exact sequence of based sets). r On the one hand, πn(Xr,Xn, x) = 0 implies in surjective, as required for connectedness. r Suppose conversely that X∗ is connected. Then in is surjective and so, using (*), the map

r jn : πn(Xr,Xn−1, x) → πn(Xr,Xn, x) has to be the 0 map. We prove πn(Xr,Xn, x) = 0 by induction on n.

Let n = 1 and consider [λ] ∈ π1(Xr,X1, x) where

λ :(I, 0, 1) → (Xr,X1, x)

0 0 is a map. By φ(X∗, 0) we can choose a path λ in X1 joining λ(0) to a point of X0. Thus, the path µ = λ λ is a map µ :(I, 0, 1) → (Xr,X0, x) and [λ] = [µ] ∈ π1(Xr,X1, x), r r proving that j1 is surjective. Since j1 = 0, we have proved

π1(Xr,X1, x) = 0 for all r > 1. If n > 1, the exact sequence (∗) may be extended to the right by

r δn : πn(Xr,Xn, x) → πn−1(Xn,Xn−1, x). that is injective by exactness. Since πn−1(Xn,Xn−1, x) = 0 by induction, therefore

πn(Xr,Xn, x) = 0.

Using this property we get lots of examples of connected filtrations because it is well known that the skeletal filtration of a CW -complex X satisfies the conditions of the previous proposition therefore every CW -complex provides a connected filtered space. 206

Now we have set the background to state the GvKT in the most general form we are going to use. As said, it takes the form of giving some conditions under which Π preserves colimits. The form we have found more useful is stated in terms of preservation of coequalisers (for a Deﬁnition see the Appendix A). That is the Theorem we are refereeing to when we talk about GvKT. Nevertheless, some particular cases that are specially interesting are going to be refereed also as GvKT for this particular situation (pushouts, subspaces, etc.).

Let us give some notation for the statement of the GvKT. Let X∗ be a filtered space. We suppose given a λ cover U = {U }λ∈Λ of X such that the interiors of the sets of U cover X. For each ζ ∈ Λn we set U ζ = U ζ1 ∩ · · · ∩ U ζn . ζ ζ Then, if , Ui = U ∩ Xi for each i ∈ N, the filtration ζ ζ ζ ζ U∗ = U0 ⊆ U1 ⊆ · · · ⊆ U is called the induced filtration of U ζ . In particular, when X is covered by the interiors of U λ, it can be expressed as a coequaliser

F a F ν / λ c / ν∈Λ2 U∗ / λ∈Λ U∗ X∗ b where a, b and c are the inclusions. Let us give a picture for the simpler cases. For two sets U = {U, V }, we have the diagram

U ∩ V / U

V / X that we are going to use often (see Theorem 7.4.6) and for three sets U = {U, V, W }, we have the diagram

V ∩ W w ww ww ww w{ w U ∩ V / V

U ∩ W / W ss w ss ww ss ww ss ww yss w{ w U / X Going back to the general case, the crossed complexes in the following Π-diagram of the cover are well deﬁned: F a F ν / λ c / ν∈Λ2 ΠU∗ / λ∈Λ ΠU∗ ΠX∗ b F Here denotes disjoint union (which is the same as coproduct in the category of crossed complexes); a, b are λ µ λ λ µ µ 2 determined by the inclusions aν : U ∩ U → U , bν : U ∩ U → U for each ν = (λ, µ) ∈ Λ ; and c is λ determined by the inclusions cλ : U → X for all λ ∈ Λ. 207

Theorem 7.4.3 (GvKT). Suppose that for every ﬁnite intersection U ζ of elements of U, the induced ﬁltration ζ U∗ is connected. Then (Con) X∗ is connected and (Iso) in the above Π-diagram c is the coequaliser of a, b in the category Crs of crossed complexes.

An immediate particular case is

λ Corollary 7.4.4 Let {X∗ }λ∈Λ be a family of ﬁltered spaces. Let G λ X∗ = X∗ λ

λ F λ F λ λ be the sum of all the X∗ , with total space X = λ X and ﬁltration Xn = λ Xn . Suppose each X∗ is connected. Then

(Con) X∗ is connected, and F λ (Iso) ΠX∗ is isomorphic to λ ΠX∗ , the free product (coproduct) of crossed complexes.

Proof We apply the GvKT to the case U λ = Xλ. Then U λ ∩ U µ = ∅ and the coequaliser is just the coproduct. 2

As always is the case with homotopy functors, we can change the spaces U λ up to homotopy type. A particularly useful application of Theorem 7.4.3 is to CW -complexes.

λ Corollary 7.4.5 Let X∗ be the skeletal ﬁltered space of a CW -complex X, and let X = {X }λ∈Λ be a cover of X by subcomplexes. Then the Π-diagram of the cover X :

F a F ν / λ c / ν∈Λ2 ΠX∗ / λ∈Λ ΠX∗ ΠX∗ b is a coequaliser diagram of crossed complexes.

Proof There is a standard method of assigning to each subcomplex Y of X a neighbourhood UY of Y in X and a retraction rY : UY → Y such that (i) Y is a strong deformation retract of UY ; (ii) if Y ⊆ Z are subcomplexes of X, then UY ⊆ UZ and rZ | UY = rY ;

(iii) if Y1,...,Yn are subcomplexes of X, then UY1∩···∩Yn = UY1 ∩ · · · ∩ UYn . The method of constructing the UY , rY is by induction on the dimension of the cells of X \ Y which meet Y. λ λ We now set U = UXλ , λ ∈ Λ. Then U = {U }λ∈Λ is a family whose interiors cover X and for which the induced ﬁltration U ζ of each ﬁnite intersection of its elements is connected. The result follows from Theorem 7.4.3. 2

In the case where we have just two elements in Λ, the coequaliser is just a pushout, and that gives a more familiar form to the GvKT

Theorem 7.4.6 (the pushout theorem for subspaces) Let X∗ be a ﬁltered space, U, V ⊆ X and A = U ∩ V . Suppose that X is the union of the interiors of the sets U, V ; and also that the ﬁltrations U∗,V∗,A∗ induced by the one on X (An = A ∩ Xn,Vn = V ∩ Xn,Un = U ∩ Xn, for all n ≥ 0) are connected. Then 208

(Con) X∗ is connected, and (Iso) the diagram / ΠA∗ ΠU∗

/ ΠV∗ ΠX∗ induced by inclusions is a pushout of crossed complexes.

This pushout form of GvKT can be generalised to allow the case when U and V are no longer subspaces of X but X is the adjunction space of V and a map f : A → U.

Theorem 7.4.7 (the pushout theorem for coﬁbrations) Suppose that the commutative diagram of ﬁltered spaces

f / A∗ U∗

i ı / V∗ X∗ f is such that for n ≥ 0, the maps in : An → Vn are closed coﬁbrations, An = A ∩ Vn, and Xn is the adjunction space Un ∪fn Vn.

Suppose also that the ﬁltrations U∗,V∗,A∗ are connected. Then (Con) X∗ is connected, and (Iso) the induced diagram / ΠA∗ ΠU∗

/ ΠV∗ ΠX∗ is a pushout of crossed complexes.

Proof This is a deduction of standard kind from Theorem 7.4.6 using mapping cylinders. 2

We are going to illustrate the use of Theorem 7.4.7 in a couple of situations. A direct application of Theorem 7.4.7 is to quotient ﬁltrations.

Theorem 7.4.8 Let V∗ be a filtered space, A ⊆ V , and X = V/A. We define the filtrations A∗, and X∗ by An = Vn ∩ A, and Xn = Vn/An, n ≥ 0. Suppose that each An → Vn is a closed cofibration, and both A∗,V∗ are connected. Then

(Con) X∗ is connected, and (Iso) we have a pushout of crossed complexes

/ ΠA∗ 0

i∗ / ΠV∗ ΠX∗. 209

Proof All we have to do is to apply Π to the diagram

f / A∗ {∗}

i ı / V∗ X∗ f that satisﬁes the conditions of Theorem 7.4.7. 2

Applying to this result the fact that the dimension functors (−)n preserve colimits, we get some results on homotopy groups. Let us ﬁrst ﬁx some notation.

Definition 7.4.9 A filtered space X∗ is reduced if X0 consists of a single point, i.e. X0 = {∗}; then ∗ is taken as base point of each Xn, n ≥ 0, and the relative homotopy groups of X∗ are abbreviated to πn(Xn,Xn−1). The base point in X0 is nondegenerate if each inclusion X0 → Xn, is a closed cofibration for all n ≥ 1.

Corollary 7.4.10 Let V∗,A∗ and X∗ be ﬁltered space as in Theorem 7.4.8. If V∗ is reduced, then we have π (V ,V ) π (X ,X ) = n n n−1 n n n−1 N a where N is the π1V1-submodule generated by all elements {u − u | u ∈ πn(Vn,Vn−1), a ∈ i∗π1A1} and i∗πn(An,An−1).

We can also mix Theorem 7.4.8 with Corollary 7.4.4 in the case of pointed spaces.

λ Theorem 7.4.11 Let {X∗ }λ∈Λ be a family of reduced, ﬁltered spaces each with non-degenerate base-point. Let _ λ X∗ = X∗ λ

λ W λ W λ λ be the wedge of all the X∗ , with total space X = λ X and ﬁltration Xn = λ Xn . Suppose each X∗ is connected. Then

(Con) X∗ is connected, and λ (Iso) ΠX∗ is isomorphic to ∗λΠX∗ , the free product (coproduct) of crossed complexes over groups.

Proof We have an inclusion of ﬁltered spaces G 0 λ X = X0 → X∗ λ where X0 is just a space, i.e. it has the constant ﬁltration consisting of the same space for all dimensions n ≥ 0. As in Theorem 7.4.8, we have a pushout of crossed complexes / A0 0

F λ / λ ΠX∗ ΠX∗

0 where A0, 0 denote the crossed complexes which are the trivial modules over the discrete groupoids on X and {∗} respectively. This pushout diagram determines ΠX∗ as the required free product. 2 210

Notice that this free product is the coproduct in the category of crossed complexes over groups, not in the category of crossed complexes over groupoids.

7.5 Induced modules and relative homotopy groups

The purpose of this section is to give some computations of homotopy groups that can be easily obtained from our main theorem in this Section (Theorem 7.5.6, the GvKT for the fundamental crossed complex of a filtered space in the case the filtration is reduced to two stages). This will show an immediate payoff for the general theorem. Recall that the proof of the general theorem requires a diversion to another category whose objects have underlying structure based on the geometry of cubes. Thus this section gives some justification for the long march to the proof of the general result. Moreover, the whole theory has other implications, such as homotopy classification results, which we give later. It will be clear from Part I that a major aspect of this work is to tie in the fundamental group and higher homotopy groups. This contrasts with previous approaches, where the action of the fundamental group is often obtained by passing to the universal covering space. It was an aesthetic objection to this diversion to obtain the fundamental group of the circle which led to the groupoid work in [28] and so to the present work. It is also unclear at present how to obtain the results of Part I by covering space methods. To relate the homotopy groups of a pair of spaces to the fundamental crossed complex of a filtered space we associate to a pair of spaces (X,A) a special filtration as follows:

Deﬁnition 7.5.1 For any pair of spaces (X,A) and dimension n ≥ 0, we have a ﬁltration of X,

En(X,A) = A ⊆ · · · ⊆ A ⊆ X ⊆ · · · ⊆ X ⊆ · · · , having A in dimensions < n and X in dimension ≥ n.

The fundamental crossed complex of En(X,A) is clearly related to the homotopy groups of the pair (X,A) in the following way:

Proposition 7.5.2 Consider a pair (X,A) and its related ﬁltration En(X,A). Then the fundamental crossed complex of the ﬁltration En(X,A) is the crossed complex En(πn(X,A), π1(A)) associated to the π1(A)-module πn(X,A).

the two stage ﬁltered space

En(X,A) = A ⊆ · · · ⊆ A ⊆ X ⊆ · · · ⊆ X ⊆ · · · , having A in dimensions ≤ n and X in dimension ≥ n + 1. It is clear that the fundamental crossed complex of this ﬁltered space is the two stage crossed complex

En(πn(X,A), π1(A)) = ··· / 0 / πn(X,A) / 0 / ··· / 0 / π1(A)

All we need to make that appropriate for use of the GvKT is to translate the connectivity of En(X,A) into conditions on the pair (X,A) and see what form the GvKT takes in this case.

Now we relate the connectivity of En(X,A) to the n-connectivity of the pair (X,A). Let us recall the deﬁnition. 211

Deﬁnition 7.5.3 A based pair of spaces (X, A, x) is said to be n-connected if πi(X, A, x) = 0 for all 0 < i ≤ n.

A pair of spaces (X,A) is said to be n-connected if the induced maps from π0A to π0X are surjective (A meets all path components of X) and the based pair (X, A, x) is n-connected for all x ∈ A. 2

Proposition 7.5.4 Consider a pair (X,A) and its related ﬁltration En(X,A). Then En(X,A) is a connected ﬁltration if and only if (X,A) is an n-connected pair.

Proof This is a direct application of Proposition 7.4.2 2

As in the case of ﬁltered spaces there is another characterization of n-connectivity.

Proposition 7.5.5 The following conditions are equivalent for a point x ∈ A:

(i) (X,A) is n-connected;

i∗ (ii) The induced morphism πr(A, x) −→ πr(X, x) is an isomorphism for 1 6 r < n − 1 and is an epimorphism for r = n − 1.

Proof As in Proposition 7.4.2, this is just a standard application of the homotopy exact sequence of the based pair: ∂ i∗ j∗ ∂ · · · → πr+1(X, A, x) −→ πr(A, x) −→ πr(X, x) −→ πr(X, A, x) −→ · · · 2

Using this result, we can deduce from the general theorems in the Section 7.4, results on homotopy groups of pairs. By the end of the Section we shall prove two classical results: the Brouwer Degree Theorem and the relative Hurewicz Theorem. First, we translate the pushout theorems for ﬁltered connected spaces (Theorems 7.4.6 and 7.4.7) to pairs of spaces. The ﬁrst one considers the case of subspaces says:

Theorem 7.5.6 Let X be a space, U, V ⊆ X, A = U ∩ V . Let X be the union of the interiors of U and V ; U, V, A be path-connected and (V,A) be (n − 1)-connected. Then: (Con) the pair (X,U) is (n − 1)-connected, and

(Iso) for n ≥ 3 the π1U-module πn(X,U) is

πn(X,U) = λ∗πn(V,A), the module induced from the π1A-module πn(V,A) by the map induced by the inclusion λ : π1A → π1U.

Proof This is just Theorem 7.4.6 applied to En−1(X,A). 2

And the second one applies to closed coﬁbrations

Theorem 7.5.7 Suppose that in the commutative square of based spaces

f A / U

i ı V / X = U ∪f V f 212

the map i is a closed coﬁbration and X is the adjunction space U ∪f V. Suppose also that U, V, A are path- connected and (V,A) is (n − 1)-connected. Then: (Con) the pair (X,U) is (n − 1)-connected, and

(Iso) for n ≥ 3 the π1U-module πn(X,U) is

πn(X,U) = λ∗πn(V,A), the module induced from the π1A-module πn(V,A) by λ = f∗ : π1A → π1U.

Proof Can be obtained either from the previous theorem using mapping cylinder arguments or directly from

Theorem 7.4.7 applied to En−1(X,A). 2

Of course the corresponding results for n = 2, with ‘module’ replaced by ‘crossed module’, have been given in Part I, and a number of consequences drawn. Here our intention is to move to applications by studying several situations where Theorem 7.5.7 can be applied. We consider diﬀerent situations around the pair (CA, A) where CA denotes the cone on the space A. In this pair, CA is contractible and so has zero homotopy groups. By the homotopy exact sequence of the pair, the boundary map ∂ : πr(CA, A, x) → πr−1(A, x) is an isomorphism for all x ∈ A. Thus the pair (CA, A) is n-connected if and only if A is (n − 1)-connected, i.e. if A is connected and πr(A, x) = 0 for 1 ≤ r < n. First we consider how to compute the lower homotopy groups of Sn, using suspension and induction. The suspension is just a quotient of the cone.

Theorem 7.5.8 (the suspension theorem) For a space A, consider SA the (unreduced) suspension of A. If A is (n − 2)-connected, for n ≥ 3, then (Con) SA is (n − 1)-connected and ∼ (Iso) πnSA = πn−1A.

Proof We deﬁne V = CA the cone on A, U = {∗} a point and f the constant map. Then X = U ∪f V is the (unreduced) suspension of A, and we can consider the diagram

f A / {∗}

i ı CA / X = SA. f

Since A is (n−2)-connected if and only if (CA, A) is (n−1)-connected, we can apply Theorem 7.5.7, getting that (X, ∗) is (n − 1)-connected and ∼ πnSA = πn(CA, A). ∼ Using again the homotopy exact sequence of this pair, we have πn(CA, A) = πn−1A. 2

Corollary 7.5.9 (Brouwer Degree Theorem) For n > 1, Sn is (n − 1)-connected and

n ∼ πn(S , 1) = Z. 213

Proof Recall that in Part I we have seen that if n = 2 and A is a path-connected space then SA is 1-connected and ∼ ab π2(SA, x) = π1(A, x) .

1 2 ∼ Given the value of π1(S , 1) as Z (a result proved in Section 1.7), we deduce π2(S , 1) = Z. The induction step follows easily from the Theorem. 2

This is actually a fairly deep result: that the sphere Sn is (n − 1)-connected means that any map Sr → Sn for r < n is nullhomotopic, while the determination of Π(Sn, 1) includes the Brouwer degree theorem, that the maps Sn → Sn are classiﬁed up to homotopy by an integer, called the degree of the map. This was one of the early triumphs in homotopy classiﬁcation results, and proofs of these results have to use some kind of subdivision argument, often through the route of simplicial approximation.

Now we study the space got by adjoining the cone on a subspace. The result we get is a version of the classical relative Hurewicz Theorem.

Theorem 7.5.10 (Relative Hurewicz Theorem) Let (V,A) be a pair of spaces and deﬁne X = CA ∪ V . We deduce that if n ≥ 3, A and V are path connected and (V,A) is (n − 1)-connected, then (Con) X is (n − 1)-connected, and

(Iso) πn(X, x) is isomorphic to πn(V, A, x) factored by the action of π1(A, x).

Proof We would like to apply Theorem 7.5.6 to the diagram of inclusions

A / CA

ı V / X = CA ∪ V f

0 1 but the subspaces do not satisfy the interior condition. We change the subspaces to A = A × [0, 2 [ ⊆ CA and V 0 = V ∪ A0. Those subspaces have the same homotopy type than A and V (moreover the pair (V 0,A0) has the homotopy type of (V,A)) and we can apply Theorem 7.5.6 to the diagram of inclusions

A0 / CA

ı V 0 / X = CA ∪ V f getting that X is (n − 1)-connected and that

πn(X,CA) = λ∗πn(V,A), the module induced from the π1A-module πn(V,A) by λ = f∗ : π1A → π1CA = 0.

Since π1CA = 0, λ∗πn(V,A) is got from πn(V,A) by killing the π1A-action. (If n = 2 that would give the abelianisation).

To ﬁnish, note that using that CA is contractible, we get that πr(X, CA, x) is isomorphic to πr(X, x), by the exact sequence of the pair. 2 214

It has to be noticed that the usual version of the Relative Hurewicz Theorem involves not πn(CA ∪ V, x) but the homology Hn(V,A). It is possible to get this more usual version from the one we have just proved in a three stage process.

First, notice that πn(CA ∪ V, x) is isomorphic to Hn(CA ∪ V ) by the absolute Hurewicz Theorem.

Then it is easy to prove that Hn(CA∪ V ) is isomorphic to Hn(CA∪ V,CA) by the homology exact sequence using that CA is acyclic because is contractible.

Last, we notice that Hn(CA ∪ V,CA) is isomorphic to Hn(V,A) by excision.

Last, we see what Theorem 7.5.7 says in the case when we are attaching a cone CA via a map of the space A.

Proposition 7.5.11 Let us deﬁne X = U ∪f CA for some map f : A → U. For any n ≥ 3, if U is path connected and A is (n − 2)-connected, then (Con) (X,U) is (n − 1)-connected and

(Iso) the π1(U)-module πn(U ∪f CA, U) is isomorphic to the induced module λ∗(πn−1(A)), i.e. ∼ πn(U ∪f CA, U) = πn−1A ⊗ Z(π1U). 2

And as result we get the eﬀect of attaching n-cells on the homotopy groups of a space. The GvKT for crossed complexes has many other consequences, some of them nonabelian, which seem diﬃcult to obtain by traditional routes of homology.

Exercise 7.5.12 Let A, B, U be path-connected, based spaces. Let X = U ∪f (CA × B) where CA is the (unreduced) cone on A and f is a map A × B → U. The homotopy exact sequence of (CA × B,A × B) gives ∼ πi(CA × B,A × B) = πi−1A, i ≥ 2, and π1(CA × B,A × B) = 0.

Suppose now that n > 2 and A is (n−2)-connected. Then π1A = 0. We conclude from Theorem 7.5.7 that (X,U) is (n − 1)-connected and πn(X,U) is the π1U-module induced from πn−1A, considered as trivial π1B-module, by λ = f∗π1B → πiU. Hence πn(X,U) is the π1U-module

(*) πn−1A ⊗Z(π1B) Z(π1U).

7.6 Crossed complexes of free type

Crossed complexes of free type model the situation one gets when attaching inductively cells, i.e. in relative CW -complexes. So we begin by adapting results of the last Section (in particular Theorem 7.5.7) to compute how the homotopy groups change when adding cells. We shall ﬁnish the Section by deﬁning the crossed complexes that have the same kind of structure that the fundamental crossed module of a relative CW -complex.

Theorem 7.6.1 Suppose that X is obtained from U by attaching a family of n-cells. That is, we have the pushout diagram F f n−1 / λ Sλ U

F n / λ Eλ X 215

n−1 where f maps the base points of the family Sλ to some 0-dimensional part U0 = X=. We deduce that if n ≥ 3 then (Con) (X,U) is (n − 1)-connected, and

(Iso) πn(X, U, X0) is the free π1(U, X0)-module on the characteristic maps of the n-cells.

n n−1 Proof It is just 7.5.11 since E = CS . And induced module λ∗(πn−1(A)) is just the indicated free module. 2

Thus the case of the fundamental crossed complex of a CW -complex is got inductively through stages that are as in the Theorem. We shall try to mimic algebraically the process to deﬁne crossed complexes of free type. Let us start by considering the building blocks corresponding to ΠEn and ΠSn−1.

Deﬁnition 7.6.2 We shall write F(n) for the crossed complex freely generated by one generator cn in dimension n. So - F(0) is {1} in dimension n = 0 and trivial elsewhere; - F(1) is a crossed complex associated to the groupoid I which has only two objects 0, 1 and non-identity −1 elements c1 : 0 → 1 and its inverse c1 : 1 → 0. Thus F(1) has {0, 1} in dimension n = 0, I in dimension n = 1, the trivial crossed module 1I in dimension n = 2 and trivial elsewhere

- F(n) for n ≥ 2 is in dimensions n and n − 1 an infinite cyclic group with generators cn and cn−1 = δcn respectively, and is otherwise trivial. The only non trivial δ is defined by δn(cn) = cn−1. Notice that F(n) is just another name for En(Z). Also, we shall write S(n − 1) for the subcomplex of F(n) which agrees with F(n) up to dimension n − 1 and is trivial otherwise. Thus the only difference between F(n) and S(n − 1) is at dimension n, where F(n)n is isomorphic to Z and S(n − 1)n is trivial. 2

Remark 7.6.3 1.- Notice that this definition satisfies what we would assume is a definition of free in this context, i.e. for any crossed complex C and any element x ∈ Cn there is a unique morphism of crossed modules

xˆ : F(n) → C so thatx ˆ(cn) = x and viceversa. 1 1 ∼ 1 ∼ 2.- It is a straightforward consequence of convexity of E that Π(E∗) = F(1), while the fact that Π(S∗) = S(1), is in essence the fact that the fundamental group of S1 is isomorphic to Z and this has been proved in Corollary 7.5.9. n ∼ n ∼ 3.- It is also clear that Π(E∗ ) = F(n) and Π(S∗ ) = S(n) for n ≥ 3.

Now we consider the situation when the free generators are added in increasing order of dimension.

Definition 7.6.4 We now define a particular kind of morphism j : A → C called a crossed complex morphism n−1 n of relative free type. Let A be any crossed complex. A sequence of morphisms jn : C → C may be defined 0 λ n−1 with C = A by choosing any family of morphisms fn : S(n) → C for λ ∈ Λn, and forming the pushout

F (f λ) S(n − 1) n / n−1 λ∈Λn C

jn F F(n) / n λ∈Λn C . 216

Let C = colim Cn, and let j : A → C be the canonical morphism. We call j : A → C a crossed complex n morphism of relative free type. The images x of the elements cn in C are called basis elements of C relative to A. We can conveniently write n C = A ∪ {x }λ∈Λn,n≥0, and may abbreviate this in some cases, for example to C = A ∪ xn ∪ xm, analogously to standard notation for CW -complexes. Let us describe the structure we get on C at each dimension.

C0 is the disjoint union of A0 and Λ0;

C0 = A0 t Λ0 in Sets

∗ ∗ C1 is the coproduct of C0-groupoids A1 and F (Λ1), where A1 is the groupoid obtained from A1 by adjoining the objects of C not already in A, and F (Λ1) is the free groupoid on Λ1 considered as a graph over C0 via the λ maps f1 ; ∗ C1 = A1 t F (Λ1) in Gpds

∗ ∗ C2 is the coproduct of crossed C1-modules A2 and F (Λ2), where A2 is the C1-crossed module induced from the A1-crossed module A2 by the morphism of groupoids A1 → C1, and F (Λ2) is the free crossed C1-module λ on Λ2 via the maps f2 ; ∗ C2 = A2 ∗ F (Λ2) in XModC1

∗ ∗ Cn, for n ≥ 3, is the direct sum of π1C1-modules An and F (Λn), where An is the module induced from the π1A1-module An by the morphism of groups π1A1 → π1C1, and F (Λn) is the free π1C1-module on Λn.

∗ Cn = An ⊕ F (Λn) in Modπ1C1

λ The boundary maps are in all cases induced by the boundary maps in A and by the maps fn .

Thus at each dimension Cn is the coproduct in the suitable category of the n-dimensional part of A (appro- priately modiﬁed) and a free structure with as many generators as n-cells we are attaching.

We remark that for A = ∅ we get by this construction the crossed complexes of free type which were considered in [43] under the name “free crossed complexes”. (The reason for avoiding the term free crossed complex is that such crossed complexes do not seem to arise from adjoints of a forgetful functor.)

It is also clear from the GvKT that for the skeletal ﬁltration X∗ of a CW -complex X, the crossed complex ΠX∗ is of free type; if Y∗ is a subcomplex of X∗ then the induced morphism ΠY∗ → ΠX∗ is of relative free type. Reduced crossed complexes of free type are called ‘homotopy systems’ in [175], and ‘free crossed chain complexes’ in [10, 9].

Of course, the advantage of a free crossed complex is because if C is a free crossed complex on X∗, then a morphism f : C → D can be constructed inductively provided one is given the values fnx ∈ Dn, x ∈ Xn, n > 0 provided the following geometric conditions are satisﬁed: α α (i) δ f1x = f0δ x, x ∈ X1, α = 0, 1;

(ii) βfn(x) = f0(βx), x ∈ Xn, n > 2;

(iii) δnfn(x) = fn−1δn(x), x ∈ Xn, n > 2.

Notice that in (iii), fn−1 has to be deﬁned on all of Cn−1 before this condition can be veriﬁed. 217

If further D is free, then to specify fn(x) we simply have to give the expression of fn(x) in terms of the basis for D. Later we will see that homotopies can be speciﬁed similarly. 218 Chapter 8

Crossed complexes: tensor product and homotopies

This Chapter is built around the notion of monoidal closed category. This will raise conceptual difficulties for those not used to the ideas, and in the case of crossed complexes it also raises technical difficulties, since there is an elaborate set of formulae for the so called ‘tensor product’. So we give some background and introduction. Sometimes we use a formal description, but the fact that we can if necessary get our hands dirty and write down some complex formulae and calculate with them, is one of the aspects of the theory that gives power to the category of crossed complexes. The start of these ideas is that a function of two variables f : R × R → R can also be regarded as a variable function of one variable. This is the basis of partial differentiation. In general, this transforms into the idea that if ZY denotes the set of functions from the set Y to the set Z, then we have a bijection of sets

e : ZX×Y → (ZY )X given by e(f)(x)(y) = f(x, y), x ∈ X, y ∈ Y. This corresponds to the exponential law for numbers mnp = (mn)p, and so the previous law is called the exponential law for sets. Because there is a bijection X × Y → Y × X this also means we can set up bijections between the functions

X → ZY ,Y → ZX ,X × Y → Z.

This becomes particularly interesting in its interpretation when Y = I = [0, 1], the unit interval, since the functions I → Z can be thought of as paths in Z, and so the set of these functions is a kind of space of paths; in practice we will want to have topologies on these sets and speak only of continuous functions, but let us elide over that for the moment. The functions X → Z we can intuitively call ‘configurations of X in Z’. A function X × I → Z we can think of as a deformation of configurations. This can be seen alternatively as a path in the configuration space ZX , or as a configuration X → ZI in the path space of Z. These alternative points of view have proved strongly useful. It is useful to rephrase the exponential law slightly more categorically, so as make analogies for other

219 220 categories, so we write it also as

e : Set(X × Y,Z) =∼ Set(X, SET(Y,Z)).

Here the distinction between Set and SET, i.e. between external and internal to the category, is less clear than it will be in our other examples. Now suppose that X,Y,Z are topological spaces, and Top(Y,Z) denotes the set of continuous maps Y → Z. We would like to make this set into a topological space TOP(Y,Z) so that the exponential correspondence gives a natural bijection Top(X × Y,Z) =∼ Top(X, TOP(Y,Z)). However this turned out not to be at all straightforward, and in the end a reasonable solution was found by restricting to what are called ‘compactly generated spaces’, and working entirely in this category. An account of this theory is given in [28], and we assume this as known. The existence of the exponential law as above is summarised by saying that the category Top is a cartesian closed category. Here ‘cartesian’ refers to the fact that we use the categorical product in the category, and ‘closed’ means that there is a space TOP(Y,Z) for all spaces Y,Z in the category Top. The space TOP(Y,Z) is also called the ‘internal hom in Top’. It is a deduction from the exponential law that there is also a natural homeomorphism

TOP(X × Y,Z) =∼ TOP(X, TOP(Y,Z)).

We leave the proof of this to the reader. There are a couple of special characteristics to this example. First, the underlying set of the space TOP(Y,Z) is the set Top(Y,Z), but in other situations there is no reason why this should be so. We shall come later to this point. Second, the product we are using is the categorically deﬁned product in the category. There are analogous laws which do not involve the cartesian product in the category.

For example, if ModR denotes the category of left modules over a commutative ring R with morphisms the R-linear maps, then we can for R-modules M,N form an R-module structure on the set ModR(M,N) to form an R-module which we write MODR(M,N). This we call the internal hom in the category ModR. For another R-module L we can then consider

ModR(L, MODR(M,N)).

However this set is bijective with the set of R-bilinear maps (L, M) → N, by which is meant the functions L × M → N which are linear in each variable. Then we have an exponential law ∼ BiLinR((L, M); N) = ModR(L, MODR(M,N)), where the left hand side denotes the set of bilinear maps.

A standard construction is the ‘universal bilinear map’ (L, M) → L ⊗R M so as to obtain a natural bijection ∼ BiLinR((L, M); N) = ModR(L ⊗R M,N) and hence a natural bijection ∼ ModR(L ⊗R M,N) = ModR(L, MODR(M,N)).

However the tensor product construction, i.e. the bifunctor,

− ⊗R − : ModR × ModR → ModR, 221

does not give a categorical product in the category ModR. To describe this situation, category theorists have developed the notion of ‘monoidal closed category’. Notice that the exponential law is an adjoint relationship. In the last situation it states that for all M the functor −−⊗RM is left adjoint to MODR(M, −−). This has some immediate consequences on the preservation of colimits and limits by these functors, and these consequences are valuable.

If M,N are left R-modules, then M ⊗R N can be constructed as the free R-module F on elements m ⊗ n for m ∈ M, n ∈ N factored by the relations

(m + m0) ⊗ n = (m ⊗ n) + (m0 ⊗ n),

m ⊗ (n + n0) = (m ⊗ n) + (m ⊗ n0), rm ⊗ n = m ⊗ rn, for all m, m0 ∈ M, n, n0 ∈ N, r ∈ R. Notice that the two ﬁrst families of relations have some consequences like

m ⊗ 0 = 0 ⊗ n = 0

(−m) ⊗ n = m ⊗ (−n) = −(m ⊗ n) while the third family of relations can be used to deﬁne an structure of R-module by the action

r(m ⊗ n) = rm ⊗ n = m ⊗ rn.

This gives as a consequence the linearity on both variables of the tensor product

(rm + r0m0) ⊗ n = r(m ⊗ n) + r0(m0 ⊗ n),

m ⊗ (sn + s0n0) = s(m ⊗ n) + s0(m ⊗ n0).

There is also the universal bilinear map

M × N → M ⊗R N given by (m, n) 7→ m ⊗ n.

An element of M ⊗R N is thus an R-linear combination of decomposable elements of the form m ⊗ n. In general, it is not quite so obvious what are the actual elements of M ⊗R N for specific M,N,R. Nonetheless, the tensor product of R-modules plays an important role in module theory. One reason is that whereas a bilinear map does not have a defined notion of kernel, a morphism M ⊗R N → P to an R-module P does have a kernel. This process of using a universal property to replace a function with complicated properties by a morphism is a powerful procedure in mathematics. Both as a remainder and as an introduction to the more involved crossed complex case, this process is described in Section 8.1 for the category Mod of modules over groupoids. The main part of the Chapter revolves about the monoidal closed category structure of Crs. The natural way to this structure probably is not to define it directly, but through the isomorphism with the category of ω-groupoids and the natural definition of product in that category. That we do in Chapter 16. Here, we state directly the Definition that results from this detour risking that this could make the conditions for the product in Crs somehow artificial. Nevertheless, it is important to get acquainted soon with the formulae for the tensor product because they are going to be used frequently in the applications of the next few chapters. 222

First a few words relating the closed category structure with homotopy. We have already observed that in any crossed complex C, the set of n-dimensional elements Cn is bijective to the set of morphisms of crossed complexes Crs(F(n),C), where F(n) is the free crossed complex in dimension n (see Remark following Deﬁnition 7.6.2). A monoidal closed category structure on Crs is given by an internal hom construction CRS(−, −), that is going to be a crossed complex having Crs(−, −) as set of objects, a tensor product of crossed complexes − ⊗ − construction and a natural isomorphism

Crs(C ⊗ D,E) =∼ Crs(C, CRS(D,E)). for all crossed complexes C,D,E. When we take C = F(n), we have ∼ ∼ Crs(F(n) ⊗ D,E) = Crs(F(n), CRS(D,E)) = CRSn(D,E)

Thus the elements of CRSn(D,E) are ‘n-fold left homotopies’ D → E. In particular, in dimension 1, we may deﬁne the set of homotopy classes of morphisms of crossed complexes

[D,E] = π0(CRS(D,E)).

We begin in Section 8.2 the description of the monoidal closed category structure by defining these elements of CRSn(D,E) (with special emphasis on the homotopies CRS1(D,E)). The rules for addition, action and boundaries come from the geometry associated to the free crossed complexes. Following the development of the tensor product for R-modules explained before, the set of morphisms of crossed modules C → CRS(D,E) is bijective to the set of bimorphisms (C,D) → E, being bimorphisms the concept playing the role of bilinear maps. Then, we can form the tensor product of crossed modules, as in the case of R-modules, by taking free objects and quotienting out by the appropriate relations. The system of relations defining the tensor product is quite complicated and difficult to understand because it comes naturally only after changing the focus from crossed complexes to ω-groupoids. Nevertheless, we think that is worth defining directly these constructions in the category Crs of crossed complexes because is in that category where the applications are manifold. So, we spend a few pages, Section 8.3, trying to make a bit more palatable the tensor product construction. We explain how it decomposes going from low dimensions upward. We end the algebraic part of this Chapter by proving in Section 8.4 that the tensor product preserves freeness.

The second part of the Chapter deals with the relations of the monoidal closed category of Crs and the fundamental crossed complex functor Π: FTop → Crs. We start by giving in Section 8.5 a structure of monoidal closed category to FTop the category of ﬁltered topological spaces and ﬁltered maps that is a straightforward generalisation of the cartesian closed category structure of Top already mentioned in this introduction. The way the two structures of monoidal closed category on FTop and Crs are related is explained in Section 8.6. As before, we leave the proofs for Chapter 16 in Part III of the book. The main result is Theorem 8.6.1 223

stating how the functor Π behaves with respect to tensor products. In particular, if X∗,Y∗ are ﬁltered spaces, then there is a natural transformation

θ : Π(X∗) ⊗ Π(Y∗) → Π(X∗ ⊗ Y∗) which is an isomorphism if X∗,Y∗ are CW -complexes. Using that both categories are monoidal closed, homotopies can be interpreted in both categories as maps from I ⊗ −. Thus an immediate consequence of Theorem 8.6.1 is that the fundamental crossed complex is a homotopy functor. This and the analysis of the cone of a crossed complex leads in Section 8.7 to computations on the fundamental crossed complex of an n-simplex providing a version of the simplicial Homotopy Addition Lemma (Theorem 8.7.4). A similar result is true for n-cubes giving a cubical Homotopy Addition Lemma (Proposition 8.7.6).

8.1 Modules over groupoids: Monoidal closed structure.

There are well known deﬁnitions of tensor product and internal hom functor for Abelian groups (without operators). If one allows operators from arbitrary groups the tensor product is easily generalised (the tensor product of a G-module and an H-module being a (G×H)-module) but the adjoint construction of internal hom functor does not exist, basically because the group morphisms from G to H do not form a group. To rectify this situation we allow operators from arbitrary groupoids and we give a discussion of the monoidal closed category structure of Mod the category of modules over groupoids introduced in Deﬁnition 7.2.7. As is customary, we write M for the G-module (M,G) when the operating groupoid G is clear from the context. Also, to simplify notation, we will assume throughout this chapter that the Abelian groups M(p) for p ∈ G0 are all disjoint; any G-module is isomorphic to one of this type. Perhaps it is more natural in this case to start describing the internal hom functor in Mod.

Deﬁnition 8.1.1 Let (M,G), (N,H) be modules, to construct the internal hom MOD((M,G), (N,H)) we consider the set of morphisms of modules

Mod((M,G), (N,H)) = {(θ, f):(M,G) → (N; H) | (θ, f) is a morphism of modules}

and we have to give this set the structure of module over a groupoid. Notice that θ is given by a family {θp}p∈G0 where

θp : Mp → Nf(p) a f(a) are group morphisms satisfying θp(x ) = θp(x) . We shall use the internal hom groupoid GPDS(G, H) described in the Appendix B whose objects are functors f : G → H and whose morphisms are natural transformations φ : f → f 0. Notice also that these natural 0 transformation φ are given by a family {φ(p)}p∈G0 where φ(p) ∈ H(f(p), f (p)) and the diagram

φ(p) f(p) / f 0(p)

f(p) f 0(a) f(q) / f 0(q) φ(q) commutes for all a ∈ G(p, q). 224

For a ﬁxed functor f : G → H, we deﬁne the set of morphisms of modules over f, i.e.

Modf ((M,G), (N,H)) = {(θ, f):(M,G) → (N; H) | (θ, f) is a morphism of modules}

It is easy to see that the Modf ((M,G), (N,H)) form an Abelian group under element-wise addition, so all morphisms Mod((M,G), (N,H)) form a family of Abelian groups indexed by the set of objects of the groupoid GPDS(G, H).

MOD((M,G), (N,H)) = {Modf ((M,G), (N,H))}f∈Gpds(G,H). It remains to describe the action of GPDS(G, H) on MOD((M,G), (N,H)), i.e. for each f, g ∈ Gpds(G, H) we need a map

Modf ((M,G), (N,H)) × GPDS(G, H)(f, g) → Modg((M,G), (N,H)) So let θ be such that (θ, f) is a morphism of modules and φ : f → g be a natural transformation. We deﬁne

(θ, f)φ = (θφ, g)

φ φ where θ = {θp }p∈G0 is given by the family of morphisms

φ θp : Mp → Ng(p),

φ and θp is deﬁned as the composition φ(p) θp ( − ) Mp −→ Nf(p) −→ Ng(p), i.e. θφ(x) = θ(x)φ(p). They give a morphism because

φ a a φ(p) f(a)φ(p) φ(p)g(a) φ g(a) θp (x ) = θp(x ) = θp(x) = θp(x) = θp (x) . It is not diﬃcult to prove that this deﬁnition gives an action giving a structure of module

MOD(M,N) = (Mod((M,G), (N,H)), GPDS(G, H)) that is the internal hom functor in Mod.

It is quite straightforward to see that, as in the group case, we can characterise the elements of this internal hom functor in terms of ‘bilinear’ maps.

Deﬁnition 8.1.2 A bilinear map of modules (M,G) × (N,H) → (P,K) is given by a couple of maps (θ, f) where f : G × H → K is a map of groupoids and θ : M × N → P is given by a family of bilinear maps

θpq : M(p) × N(q) → Pf(p,q) that preserve actions, i.e.

a b f(a,b) θpq(x , y ) = θpq(x, y) .

Proposition 8.1.3 There is a natural bijection between bilinear maps M × N → P and morphisms of modules from M to MOD(N,P )).

Proof Let us consider an element (θ, f) ∈ Mod(M, MOD(N,P )) then we can deﬁne

fˆ(p, q) = f(p)(q) and θˆ(x, y) = θ(x)(y).

It is easy to see that (θ,ˆ fˆ) is a bilinear map and that this assignation is a natural bijection. 2

Now the tensor product is just deﬁned as the one that transforms these bilinear maps in morphisms of modules. 225

Deﬁnition 8.1.4 The tensor product in Mod of modules (M,G), (N,H) is the module

(M ⊗ N,G × H) where, for p ∈ G0, q ∈ H0,M ⊗ N(p, q) = M(p) ⊗Z N(q) and the action is given by

(x ⊗ y)(a,b) = xa ⊗ yb.

Remark 8.1.5 The module M ⊗ N is the G × H-module generated by all elements

{m ⊗ n | m ∈ M, n ∈ N} subject to the relations (m + m0) ⊗ n = (m ⊗ n) + (m0 ⊗ n), m ⊗ (n + n0) = (m ⊗ n) + (m ⊗ n0), (m ⊗ n)(g,h) = mg ⊗ nh. Thus to deﬁne a morphism M ⊗ N → P all we need is a bilinear map M × N → P

Proposition 8.1.6 There is a natural bijection between bilinear maps M × N → P and morphisms of modules from M ⊗ N to P .

Proof Let us consider a bilinear map (θ, f): M × N → P then we can deﬁne

fˆ(p, q) = f(p)(q) and θˆ(x ⊗ y) = θ(x, y).

It is easy to see that (θ,ˆ fˆ) is a morphism of modules ant that this assignation is a natural bijection. 2

The tensor product gives the category Mod a symmetric monoidal structure with unit object the module (Z, 1), where 1 denotes the trivial group seen as a groupoid. Let us see that both the tensor product and the internal morphisms just deﬁned give Mod the structure of symmetric monoidal closed category.

Proposition 8.1.7 There is a natural bijection

MOD(L ⊗ M,N) =∼ MOD(L, MOD(M,N)).

Proof It is straightforward to verify the natural bijection

Mod(L ⊗ M,N) =∼ Mod(L, MOD(M,N)), where L is a G-module. These families of groups are modules over GPDS(G×H,K) =∼ GPDS(G, GPDS(H,K)) and the actions agree, giving a natural isomorphism of modules

MOD(L ⊗ M,N) =∼ MOD(L, MOD(M,N)).

2 226

8.2 The monoidal closed category Crs of crossed complexes

Just as the internal morphisms gave a correspondence from morphisms in the internal hom construction to bilinear maps and then to morphisms of the tensor product, as in ∼ ModR(C ⊗ D,E) = ModR(C, MODR(D,E)), so we obtain an internal hom CRS(D,E) for crossed complexes D,E as part of the exponential law

Crs(C ⊗ D,E) =∼ Crs(C, CRS(D,E)), for crossed complexes C,D,E where CRS(D,E) is of course again a crossed complex, the internal hom in the category Crs. Crossed complexes have structure in a range of dimensions, whereas R-modules have structure in just one dimension, so the description of the internal hom in Crs has to be much more complicated than of that in ModR, and indeed this complication is part of its value in modelling complicated geometry. We define the internal hom for crossed complexes as giving a ‘home’ for the notion of ‘higher dimensional homotopy’, and then explain the tensor product for crossed complexes. We will be using this process to define our tensor product of crossed complexes, and this means defining the notion of bimorphism b :(C,D) → E for crossed complexes C,D,E. We have to specify the algebraic expression of the geometric properties which we want this notion to reflect. Unfortunately, or perhaps fortunately, these geometric properties are quite complicated, since they depend on the cellular subdivision of products Em × En of cells Em where we have

0 1 0 1 m 0 m−1 m E = {1}, E = e± ∪ e , E = e ∪ e ∪ e , m ≥ 2, 0 0 where e− = −1, e+ = 1. Thus in general the product of these cells has a cell structure with 9 cells. The picture for the cylinder E1 × E2 is as follows. [Cylinder picture, horizontally for E1 direction] We cannot draw the picture for E2 × E2, but that structure contains two solid tori, one of which is pictured as follows [torus picture] and which can be seen as the above cylinder with the two ends identiﬁed. Note that the boundary of E2 × E2 is homeomorphic to a 3-sphere. This can be represented as the set of points (x, y, z, w) ∈ R4 such that x2 + y2 + z2 + w2 = 1 and one of the solid tori is represented by the subset of S3 of points such that x2 + y2 ≤ 1/2, whence z2 + w2 ≥ 1/2.

The corresponding algebraic expression for the boundary of the solid cylinder e1 ×e2 should involve the cells 1 1 0 2 0 2 e × e , e− × e , e+ × e . Our conventions set the base point of the cylinder at (1, 1), i.e. at the ‘top’ end of the cylinder. In the end we take the boundary to be

1 δ(e1 × e2) = −(e1 × e1) − (1 × e2) + (−1 × e2)e ×1, where the conventions as to sign and order of the terms come from some other considerations which we explain later. When we come to take the boundary in the solid torus in E2 × E2 we get a similar formula, except that now −1 × e2, 1 × e2 are identiﬁed to 1 × e2 and so the formula becomes

1 δ(e1 × e2) = −(e1 × e1) − (1 × e2) + (1 × e2)e ×1, 227 which relates to our picture of the solid torus. Another complication is when we glue two cylinders together as in [gluing picture]. The base point of the whole cylinder is at the right hand end, but the base point of the first cylinder is half way along. Thus the algebraic formulae have to reflect this. Finally, we have to distinguish the formulae for Em × En for m, n odd, even, and equal to 0,1,or > 2. All these complications are reflected in the notion of a bimorphism. They give quite different formulae for dimensions 0, 1 and for dimensions ≥ 2. Also in the general case for elements c⊗d where c or d have dimensions 0, 1 and for dimensions ≥ 2.

8.2.1 The internal hom structure of Crs(C,D)

Recall from the introduction to this Chapter, that the elements of CRSn(C,D) can be seen as n-fold homotopies

F(n) ⊗ C → D reflecting the geometry of F(n). In particular, F(1) = I is the unit interval, giving a cylinder construction. An advantage of this procedure is that the elements of the internal hom crossed complex CRS(B,C) in dimension n have a clear interpretation. What is not so clear is that these elements taken altogether can be given the structure of crossed complex. This difficulty is overcome in Chapter 16 in Part III by working with a different but equivalent structure, that of ω-groupoids, which is based on cubes. So in this section, we are not giving the full justification of the results, but hope to explain their intuitive content. We begin the definition of CRS(C,D) from the bottom dimension upwards.

In dimension 0, CRS(C,D)0 is a set deﬁned as

CRS0(C,D) = Crs(C,D).

For dimension 1, we use the concept of left (1-)homotopy, having many points of contact with the homotopy between morphisms of chain complexes.

Deﬁnition 8.2.1 Let C,D be crossed complexes and let

f 0, f 1 : C → D

0 1 be morphisms of crossed complexes (i.e. elements of CRS0(C,D)). A left (1-)homotopy from f to f

H : f 0 ∼ f 1

0 0 1 is a “map of degree 1 from C to D over f0 starting at f and ending at f ”. Let us indicate what is the meaning of this sentence 1.- “a map of degree 1” from C to D For each n ≥ 0, we have a map

Hn : Cn → Dn+1. This sequence of maps can be written 228

/ δn+1 / δn / δn−1 / δ3 / δ2 / s,t / ··· Cn+1 Cn Cn−1 ······ C2 C1 C0 z z z z } } z zz zz zz } } zz z H z z }} }} Hn+1 z Hn zz n−1 zz H2 zz H1 } H0 } zz z z z } } f0 z zz zz zz }} }} zz zz zz zz }} }} }zz z| δ z| z| }~ }~ / n+1 / δn / δ4 / δ3 / δ2 / s,t / ··· Dn+1 Dn ······ D3 D2 D1 D0

0 2.- All Hn for n ≥ 0 have to be “over f0 ”, i.e. Hn is a family of maps {Hn(x)}x∈C0 deﬁned as follows: 0 1 • In dimension 0, for each x ∈ C0, H0(x) connects f and f , i.e.

1 0 H0(x): f0 x → f0 x

0 • For n ≥ 1, the map is over f0 , i.e., for each x ∈ C0,

0 Hn(x): Cn(x) → Dn+1(f0 x).

3.- Moreover, for n ≥ 1, we ask the Hn to preserve operation and action in the best possible way: • For n ≥ 2,

0 c1 - Hn preserve action over f1 , i.e. if c ∈ Cn and c1 ∈ C1, if c is deﬁned, then

0 c1 f (c1) Hn(c ) = Hn(c) 1 and

0 0 - Hn are linear, i.e. for c, c ∈ Cn, if c + c is deﬁned, then

0 0 Hn(c + c ) = Hn(c) + Hn(c ).

0 0 0 • When n = 1, H1 is a (left?) derivation over f , i.e. for c, c ∈ C1, if cc is deﬁned, then

0 f 0(c0) 0 H1(cc ) = H1(c) 1 + H1(c ).

(see Remark 8.2.2 for more details)

To have maps Hn satisfying the preceding properties is just to have a homotopy. 4.- The homotopy H is “from f 0 to f 1” if in the diagram

δn / Cn Cn−1 zz zz zz z Hn z zz z f 0 f 1 z zz n n zz zz zz Hn−1 z| z }zz δn+1 / Dn+1 Dn the “diﬀerence” between the two vertical maps is given by the sum of the two triangles. This requires some care with the base point and to treat as diﬀerent the cases n ≥ 2 and n = 1 • For n ≥ 2, 1 0 −H0(tc) fn(c) = [fn(c) + Hn−1(δnc) + δn+1(Hnc)] , 0 where fn(c) + Hn−1(δnc) + δn+1(Hnc) comes from the diagram and the action is used to change base point from 0 1 f0 (tc) to f0 (tc), and 229

• for n = 1, 1 0 −1 f1 (c) = H0(sc)f1 (c)δ2(H1c)H0(tc) .

With all this preliminaries, we deﬁne the groupoid CRS1(C,D) as the one having Crs0(C,D) = Crs(C,D) as objects, the morphisms from f : C → D to g : C → D are the homotopies, i.e.

CRS1(C,D)(f, g) = {H : f ∼ g | homotopies from f to g}.

The composition is given as follows: Let H : f 0 ∼ f 1 and K : f 1 ∼ f 2 be left homotopies, then we deﬁne H + K : f 0 ∼ f 2 by ( K0(tc) Kn(c) + Hn(c) if c ∈ Cn, n > 1, (H + K)n(c) = H0(c) + K0(c) if c ∈ C0.

It is an easy exercise to check that CRS1(C,D) is a groupoid. This implies immediately that homotopy of morphisms as given explicitly above is an equivalence relation on morphisms C → D. The quotient set is called [C,D].

0 0 Remark 8.2.2 Let us expand a bit on the fact that H1 is an f -derivation. Note that C1 operates on D2 via f and so we can form the semidirect product groupoid C1nD2 with projection pr1 to C1. This groupoid has objects 0 0 0 0 f 0c0 0 C0 and arrows pairs (c, d) ∈ C1 × D2, such that f0 δ1(c) = t(d), with composition (c, d)(c , d ) = (cc , d d ). This can be seen in the picture

•d0 0 0 •df c d0 f 0c0 = •d •d cc0 • / • c c0 • / • / •

0 0 It is then easily seen that an f -derivation H1 is determined completely by a morphism H1 : C1 → C1 n D2 0 0 such that pr1H1 = 1C1 . A corollary is that if C1 is a free groupoid, then an f -derivation is completely determined by its values on a set of free generators of C1.

Remark 8.2.3 Notice that the deﬁnition of the derivation has been on the left. We could have written the derivation rule on the right, i.e. 0 0 f 0(c0) H1(cc ) = H1(c ) + H1(c) 1 .

Exercise 8.2.4 Prove directly that homotopy between morphisms of crossed complexes is an equivalence relation.

Exercise 8.2.5 Deﬁne the notion of homotopy equivalence f : C → D of crossed complexes. Recall that a morphism f : C → D of crossed complexes induces morphisms of the fundamental groupoids and homology groups. Prove that a homotopy equivalence of crossed complexes induces an equivalence of fundamental groupoids. What can you say about the induced morphism of homology groups?

Now we turn to the general structure, deﬁning CRSm(C,D)(f) for dimension m ≥ 2 and f ∈ Crs(C,D); providing it with a structure of CRS1(C,D)-module, and deﬁning the “boundary” maps. 230

Deﬁnition 8.2.6 Let C,D be crossed complexes and let m > 2. Then an m-fold homotopy C → D is a pair (H, f), where f : C → D is a morphism of crossed complexes (the base morphism of the homotopy) and H is a map of degree m from C to D given by functors Hn : Cn → Dn+m for each n > 0 that are morphism of modules over the morphism f1 of groupoids, i.e., Relations with actions for n ≥ 2

Hn preserve action, i.e. if c ∈ Cn and c1 ∈ C1, then

c1 f1(c1) Hn(c ) = Hn(c) .

Relations with operations for n ≥ 1 0 0 Hn are linear for n ≥ 2, i.e. if c, c ∈ Cn and c + c is deﬁned, then 0 0 Hn(c + c ) = Hn(c) + Hn(c ).

0 0 H1 is a derivation over f, i.e. if c, c ∈ C1 and c + c is deﬁned, then

0 0 f1(c ) 0 H1(cc ) = H1(c) + H1(c );

Thus, in each dimension, H and f preserve structure in the only reasonable way. (However, there is no requirement that H should be compatible with the boundary maps δ : Cn → Cn−1 and δ : Dn → Dn−1.) We deﬁne

CRS(C,D)m(f) = {H | (H, f) are m-fold homotopies}.

Remark 8.2.7 For m ≥ 2 there is no diﬀerence between deﬁnition on the left (as given) and on the right, because Hn takes images in abelian groupoids for n ≥ 1.

Let us see how this family of sets get the structure of a crossed complex.

Deﬁnition 8.2.8 The operations, action and boundary maps on CRSm(C,D) are given by:

Operations on CRSm(C,D).- If (H, f), (K, f) are m-fold homotopies C → D over the same base morphism f, where m > 2, we deﬁne (H + K)(c) = H(c) + K(c) for all c ∈ C. 0 0 1 Actions on CRSm(C,D).- If (H, f ) is an m-fold homotopy C → D and if K : f ∼ f is a left homotopy, then we deﬁne HK (c) = H(c)K(tc) for all c ∈ C. Then (HK , f 1) is a morphisms of modules.

Boundaries on CRSm(C,D).- If (H, f) is an m-fold homotopy with m ≥ 2, we deﬁne the boundary δ(H, f) = (δH, f) where δH is the (m − 1)-fold homotopy given by   m+1 δ(H(c)) + (−1) H(δc) if c ∈ Cn(n > 2), (δH)(c) = (−1)m+1H(sc)f(c) + (−1)mH(tc) + δ(H(c)) if c ∈ C ,  1  δ(H(c)) if c ∈ C0.

For 1-homotopies the boundaries are the source and the target already deﬁned (the initial and ﬁnal morphisms). 231

Proposition 8.2.9 CRS(C,D) is a crossed complex.

Proof We could check this result directly, but we shall get later an indirect proof. 2

The fact that CRS(C,D) is a crossed complex contains a lot of information. The formulae for m-fold homotopies are exactly what is needed to express the geometry of the cylinder n I∗ × E because a 1-homotopy can be seen as a morphism I ⊗ C → D. We shall concentrate from now on 1-homotopies.

Remark 8.2.10 An important observation which we will use later is that if g, f are given and c ∈ Cn then δn+1Hn(c) is determined by H0tc and Hn−1δn(c). This is a key to later inductive constructions of homotopies.

8.2.2 The bimorphisms as an intermediate step

With the structure of crossed complex on CRS(C,D) just described, we may study the crossed complex morphisms Crs(C, CRS(D,E)), see how they are deﬁned and reorganise the data. Such a morphism is given by a family of maps fm : Cm → CRSm(D,E) commuting with the boundary maps. For each c ∈ Cm, fm(c) is a homotopy, thus a family of maps fm(c)n : Dn → Em+n satisfying some conditions. We can reorganise these maps, getting a family

fm,n : Cm × Dn → Em+n and see what the diﬀerent conditions mean for these maps. That gives the notion of bimorphism.

Deﬁnition 8.2.11 A bimorphism θ :(C,D) → E for crossed complexes C,D,E is a family of maps

θmn : Cm × Dn → Em+n so that, for every c ∈ Cm, the map θm(c) = {θmn(c, −)}n∈N is an m-homotopy. That means that θmn have to satisfy the following conditions, where c ∈ Cm, d ∈ Dn, c1 ∈ C1, d1 ∈ D1: • Source and target They preserve target and, whenever appropriate, source

t(θ(c, d)) = θ(tc, td) for all c ∈ C, d ∈ D. s(θ(p, d)) = θ(p, sd) if m = 0, n = 1 , s(θ(c, q)) = θ(sc, q) if m = 1, n = 0 .

• Actions They preserve the action in dimensions ≥ 2

θ(c, dd1 ) = θ(c, d)θ(tc,d1) if m > 0, n > 2 , θ(cc1 , d) = θ(c, d)θ(c1,td) if m > 2, n > 0 .

• Operations They preserve operations in the best possible way:

- For m 6= 1 or n 6= 1, the θmn are homomorphisms

θ(c, d + d0) = θ(c, d) + θ(c, d0) if m = 0, n > 1 or m > 1, n > 2 , θ(c + c0, d) = θ(c, d) + θ(c0, d) if m > 1, n = 0 or m > 2, n > 1. 232

- Whenever m = 1 or n = 1 they behave like derivations

0 θ(c, d + d0) = θ(c, d)θ(tc,d ) + θ(c, d0) if m > 1, n = 1 , 0 θ(c + c0, d) = θ(c0, d) + θ(c, d)θ(c ,td) if m = 1, n > 1 .

• Boundaries These are a bit mysterious. - In general, they behave like in chain complexes

m δm+n(θ(c, d)) = θ(δmc, d) + (−1) θ(c, δnd) if m > 2, n > 2.

- When one of the elements has dimension 1, we have to take account of the action to make elements compatible   θ(c,td)  − θ(c, δnd) − θ(tc, d) + θ(sc, d) if m = 1, n > 2 , m+1 m θ(tc,d) δm+n(θ(c, d)) = (−1) θ(c, td) + (−1) θ(c, sd) + θ(δmc, d) if m > 2, n = 1 ,   − θ(tc, d) − θ(c, sd) + θ(sc, d) + θ(c, td) if m = n = 1 .

- And, whenever one of the elements has dimension 0, we forget about the 0-dimensional part. ( θ(c, δnd) if m = 0, n > 2 , δm+n(θ(c, d)) = θ(δmc, d) if m > 2, n = 0 .

You should look at these carefully and note (but not necessarily learn!) the way these formulae reflect the geometry and algebra of crossed complexes, which allow for differences between the various dimensions, and also for change of base point. Let us point out that we have temporarily used additive notation throughout the definition for all dimensions (including 1 and 2) to reduce the number of formulae. The bimorphisms are used as an intermediate step in the construction of the tensor product due to the following property

Theorem 8.2.12 For crossed complexes C,D,E, there are natural bijections from Crs(C, CRS(D,E)), to the set of bimorphisms θ :(C,D) → E.

8.2.3 The tensor product of two crossed complexes

Following the pattern in the tensor product of R-modules, we now try to internalise the concept of bimorphism. We want to construct a crossed complex, the tensor product C ⊗ D of two crossed complexes, and a universal bimorphism Υ:(C,D) → C ⊗ D.

Essentially, this means that C ⊗ D is generated by elements c ⊗ d, with c ∈ Cm and d ∈ Dn, subject to the relations given by the rules of bimorphisms with θ(c, d) replaced by c ⊗ d. Thus (C ⊗ D)p has to decompose in pieces (C ⊗ D)m,n with m + n = p each one internalising the rules corresponding to θmn Let us start by making clear the meaning for the groupoid part of C ⊗ D. For p = 0, we deﬁne

(C ⊗ D)0 = C0 × D0 233 as sets.

For p = 1, the groupoid (C ⊗ D)1 over (C ⊗ D)0 has two parts, namely (C ⊗ D)1,0 = C1 × D0 and (C ⊗ D)0,1 = C0 × D1. Then (C ⊗ D)1 is their coproduct as groupoids over (C ⊗ D)0, we write

(C ⊗ D)1 = C1 # D1.

This groupoid may be seen also as generated by the symbols

{c ⊗ q | c ∈ C1} ∪ {p ⊗ d | d ∈ D1} for all p ∈ C0 and q ∈ D0 subject to the relations given by the products on C1 and on D1. We shall go back to this in Subsection 8.3.1.

Also, we shall prove in Subsection 8.3.2 that the image of δ2(C ⊗ D)2 in (C ⊗ D)1 is generated by all the elements

0 −1 0 −1 {δc ⊗ q | c ∈ C2} ∪ {p × δd | d ∈ D2} ∪ {(c ⊗ q)(p ⊗ d)(c ⊗ q ) (p ⊗ d) | c ∈ C1, d ∈ D1} for all p ∈ C0 and q ∈ D0 (Notice that the last family of relations is given by the commutators of the generators of (C ⊗ D)1,0 = C1 × D0 and (C ⊗ D)0,1 = C0 × D1).

Once this has been recorded, we can proceed with the deﬁnition of (C ⊗ D)p for p ≥ 2.

Deﬁnition 8.2.13 Let C,D be crossed complexes. For any c ∈ Cm, d ∈ Dn we consider the symbol c ⊗ d. Whenever m + n ≥ 2, we deﬁne its source and target

s(c ⊗ d) = t(c ⊗ d) = tc ⊗ td.

(Notice that for elements of dimension 0 we deﬁne t(p) = p and t(q) = q.

For p ≥ 2, we consider Fp the free (C ⊗ D)1-module (or crossed module if p = 2) on

{c ⊗ d | c ∈ Cm, d ∈ Dn, m, n ∈ N, m + n = p}.

To get (C ⊗ D)p we have to quotient out by some relations with respect to the additions and actions. Notice that all relations are “dimension preserving”. There are two essentially diﬀerent cases. • When both m, n 6= 1, we do not have to worry about source and target (both are the same), and the relations are easier: - Operation: The relations to make ⊗ compatible with operations are

c ⊗ (d + d0) = c ⊗ d + c ⊗ d0 if n > 2

(c + c0) ⊗ d = c ⊗ d + c0 ⊗ d if m > 2.

- Action: The relations to make ⊗ compatible with operations are

(c ⊗ d)(tc⊗d1) = c ⊗ dd1 if m > 0, n > 2 , (c ⊗ d)(c1⊗td) = cc1 ⊗ d if m > 2, n > 0 . and it is compatible with relations.

- Cokernel. When m + n ≥ 3, we have to kill the action of δ2(C ⊗ D)2 ⊆ (C ⊗ D)1 234

• When one element has dimension 1. - Then the operation has to be related with the action because the groupoid part acts on itself by conjugation.

0 c ⊗ dd0 = (c ⊗ d)(tc⊗d ) + c ⊗ d0 if m > 1, n = 1 , 0 cc0 ⊗ d = c0 ⊗ d + (c ⊗ d)(c ⊗td) if m = 1, n > 1 .

- Cokernel. When m + n ≥ 3, we have to kill the action of δ2(C ⊗ D)2 ⊆ (C ⊗ D)1.

With this, we get (C ⊗ D)p as the quotient of Fp by all these relations. To ﬁnish the structure of C ⊗ D as a crossed complex, the boundaries are deﬁned on generators with formulae varying according to dimensions. • When both have dimension ≥ 2

m δm+n(c ⊗ d) = δmc ⊗ d + (−1) (c ⊗ δnd)

• When one has dimension 1 and the other one has dimension ≥ 1

  (c⊗td)  − (c ⊗ δnd) − (tc ⊗ d) + (sc ⊗ d) if m = 1, n > 2 , m+1 m (tc⊗d) δm+n(c ⊗ d) = (−1) (c ⊗ td) + (−1) (c ⊗ sd) + (δmc ⊗ d) if m > 2, n = 1 ,   − (tc ⊗ d) − (c ⊗ sd) + (sc ⊗ d) + (c ⊗ td) if m = n = 1

• When one has dimension 0 ( (c ⊗ δnd) if m = 0, n > 2 , δm+n(c ⊗ d) = (δmc ⊗ d) if m > 2, n = 0 . and these deﬁnitions are compatible with the relations.

Remark 8.2.14 Notice that if we denote by Fm,n the free (C ⊗ D)1-module on {c ⊗ d | c ∈ Cm, d ∈ Dn} for some ﬁxed m, n ∈ N, Fp is the coproduct of {Fm,n}m+n=p.

Since the relations with respect to the additions and actions we are using to get (C ⊗ D)p preserve the decomposition of Fp as the coproduct of Fm,n,(C ⊗ D)p also decomposes as coproduct of the quotient of Fm,n respect to the corresponding relations. We shall call (C ⊗ D)m,n this quotient.

Remark 8.2.15 There is an alternative way of defining Fm,n that works when m, n 6= 0, m + n ≥ 3. 0 We could define Fm,n as the free abelian groupoid on {c ⊗ d | c ∈ Cm, d ∈ Dn} and quotient out by the 0 relations on operations included in the previous definition getting an abelian groupoid Cm,n. This quotient is isomorphic to (C ⊗ D)m,n as abelian groupoid. 0 Next we define the (C ⊗ D)1-action on Cm,n by the formulae in the previous definition (notice that the 0 definition is different when m = 1 or n = 1). It is not difficult to prove that this gives an action and that Cm,n is the isomorphic to (C ⊗ D)m,n as (C ⊗ D)1-modules

Exercise 8.2.16 Check the rule δδ(c ⊗ d) = 0 for some low dimensional cases, such as dim(c ⊗ d) = 3, 4, seeing how the crossed module rules come into play. 235

Those familiar with the tensor product of chain complexes may note that in that theory the single and simple formula we need is ∂(c ⊗ d) = (∂c) ⊗ d + (−1)mc ⊗ (∂d) where dim c = m. So it is not surprising that the tensor product of crossed complexes has much more power than that of chain complexes, and can handle more complex geometry. The specific conventions in writing down the formulae for this tensor product of crossed complexes come from another direction, which is explained fully in Chapter 16 in Part III, namely the relation with cubical ω-groupoids with connection. The tensor product there comes out simply, because it is based on the formula Im × In =∼ Im+n. The distinction between that formula and that for the product of cells as above lies at the heart of many difficulties in basic homotopy theory. The relation between ω-groupoids and crossed complexes gives an algebraic expression of these geometric relationships. Using this definition, it can be proved that the tensor product gives a symmetric monoidal structure to Crs the category of crossed modules by defining the maps on generators and checking that they preserve the relations.

Theorem 8.2.17 With the bifunctor −⊗−, the category Crs of crossed complexes has a structure of a symmetric monoidal category, i.e.

i) For crossed complexes C,D,E, there are natural isomorphisms of crossed complexes

(C ⊗ D) ⊗ E =∼ C ⊗ (D ⊗ E),

ii) for all crossed complexes C,D there is a natural isomorphism of crossed complexes

T : C ⊗ D → D ⊗ C satisfying the appropriate axioms.

Proof The existence of both isomorphisms could be established directly, giving the values on generators in the obvious way: i) is given by (c ⊗ d) ⊗ e 7→ c ⊗ (d ⊗ e) and mn ii) is given by T (c ⊗ d) = (−1) d ⊗ c if c ∈ Cm and d ∈ Dn. and then checking that the relations on generators c ⊗ d in Definition (8.2.13) are preserved by both maps. The necessary coherence and naturality conditions are obviously satisfied. But to check all the cases even for such simple maps seems tedious. An alternative approach is to go via ω-groupoids where the tensor product fits more closely to the cubical context. 2

This proof of commutativity is somehow unsatisfactory because, although it is clear that c ⊗ d 7→ d ⊗ c does not preserve the relations in Definition 8.2.13, the fact that c ⊗ d 7→ (−1)mnd ⊗ c does preserve them seems like a happy accident. A better explanation is provided by the transposing functor T (see Sections 16.1 and 16.2). Note that while the tensor product can be defined directly in terms of generators and relations and this can sometimes prove useful, such a definition may make it difficult to verify essential properties of the tensor product, such as that the tensor product of free crossed complexes is free. We shall prove that later (Section 8.4) using the adjointness of ⊗ and the internal hom functor as a necessary step to prove that − ⊗ C preserves colimits. 236

Nevertheless, this Deﬁnition is interesting for its relation to tensor product of ﬁltered spaces that we shall study in Section 8.6.

Theorem 8.2.18 For crossed complexes C,D,E, there are natural exponential law

Crs(C ⊗ D,E) =∼ Crs(C, CRS(D,E)), giving the category Crs of crossed complexes a structure of monoidal closed category. Moreover, they produce isomorphisms of crossed complexes

CRS((C ⊗ D),E) =∼ CRS(C, CRS(D,E)).

It is also important that we have to use crossed complexes of groupoids to make sense of the exponential law in Crs. This is analogous to the fact that the category of groups has no internal hom, while that of groupoids does.

Remark 8.2.19 Consider the groupoid I having one arrow ι : 0 → 1 so that t(ι) = 1 (we have seen that this groupoid is Π(E1). A ‘1-fold left homotopy’ of morphisms f 0, f 1 : C → D is seen to be a morphism I ⊗ C → D which takes the values of f 0 on 0 ⊗ C and f 1 on 1 ⊗ C . The existence of this ‘cylinder object’ I ⊗ C allows a lot of abstract homotopy theory [116] to be applied immediately to the category Crs. This is useful in constructing homotopy equivalences of crossed complexes, using for example the gluing lemma [116, Lemma 7.3].

8.3 Analysis of the tensor product C ⊗ D

The Definition of the tensor product of two crossed complexes C ⊗ D is quite complex. We are going to devote this Section to clarify the definition. It happens that at each dimension, the tensor product (C ⊗ D)p decomposes as the coproduct of simpler bits (C ⊗ D)m,n that can be identified to (or related with) some better known constructions. As is normal in the crossed complex situation, the description is different (and more complicated) in low dimensions.

So, ﬁrst we study the groupoid (C ⊗ D)1 that is just the coproduct over C0 × D0 of the two groupoids C1 × D0 and C0 × D1. This is very important since all the (C ⊗ D)p are modules (or crossed modules) over (C ⊗ D)1 Then, we study the crossed module part of C ⊗D. It has three parts, two of them being got from the crossed modules C2 × D0 → C1 × D0 and C0 × D2 → C0 × D1 using the induced crossed module construction for the inclusions C1 × D0 → (C ⊗ D)1 and C0 × D1 → (C ⊗ D)1.

With respect to higher dimensions, (C ⊗ D)p decomposes in many pieces (C ⊗ D)m,n for m + n = p. When both m, n ≥ 3, (C ⊗ D)m,n is just the tensor product as modules studied in Section 8.1. When one (or both) of the dimensions is 2, (C ⊗ D)m,n is the tensor product of the abelianisation. It remains the cases when m = 0 or m = 1 (and the symmetric). To identify them, we have to introduce a couple of constructions associated to a groupoid: the right regular H-module Z~ H, and the right augmentation module IH~ .

8.3.1 The groupoid (C ⊗ D)1.

In order to become more familiar with the deﬁnition of the tensor product of crossed complexes, in this Section we are going to do the computations with some detail in low dimensions. 237

Notice ﬁrst that it is clear from the deﬁnition that to construct (C ⊗ D)p we only need to know {Cm}m≤p and {Dn}n≤p since there are no relations among the generators in (C ⊗ D)p coming from higher dimensions. Let us see what this means for low dimensions. The case p = 0 is immediate. Let us start with the case p = 1.

Proposition 8.3.1 For any pair of crossed complexes C,D ∈ Crs the groupoid (C ⊗ D)1 of their tensor product is the following pushout in the category of groupoids / 1C1 × 1D1 C1 × 1D1

/ 1C1 × D1 (C ⊗ D)1 where, for any groupoid G, 1G denotes the trivial sub-groupoid consisting of all identity elements of G. It is easy to see that this pushout is the coproduct of C1 ×1D1 and 1C1 ×D1 in the category of groupoids over C0 ×D0

Let us give a description of this groupoid. By the previous Proposition, it is actually a construction in the category of groupoids. So, let us consider a pair of groupoids G and H and let us deﬁne

G # H = G × 1H ∗ 1G × H, the coproduct in the category of groupoids over G0 × H0.

The groupoid G # H is generated by all elements (1p, h), (g, 1q) where g ∈ G, h ∈ H, p ∈ G0, q ∈ H0. We will sometimes write g for (g, 1q) and h for (1p, h). This may seem to be willful ambiguity, but when composites are specified in G # H, the ambiguity is resolved; for example, if gh is defined in G # H, then g must refer to (g, 1q), where q = sh, and h must refer to (1p, h), where p = tg. This convention simplifies the notation and there is an easily stated solution to the word problem for G # H. Every element of G # H is uniquely expressible in one of the following forms:

(i) an identity element (1p, 1q);

(ii) a generating element (g, 1q) or (1p, h), where p ∈ G0, q ∈ H0, g ∈ G, h ∈ H and g, h are not identities;

(iii) a composite k1k2 ··· kn(n > 2) of non-identity elements of G or H in which the ki lie alternately in G and H, and the odd and even products k1k3k5 ··· and k2k4k6 ··· are deﬁned in G or H.

−1 For example, if g1 : m → p, g2 : p → q in G, and h1 : r → s, h2 : s → t in H, then the word g1h1g2h2g2 represents an element of G # H from (m, r) to (p, t). Note that the two occurrences of g2 refer to diﬀerent elements of G # H, namely (g2, 1s) and (g2, 1t). This can be represented as a path in a 2-dimensional grid as follows

(m, r) (m, s)(m, t)

h (p, r) 1 / (p, s) (p, t)

−1 g2 g2 h (q, r)(q, s) 2 / (q, t)

The similarity with the free product of groups is obvious and the normal form can be veriﬁed in the same way; for example, one can use ‘van der Waerden’s trick’. We omit the details. 238

8.3.2 The crossed module (C ⊗ D)2 → (C ⊗ D)1.

To identify the crossed module in the title for crossed complexes C,D, we need to use two constructions from the theory of crossed modules: the coproduct of crossed modules over the same base and the induced crossed module.

In the case when G is a group, the construction of the coproduct M ◦G N of crossed G-modules M and N has been studied in Part I. This construction works equally well when G is a groupoid. The family of groups M acts on N via G, so one can form the semidirect product M n N. It consists of a semidirect product of groups

Mp n Np at each vertex p of G and it is a pre-crossed module over G. One then obtains the crossed G-module M ◦G N from M n N by factoring out its Peiﬀer groupoid.

Now, recall that (C ⊗ D)2 as (C1 # D1)-crossed module is the coproduct

(C ⊗ D)2 = (C ⊗ D)2,0 ◦ (C ⊗ D)1,1 ◦ (C ⊗ D)0,2 where these last crossed modules have been deﬁned in Remark 8.2.14.

Since C2 is a crossed module over the groupoid C1, C2 × D0 is a crossed module over C1 × D0. Using embedding

µ1 : C1 × D0 → C1 # D1 we get an induced crossed module

Cˆ2 = µ1∗(C2 × D0). It is not diﬃcult to see that ∼ (C ⊗ D)2,0 = Cˆ2 as (C1 # D1)-crossed modules. In the same way, we identify ∼ (C ⊗ D)2,0 = Dˆ2 where Dˆ2 = µ2∗(C0 × D2).

It remains to identify (C ⊗ D)1,1. To do this, we only need to consider the case when C and D are just groupoids. To make things more clear we restrict ourselves to crossed complexes associated to groupoids since the higher dimensional part does not intervene. So Cn = Dn = {0} for all n ≥ 2. Then we know (C ⊗ D)p = {0} for all p ≥ 3 and we have computed that (C ⊗ D)0 = C0 × D0 and (C ⊗ D)1 = C1 # D1. Also, to make notation easier, let us write G and H for the groupoids C1 and D1. Notice that there is a canonical morphism

σ : G # H → G × H induced by the inclusions 1G × H → G × H and G × 1H → G × H. This morphism is deﬁned on a word k1k2k3 ··· , by separating the odd and even parts, i.e.

σ(k1k2k3 ··· ) = (k1k3 ··· , k2k4 ··· ).

That is, the map σ introduces a sort of commutativity between G and H. The kernel of σ will be called the Cartesian subgroupoid of G # H and will be denoted by G 2 H, i.e.

Ker σ = G 2 H. 239

It consists of all identities and all words k1k2 ··· kn for which both odd and even products are trivial. Clearly, it is generated by all ‘commutators’ [g, h] = g−1h−1gh, where g ∈ G, h ∈ H and g, h are not identities. (Note that [g, h] is uniquely deﬁned in G # H for any such pair of elements g, h, but the two occurrences of g (or of h) do not refer to the same element of G # H.)

Proposition 8.3.2 The Cartesian subgroupoid G 2 H of G # H is freely generated, as a groupoid, by all elements [g, h] where g, h are non-identity elements of G, H, respectively. Thus, G 2 H is the disjoint union of free groups, one at each vertex, and the group at vertex (p, q) has a basis consisting of all [g, h] with tg = p and th = q (g and h not identity elements).

Proof In the notation introduced above the ‘commutators’ [h, g] satisfy the same formal identities as in the group case: [h, g] = [g, h]−1,

h1 [hh1, g] = [h, g] [h1, g],

g1 [h, gg1] = [h, g1][h, g] whenever gg1, hh1 are deﬁned in G, H. These identities are to be interpreted as equations in G # H, with the h1 −1 −1 −1 obvious meaning for conjugates: [h, g] means h1 h g hgh1, which represents a unique element of G # H. Now G 2 H is an intransitive free groupoid with basis consisting of all [g, h](g ∈ G, h ∈ H, g, h 6= 1) (see Gruenberg [98], Levi [123]). 2

Theorem 8.3.3 The tensor product of the groupoids G and H, considered as crossed complexes of rank 1, is the crossed complex G ⊗ H = (· · · → 0 → · · · → 0 → G 2 H → G # H) with g ⊗ h = [h, g], p ⊗ h = (1p, h), g ⊗ q = (g, 1q) for g ∈ G, h ∈ H, p ∈ G0, q ∈ H0.

Proof G 2 H is a normal subgroupoid of G # H, so

δ : G 2 H → G # H is a crossed module which we view as a crossed complex C, trivial in dimension > 3. One verifies easily that the equations θ(g, h) = [h, g], θ(g, ·) = g, θ(·, h) = h define a bimorphism θ :(G, H) → C; the equations in Definition 8.2.11(iii) reduce to the standard commutator identities

h1 [hh1, g] = [h, g] [h1, g],

g1 [h, gg1] = [h, g1][h, g] , and the rest are trivial. It follows that if φ :(G, H) → D is any bimorphism, there is a unique morphism of groups φ2 : G 2 H → D2 such that φ2([h, g]) = φ(g, h) for all g ∈ G, h ∈ H. (Note that the deﬁnition of bimorphism implies that φ(g, h) = 1 if either g = 1 or h = 1.) There is also a unique morphism φ1 : G # H → D1 such that φ1(g) = φ(g, ·) and φ1(h) = φ(·, h) for all g ∈ G, h ∈ H. These morphisms combine to give a morphism

φ : C → D 240 of crossed complexes as we show below, and this proves the universal property making C the tensor product of G and H, with g ⊗ h = [h, g]. We need to verify that φ : C → D is a morphism of crossed modules. This amounts to (i) φ is compatible with δ : G 2 H,→ G # H. Now

δφ2([h, g]) = δφ(g, h) = −φ(·, h) − φ(g, ·) + φ(·, h) + φ(g, ·) by (8.2.11)(iv)

= [φ(·, h), φ(g, ·)] = [φ1(h), φ1(g)] = φ1[h, g] and

(ii) φ preserves the actions of G # H and D1. Now

g1 −1 φ2([h, g] ) = φ2([h, g1] [h, gg1])

= −φ(g1, h) + φ(gg1, h) = φ(g, h)φ(g1,·) by (8.2.11)(iii)

φ1(g1) = φ2([h, g]) .

There is a similar calculation for the action of h1 ∈ H, and the result follows. 2

That gives a useful description of the crossed module part of the tensor product of two crossed complexes C and D.

Theorem 8.3.4 There is an isomorphism of (C1 # D1)-crossed modules ∼ (C ⊗ D)2 = µ1∗(C2 ⊗ D0) ◦ G 2 H ◦ µ2∗(C0 ⊗ D2).

This isomorphism maps c ⊗ q 7→, p ⊗ d 7→ and c ⊗ d 7→ to(c ⊗ q)(p ⊗ d)(c ⊗ q0)−1(p0 ⊗ d)−1. So the subgroupoid

δ2(C ⊗ D)2 is generated as a groupoid by the elements

0 −1 0 −1 {δc ⊗ q | c ∈ C2, q ∈ D0} ∪ {p × δd | p ∈ C0, d ∈ D1} ∪ {(c ⊗ q)(p ⊗ d)(c ⊗ q ) (p ⊗ d) | c ∈ C1, d ∈ D1}.

The description in Theorem 8.3.3 is much easier for the case of groups. Any group G can be viewed as a crossed complex E1(G) with E1(G)0 = {·}, E1(G)1 = G, E1(G)n = 0 for n > 2. The tensor product of two such crossed complexes will have one vertex and will be trivial in dimension > 3, that is, it will be a crossed module. We use multiplicative notation for G for reasons which will appear later.

Proposition 8.3.5 The tensor product of groups G, H, viewed as crossed complexes of rank 1, is the crossed module G 2 H → G ∗ H, where G 2 H denotes the Cartesian subgroup of the free product G ∗ H, that is, the kernel of the map G ∗ H → G × H. If g ∈ G, h ∈ H, then g ⊗ h is the commutator [h, g] = h−1g−1hg = [g, h]−1 in G ∗ H.

Remark 8.3.6 This tensor product of (non-Abelian) groups is related to, but not the same as, the tensor product defined by Brown and Loday and used in their construction of universal crossed squares of groups [50]. The Brown-Loday product is defined for two groups acting compatibly on each other. It also satisfies the standard commutator identities displayed above. The relation between the two tensor products is clarified by Gilbert and Higgins in [91]. See also the results of Baues and Conduchéin [12]. 241

8.3.3 The tensor product (C ⊗ D)m,n for m, n ≥ 2

Recall that for any crossed complexes C,D,(C ⊗ D)p as (C1 # D1)-module is the coproduct of (C ⊗ D)m,n for m + n = p. Also recall that the (C1 # D1)-module (C ⊗ D)m,n is generated by c ⊗ d with three set of relations given by the operations, the actions and the cokernel condition. By the deﬁnition of the action on C ⊗ D, it is clear that if m, n ≥ 3, the cokernel relations are redundant and can be forgotten. The resulting quotient can be easily identiﬁed in terms of tensor product of modules.

Let M,N be modules over the groupoids G, H, respectively, we may form M ⊗Z N, an abelian groupoid over G0 × H0 with the group M(p) ⊗Z N(q) lying over (p, q). Then M ⊗Z N is a (G × H)-module, with diagonal action (g,h) g h (m ⊗Z n) = m ⊗Z n , and hence it is a (G # H)-module via the canonical map G # H → G × H.

Proposition 8.3.7 For m, n ≥ 3, there is an isomorphism of (C1 # D1)-modules

∼ (C ⊗ D)m,n = Cm ⊗Z Dn.

Proof We have just got to check that the sets of generators and relations of both constructions are naturally bijective. 2

Very similar is the case when one of the dimensions is 2. Then we have to take care of the relations corresponding to the crossed module part. It is clear (using the crossed module condition) that to kill the action of δ2(C ⊗ D)2 is the same as saying that [c, d] = 0 for all commutators of c ∈ C1 and d ∈ D1. Thus

Proposition 8.3.8 There is a natural isomorphism when m ≥ 3

∼ ab (C ⊗ D)m,2 = Cm ⊗Z D2 as (C1 # D1)-modules.

This is true even in the case when both parts have dimension 2.

Proposition 8.3.9 There is a natural isomorphism

∼ ab ab (C ⊗ D)2,2 = C2 ⊗Z D2 as (C1 # D1)-modules.

8.3.4 The tensor product (C ⊗ D)m,n for n = 0, 1

The only pieces of (C ⊗ D)m+n that remains to be identiﬁed are those containing one factor in dimensions 0, 1. Let us analyse ﬁrst the case (C ⊗D)m,0 when m ≥ 2. Checking the appropriate universal property we can prove the following result.

Proposition 8.3.10 The (C1 # D1)-module (C⊗D)m,0 is naturally isomorphic to the induced module µ1∗(Cm⊗ D0), where Cm ⊗ D0 is the obvious (C1 ⊗ D0)-module and µ1 : C1 ⊗ D0 ⊆ C1 # D1 is the inclusion map. 242

Let us get an alternative description when m ≥ 3. Given a groupoid H, we write Z~ H for the abelian groupoid

(family of abelian groups) over H0 in which Z~ H(q) is the free abelian group on all h ∈ H with th = q. Then Z~ H becomes a (right) H-module under composition on the right: X X k ( αhh) = αh(hk) P when αhh ∈ Z~ H(q) and k ∈ H(q, r). This is the right regular H-module.

Proposition 8.3.11 There is a natural isomorphism

∼ (C ⊗ D)m,0 = Cm ⊗Z Z~ D1 as (C1 # D1)-modules when m ≥ 3.

The second piece to have to identify is (C ⊗ D)m,1 when m ≥ 2. The right augmentation module IH~ of H is the submodule generated by all h − h0, where h, h0 ∈ H and 0 th = th . Then IH~ (q) has a basis consisting of all h − 1q where th = q.

Proposition 8.3.12 The piece (C ⊗ D)m,1 can be naturally identiﬁed as (C1 # D1)-modules with the quotient of Cm ⊗Z ID~ 1 killing the action of δ2(D2). (for m ≥ 3?)

Corollary 8.3.13 For any crossed complex C the canonical maps i0, i1 : C → C ⊗ I deﬁned by iα(c) = c ⊗ pα (α = 0, 1) are injections. 2

8.4 The tensor product of free crossed complexes is free

The exponential law in Crs of Theorem 8.2.18 has as a consequence that the tensor product of free crossed complex is a free crossed complex. We start by proving the result for the standard models.

Proposition 8.4.1 Consider the inclusions S(n − 1) → F(n) and S(m − 1) → F(m) then

S(n − 1) ⊗ F(m) ∪ F(n) ⊗ S(m − 1) → F(m) ⊗ F(n) is of relative free type.

Proof Let us remark that they diﬀer only in dimension (m + n). We have to check that the diagram

S(n + m − 1) / S(n − 1) ⊗ F(m) ∪ F(n) ⊗ S(m − 1)

F(n + m) / F(m) ⊗ F(n) given on generators by mapping xm+n 7→ xm ⊗ xn is a pushout of crossed complexes and this is easily done. 2

The proof of the general theorem builds inductively on the previous case. Crucial steps are proved using that Y ⊗ − and − ⊗ Z preserve colimits. 243

Theorem 8.4.2 If C0 → C and D0 → D are morphisms of relative free type then so also is C0 ⊗ D ∪ C ⊗ D0 → C ⊗ D, where C0 ⊗ D ∪ C ⊗ D0 denotes the pushout of the pair of morphisms

C0 ⊗ D ← C0 ⊗ D0 → C ⊗ D0.

Proof Since the tensor product − ⊗ − is symmetric and − ⊗ B has a right adjoint, the functors − ⊗ C and D ⊗ − preserve colimits. Using this fact and standard properties of pushouts, one easily proves the following four lemmas:

Lemma 8.4.3 If in a pushout square C0 / D0

C / D the morphism C0 → C is of relative free type, so is the morphism D0 → D.

Lemma 8.4.4 If in a sequence of morphisms of crossed complexes

C0 → C1 → ... → Cn → ... each morphism is of relative free type, so are the composites C0 → Cn and the induced morphism C0 → n colimnC .

Lemma 8.4.5 If in a commutative diagram

C0 / C1 / ... / Cn / ...

D0 / D1 / ... / Dn / ...

n n each vertical morphism is of relative free type, so is the induced morphism colimnC → colimnD .

Lemma 8.4.6 If the following squares are pushouts

C / D U / V

E / F W / X then so is the induced square C ⊗ W ∪ E ⊗ U / D ⊗ X ∪ F ⊗ V

E ⊗ W / F ⊗ X

These four lemmas are the basis for an inductive proof of our theorem. We start by the case proved before the Theorem. Using the fact that Y ⊗ − and − ⊗ Z preserve coproducts, one deduces the result in the case when C0 → C, 0 ` ` ` ` D → D are of the type λ S(n − 1) → λ F(n) and λ S(m − 1) → λ F(m). 244

Putting morphisms of this type in Lemma 8.4.6, and using Lemma 8.4.3, one ﬁnds that the theorem is true for morphisms of simple relative free type, that is for morphisms C0 → C, D0 → D obtained as pushouts ` ` / 0 / 0 λ S(n − 1) C µ S(m − 1) D

` ` / / λ F(n) C µ F(m) D

Next, using Lemmas 8.4.3, 8.4.4, 8.4.6 one proves the result for composites of morphisms of relative free type. A general morphism of relative free type is a colimit of simple ones, as in Lemma 8.4.4, and the full result now follows from Lemmas 8.4.4 and 8.4.6. 2

Some of the Lemmas in the above proof are useful later.

Corollary 8.4.7 If C0 → C is a morphism of relative free type and W is a crossed complex of free type, then C0 ⊗ W → C ⊗ W is of relative free type.

Corollary 8.4.8 If C is a free crossed complex and f : C → D is a morphism of crossed complexes, then a homotopy H : f ' g of morphisms is entirely determined by its values on the free basis of C.

8.5 The monoidal closed category FTop of ﬁltered spaces

We proceed a step further and consider the category FTop of ﬁltered spaces and look for a natural structure of closed category. The categorical product in FTop is given by

(X∗ × Y∗)n = Xn × Yn, n ≥ 0.

This product is convenient for maps into it, as for any categorical product. However our main example of filtered spaces, that of CW -complexes, suggests a different product as worth consideration, and this will turn out to be convenient for maps from it, to other filtered spaces.

If X∗,Y∗ are CW -ﬁltrations, then the product X × Y of the spaces (in the category of compactly generated spaces) has a natural and convenient CW -structure in which the n-dimensional cells are all products ep × eq of cells of X∗,Y∗ respectively where p + q = n. This suggests the following deﬁnition.

Definition 8.5.1 If X∗,Y∗ are filtered spaces, their tensor product X∗ ⊗ Y∗ is the filtered space given on X × Y by the family of subspaces [ (X ⊗ Y )n = Xp × Yq p+q=n where the union is simply the union of subspaces of X × Y .

Exercise 8.5.2 1. We have said that the filtration X∗ ⊗ Y∗ is not the product in the category FTop. Verify that our definition above does define the product X∗ × Y∗ in the category FTop. 2. Is there a structure of cartesian closed category on FTop? i.e. is there an internal hom that is adjoint to the cartesian product? 245

n n Notice that, for example, I∗ is the n-fold tensor product of I∗ with itself because I∗ is the CW -ﬁltered space of the standard n-cube. With the product ⊗, FTop is a monoidal category. The tensor product is also commutative.

We now show how to deﬁne an ‘internal hom’ FTOP(Y∗,Z∗) in the category FTop so as to make that category a monoidal closed category with an exponential law giving a natural bijection ∼ e : FTop(X∗ ⊗ Y∗,Z∗) = FTop(X∗, FTOP(Y∗,Z∗)).

To see how this comes about, note that a filtered map f : X∗ ⊗ Y∗ → Z∗ will map Xp × Yq to Zp+q, by definition of the filtration on the tensor product of filtered spaces. Under the exponential law for topological spaces we have ∼ Top(Xp × Yq,Zp+q) = Top(Xp, TOP(Yq,Zp+q)). This suggests the definition:

FTOP(Y∗,Z∗)p = {g ∈ Top(Y,Z) | g(Yq) ⊆ Zp+q for all q ≥ 0}.

This gives a filtration on the topological space TOP(Y,Z) and so defines the filtered space FTOP(Y∗,Z∗)p. The exponential law in the category Top now gives the exponential law ∼ e : FTop(X∗ ⊗ Y∗,Z∗) = FTop(X∗, FTOP(Y∗,Z∗)), from which one can deduce the exponential law ∼ e : FTOP(X∗ ⊗ Y∗,Z∗) = FTOP(X∗, FTOP(Y∗,Z∗)). either using the general result in the Appendix B or directly as is left as an exercise. An advantage of having this internal hom for filtered spaces is that we can apply our fundamental crossed complex functor Π to it. To say more on this, we first discuss the notion of homotopy in FTop.

The convenient definition of homotopy H : f0 ' f1 : Y∗ → Z∗ of maps f0, f1 of filtered spaces is that H is a map I × Y → Z which is a homotopy f0 ' f1 such that H(I × Yq) ⊆ Zq+1 for all q ≥ 0. This last condition is equivalent to H being a filtered map I∗ ⊗ Y∗ → Z∗. Equivalently, we can regard H also as a map

I∗ → FTOP(Y∗,Z∗), or Y∗ → FTOP(I∗,Z∗), although the latter interpretation involves the twisting map I∗ ⊗ Y∗ → Y∗ ⊗ I∗. It is also possible to consider ‘higher ﬁltered homotopies’ as ﬁltered maps

n E∗ ⊗ Y∗ → Z∗ or equivalently as maps n E∗ → FTOP(Y∗,Z∗). This will ﬁt with results on crossed complexes.

8.6 Tensor products and the fundamental crossed complex

In order to obtain the homotopy classiﬁcation Theorem 9.2.14, we need to use tensor products and homotopies of crossed complexes and its relation to homotopies of ﬁltered maps. 246

We have deﬁned the notion of homotopies for maps of ﬁltered spaces. They give 1-homotopies between the induced morphisms of fundamental crossed complexes. Again, it is possible to prove this directly, but it follows more elegantly from later more general results. In particular,

Theorem 8.6.1 If X∗ and Y∗ are ﬁltered spaces, then there is a natural morphism

θ :ΠX∗ ⊗ ΠY∗ → Π(X∗ ⊗ Y∗) such that:

i) θ is associative;

ii) if ∗ denotes a singleton space or crossed complex, then the following diagrams are commutative

∼ ∼ ΠX o = (ΠX ) ⊗ ∗ ∗ ⊗ ΠX = / ΠX ∗ KK ∗ ∗ s ∗ KKK sss KK θ θ ss ∼ KK ss ∼ = K% sy s = Π(X∗ ⊗ ∗) Π(∗ ⊗ X∗)

iii) θ is commutative in the sense that if Tc : C ⊗ D → D ⊗ C is the natural isomorphism of crossed complexes described in Theorem 8.2.17, and Tt : X∗ ⊗ Y∗ → Y∗ ⊗ X∗ is the twisting map, then the following diagram is commutative θ / ΠX∗ ⊗ ΠY∗ Π(X∗ ⊗ Y∗)

Tc Π(Tt) θ / ΠY∗ ⊗ ΠX∗ Π(Y∗ ⊗ X∗);

iv) if X∗,Y∗ are the skeletal ﬁltrations of CW -complexes, then θ is an isomorphism.

The proof is deferred to Chapter 16 where we can use the techniques of ω-groupoids. Note that the construction of the natural transformation θ could in principle be proved directly, but this would be technically diﬃcult because of the complications of the relations for the tensor product of crossed complexes. In fact θ is an isomorphism under more general conditions (see the result by Baues and Brown in [11]). In a similar spirit, let us now prove that the functor Π : FTop → Crs is a homotopy functor.

Proposition 8.6.2 There is a natural morphism of crossed complexes

ψ : Π(FTOP(X∗,Y∗)) → CRS(ΠX∗, ΠY∗) which is Π in dimension 0.

Proof It is suﬃcient to construct the morphism ψˆ as the composition in the following commutative diagram

ψˆ Π(FTOP(X ,Y )) ⊗ ΠX / ΠY ∗ ∗ VV ∗ mm6 ∗ VVVV mmm VVVV mmm VVVV mmm θ VVV* mmm Πe Π(FTOP(X∗,Y∗) ⊗ X∗) 247

where e : FTOP(X∗,Y∗)⊗X∗ → Y∗ is the evaluation morphism, i.e. the adjoint to the identity on FTOP(X∗,Y∗). 2

Corollary 8.6.3 In particular, a homotopy F : f0 ' f1 : X∗ → Y∗ in FTop induces a (left) homotopy ΠF : Πf0 ' Πf1 :ΠX∗ → ΠY∗ in Crs.

Proof This is an immediate consequence of the information given by ψ in dimension 1. 2

Similar statements hold for right homotopies of crossed complexes. A right homotopy C → D is a morphism C ⊗ I → D, or, equivalently, a morphism C → CRS(I,D). We may also deﬁne a right homotopy in FTop to be a map Y∗ ⊗ I∗ → Z∗. By Theorem 8.6.1, such a map gives rise to a right homotopy ΠY∗ ⊗ I → ΠZ∗.

8.7 The homotopy addition lemma for a simplex.

In this section we describe explicitly and algebraically a free basis and the boundary for Π∆n where ∆n is the topological n-simplex with its standard ﬁltration by dimension. This formula is called the homotopy addition lemma for a simplex. We call Π∆n the ‘n-simplex crossed complex’, and its description is used in many places later (see in particular Section 9.1). It is a feature of our exposition using crossed complexes that the homotopy addition lemma can be seen as an algebraic fact which models accurately the geometry. That happens because crossed complexes model well the geometry.

Remark 8.7.1 First it is useful to write out all the rules for the cylinder Cyl (C) = I ⊗ C, as a reference. Let C be a crossed complex. We apply the relations in the definition of tensor product of crossed complexes (Definition 8.2.13) to this case. −1 For all n > 0 and c ∈ Cn, I ⊗ C is generated by elements 0 ⊗ c, 1 ⊗ c of dimension n and ι ⊗ c, ι ⊗ c of dimension (n + 1) with the following defining relations for a = 0, 1, ι: Source and target

t(a ⊗ c) = ta ⊗ tc for all a ∈ I, c ∈ C s(a ⊗ c) = a ⊗ sc if a = 0, 1, n = 1 , s(a ⊗ c) = sa ⊗ c if a = ι, ι−1, n = 0 .

Relations with operations c0 ta⊗c0 0 a ⊗ c = (a ⊗ c) if n > 2, c ∈ C1.

Relations with additions ( 0 −1 0 a ⊗ c + a ⊗ c , if a = 0, 1, n > 1 or if a = ι, ι , n > 2, a ⊗ (c + c ) = 0 (a ⊗ c)ta⊗c + a ⊗ c0, if a = ι, ι−1, n = 1 ( −1 −(ι ⊗ c) if n = 0, (ι ) ⊗ c = −1 −(ι ⊗ c)(ι )⊗tc if n > 1. 248

Boundaries   a⊗tc −1 −(a ⊗ δc) − (ta ⊗ c) + (sa ⊗ c) if a = ι, ι , n > 2, δ(a ⊗ c) = −ta ⊗ c − a ⊗ sc + sa ⊗ c + a ⊗ tc if a = ι, ι−1, n = 1,  a ⊗ δc if a = 0, 1, n > 2.

These rules simplify if instead of the cylinder, we analyse the cone.

Deﬁnition 8.7.2 Let C be a crossed complex. The cone Cone(C) is deﬁned by

Cone (C) = (I ⊗ C)/({1} ⊗ C) , which can alternatively be seen as a pushout

{1} ⊗ C / {v}

I ⊗ C / Cone (C).

We call v the vertex of the cone.

Proposition 8.7.3 So Cone (C) is generated by elements 0 ⊗ c, ι ⊗ c of dimensions n, n + 1 respectively, and v of dimension 0 with the rules

Source and target ( 0 ⊗ tc, if a = 0, t(a ⊗ c) = v otherwise.

Relations with operations c0 0 a ⊗ c = a ⊗ c if n > 2, c ∈ C1.

Relations with additions

a ⊗ (c + c0) = a ⊗ c + a ⊗ c0. and ( −1 −(ι ⊗ c) if n = 0, (ι ) ⊗ c = −1 −(ι ⊗ c)(ι )⊗tc if n > 1.

Boundaries ( −(ι ⊗ δc) + (0 ⊗ c)ι⊗tc if n > 2, δ(ι ⊗ c) = −ι ⊗ sc + 0 ⊗ c + ι ⊗ tc if n = 1, δ(0 ⊗ c) = 0 ⊗ δc if n > 2.

The simplicity of the rules for operations and additions is one of the advantages of the form of our deﬁnition of the cone, in which the end at 1 is shrunk to a point.

We use the above to work out the fundamental crossed complex of the simplex ∆n in an algebraic fashion. We regard ∆n topologically as the topological cone

Cone (∆n−1) = (I × ∆n−1)/({1} × ∆n−1). 249

1 n The vertices of ∆ = I are ordered as 0 < 1. Inductively, we get vertices v0, . . . , vn of ∆ with vn = v being the last introduced in the cone construction, the other vertices vi being (0, vi). The fact that our algebraic formula corresponds to the topological one follows from facts stated earlier on the tensor product and on the GvKT stated in the next section. We now deﬁne inductively top dimensional generators of the crossed complex Π∆n by, in the cone complex:

σ0 = v, σ1 = ι, σn = (ι ⊗ σn−1), n ≥ 2 with σ0 being the vertex of Π∆0. Next we need conventions for the faces of σn.

We deﬁne inductively ( n−1 n ι ⊗ ∂iσ if i < n, δiσ = 0 ⊗ σn−1 if i = n.

Theorem 8.7.4 (Homotopy Addition Lemma) The following formulae hold, where un = ι ⊗ vn−1:

2 2 2 2 (8.7.1) δ2σ = −∂1σ + ∂2σ + ∂0σ ,

3 3 u3 3 3 3 (8.7.2) δ3σ = (∂3σ ) − ∂0σ − ∂2σ + ∂1σ , while for n ≥ 4

nX−1 n n un n−i n (8.7.3) δnσ = (∂nσ ) + (−1) ∂iσ . i=0

Proof For the case n = 2 we have

2 δ2σ = δ2((ι ⊗ ι)) = −ι ⊗ 0 + 0 ⊗ ι + ι ⊗ 1 2 2 2 = −∂1σ + ∂2σ + ∂0σ . For n = 3 we have:

3 2 δ3σ = δ3(ι ⊗ σ )

2 ι⊗v2 2 = (0 ⊗ σ ) − ι ⊗ δ2σ

2 u3 2 2 2 = (0 ⊗ σ ) − ι ⊗ (−∂1σ + ∂2σ + ∂0σ )

3 u3 3 3 3 = (∂3σ ) − ∂0σ − ∂2σ + ∂1σ . We leave the general case to the reader. The key points that make it easy are the rules on operations and additions of Proposition 8.7.3. 2

Remark 8.7.5 (i) Notice the formula of δ2 gives a groupoid formula, and the one of δ3 gives a formula in a crossed module which is nonabelian. (ii) There are many possible conventions for the Homotopy Addition Lemma, and that given here is unusual. However, our formula follows naturally from the geometry of the cone and our algebra for the tensor product.

(iii) It is a good exercise to prove that δ2δ3 = 0. It is not so easy to prove directly from the formula that δ3δ4 = 0, and a direct proof (given for example by G.W.Whitehead in his book [171]) does use the second law for a crossed module. Of course we know these composites are 0 since we are working in the category of crossed complexes. 250

The representation Cone (∆n−1) = ∆n gives a cellular contracting homotopy of ∆n−1, and so Π∆n is a contractible crossed complex. We shall use this fact later. We can now state the formula in terms of free generators and boundaries for the whole crossed complex Π∆n. It has a free generator σn in dimension n and also free generators ασn in dimension m for all 0 ≤ m < n and all increasing functions α :[m] → [n]. The boundary of such a ασn is given by the simplex homotopy addition lemma in dimension m.

We can also obtain a cubical homotopy addition lemma using the cube crossed complex ΠIn = I⊗n. In this n α α n crossed complex, let c = ι ⊗ · · · ⊗ ι be the n-fold tensor product of ι with itself, and for α = 0, 1 let ci = ∂i c be the element of dimension (n − 1) obtained by replacing in cn the ι in the ith place by α. The formulae for the boundary in the tensor product, using I⊗n = I ⊗ I⊗(n−1), then yield by induction:

Proposition 8.7.6 [Cubical homotopy addition lemma]  −c1 − c0 + c0 + c1 if n = 2,  1 2 1 2   n δ c = −c1 − (c0)u2c − c1 + (c0)u3c + c1 + (c0)u1c if n = 3, n  3 2 1 3 2 1   P  n+1 i 1 0 uic i=1 (−1) {ci − (ci ) } if n ≥ 4,

n 1 1 1 (where c = c and ui = ∂1 ∂2 ···ˆı ··· ∂n+1).

It should be said that this suggested ‘proof’ is not quite fair, since we are using a lot of results on crossed complexes the proofs of some of which rely on the cubical homotopy addition lemma established independently. However, this calculation shows how the results tie in, and that once we have these results established they give powerful means of calculation, some of which are inherently nonabelian, and which usually involve module operations not so easily handled by traditional methods. Chapter 9

The simplicial classifying space

In the preceding Chapters of this part II, we have being studying the category Crs of crossed complexes, a crossed complex C being like a chain complex of modules with a groupoid G as operators, but with non-Abelian features in dimensions 1 and 2, in the sense that the part C2 → C1 is a crossed module with cokernel G. In the Chapter 7 we have seen that the basic example of a crossed complex is the fundamental crossed complex πX∗ of a ﬁltered space X∗. Now we deﬁne a classifying space functor

B : Crs → Top to the category of spaces that is the “homotopy left adjoint of Π”, i. e. such that there is a natural bijection of homotopy classes

[ΠX∗,C] → [X,BC] for any CW -complex with skeletal filtration X∗. This result is Theorem 9.2.14. In doing so we shall use some techniques and results from algebraic topology, so this Chapter requires a deeper background that most of the other Chapters in the book. It has three well defined parts. First, there is a recapitulation of the simplicial methods we are going to use. Then, we use them to define the classifying space and prove Theorem 9.2.14. The third part sees some applications of this construction. The simplicial methods are described in the first Section, giving Definitions and stating results that are used later. References giving all the details are [71, 88, 93, 143]. The most important results include the closed structure of the category Simp of simplicial sets and maps and the existence of a pair of adjoint functors from Simp to the category of CW -complexes, i.e. the singular chains functor and the realization functor. Using these and the monoidal closed structure on the category Crs of crossed complexes, with tensor product − ⊗ − and internal hom Crs(−, −), described in Chapter 8, we can define the classifying space using prove the main result of this Chapter: the classification Theorem 9.2.14. Section 9.2.3 deals with the modifications needed for the pointed case. Some consequences of Theorem 9.2.14 are drawn in Section 9.3. In particular if X is a reduced CW -complex which is n-coconnected (i.e. if πi(X) = 0 for 1 < i < n), then BπX∗ models the n-type of X, where X∗ is the skeletal filtration. In the case n = 2, this in effect recovers the result of [134] that crossed modules are algebraic models of 2-types (called“3-types” in [134]). In view of the GvKT for the fundamental crossed complex proved in Chapter 7, this result also allows for the computation of the n-types of certain colimits.

251 252

A further advantage of the classifying space is that if p : E → D is a fibration of crossed complexes then the induced map Bp : BE → BD is a (Serre) fibration of spaces. This result is proved in Section 9.4 and it is convenient for relating BC with Postnikov invariants. It should be noted that we define the classifying space using simplicial rather than cubical methods. Cubical sets are more convenient than simplicial sets for dealing with homotopies and higher homotopies, and in Chapter 16 we use the cubical category of ω-groupoids in constructing the internal hom functor and tensor product on the category Crs. It is also used in constructing η, as explained before. However the simplicial theory fits much better with the algebraic topology literature.

9.1 Simplicial sets and the nerve of a crossed complex

Let us begin by a short remainder of results on simplicial sets. As stated in the Introduction, our goals are two results: the closed structure of the category Simp of simplicial sets and maps (see Subsection ??) and the realization functor studied in Subsection ??.

9.1.1 Simplicial sets

Deﬁnition 9.1.1 We deﬁne the category of natural numbers and monotonous maps ∆ having as objects n = {0, 1, ··· , n} for all n ∈ N and morphisms ∆(m, n) the monotonous maps φ : m → n for all n, m ∈ N.

Remark 9.1.2 Notice that the monotonous maps ∆(m, n) are generated by two families, the injective maps

δi : n − 1 → n deﬁned by δi(j) = j whenever j < i and δi(j) = j + 1 if j ≥ i, and the surjections

σi : n + 1 → n deﬁned by σi(j) = j whenever j ≤ i and σi(j) = j − 1 if j > i. Moreover, all relations among the elements of these families are generated by

δjδi = δiδj−1 i < j

σjσi = σiσj+1 i ≤ j   δjσj−1 i < j σ δ = 1 i = j, j + 1 j i   δi−1σj i > j + 1

Deﬁnition 9.1.3 Let C be a category. A simplicial C-object is a functor K : ∆op → C where ∆op is the opposite category to ∆ Similarly, a simplicial C-morphism between simplicial C-objects is a natural transformation Φ: K → L.

This produces the category C-Simp of simplicial C-objects and C-morphisms. In particular, we are interested in Simp, the category whose objects are simplicial sets and whose morphisms are the appropriate natural transformations. 253

Remark 9.1.4 A simplicial C object is given by a sequence {Kn = K(n)}n∈N of objects in C, and two families of morphisms in C, the faces

∂i = K(δi): Kn → Kn−1 deﬁned for i = 0, 1, ··· , n and n ∈ N and the degeneracies

si = K(σi): Kn → Kn+1 deﬁned for i = 0, 1, ··· , n and n ∈ N. Moreover, all relations among them are generated by

∂i∂j =∂j−1∂i i < j

sisj =sj+1si i ≤ j   sj−1∂i i < j ∂ s = 1 i = j, j + 1 i j   sj∂i−1 i > j + 1

Example 9.1.5 Let us see the standard example of simplicial set: the m-simplex ∆[m]. It is the free simplicial set with one generator of dimension m and it has objects ∆[m](n) = ∆(m, n) and morphisms given by composition. Notice that the non-degenerate n-simplices of this simplicial set correspond to the n-dimensional faces of the standard m-simplex. In particular, it has only one non degenerate m-simplex ιm corresponding to the identity map.

The simplicial set ∆[m] is free with one generator of dimension m in the sense that a map from ∆[m] to a simplicial set K is given by the element of Kn corresponding to ιm. As a consequence we get the following important property.

Proposition 9.1.6 There is a natural bijection for all simplicial sets K between the set of n-simplices Kn and the simplicial maps from the n-simplex standard ∼ Kn = Simp(∆[n],K).

Proof To deﬁne the map from right to left to any simplicial map f : ∆[n] → K corresponds the element f = f(ιn) ∈ Kn.

The map from right to left is given by mapping an element x ∈ Kn to the unique simplicial map x : ∆[n] → K satisfying x(ιn) = x ∈ Kn. 2

Example 9.1.7 Let us see some more examples of simplicial sets that are sub simplicial sets of ∆[m]. i.- The boundary of the m-simplex ∆[˙ m] is the sub simplicial set of ∆[m] generated by all faces of ιm.

In this case, the top dimensional non degenerate simplices are the (m − 1)-dimensional faces of ιm:

{∂iσm}i=0,··· ,m. ii.- Another sub simplicial sets of ∆[m] of relevance are the i-th m-dimensional horns Λi[m] for i = 0, ··· , m and m ∈ N. They are generated by all faces of ιm but ∂i(ιm). 254

The top dimensional non degenerate simplices of Λi[m] are the (m − 1)-dimensional simplices

{∂0ιm, ··· , ∂i−1ιm, ∂i+1ιm, ··· , ∂mιm}.

Another set of examples comes from the topological spaces.

Deﬁnition 9.1.8 For any topological space X, its singular simplicial set S(X) is given by all singular simplices, i.e. n Sn(X) = {σ : ∆ → X | σ a continuous map} faces and degeneracies are given by composition with the appropriate maps

n−1 n δi : ∆ → ∆ deﬁned as δi(vj) = vj if j ≤ i and δi(vj) = vj−1 whenever j > i and

n n+1 σi : ∆ → ∆ deﬁned as σi(vj) = vj if j < i and σi(vj) = vj+1 whenever j ≥ i. This gives a functor S : Top → Simp.

As is immediately apparent, this deﬁnition is closely related to the deﬁnition of singular homology. This relation is made more explicit via the homology groups of a simplicial set.

Deﬁnition 9.1.9 For any simplicial set K, we can associate a chain complex C(K) where Cn(K) is the abelian group generated for Kn and the boundary

dn : Cn(K) → Cn−1(K) is the linear map deﬁned on generators by

Xn i dn(σ) = (−1) ∂i(σ). i=0

The homology groups of the simplicial set K are deﬁned

H∗(K) = H∗(C(K)).

They give functors

Hn : Simp → Groups for all n ∈ N.

Proposition 9.1.10 The homology groups of the singular complex associated to a topological space are its singular homology groups. 255

9.1.2 Homotopy of simplicial sets

0 0 Deﬁnition 9.1.11 Two n-simplices of a simplicial set x, x ∈ Kn are said to be homotopic (x ∼ x ) if there 0 0 is an (n + 1)-simplex y ∈ Kn+1 such that the two last faces are x and x , i.e. ∂n(y) = x, ∂n+1y = x , and all others are degeneracies. This means 0 ∂iy = sn−1∂ix = sn−1∂ix for all i = 0, ··· , n. In particular, in order to be homotopic two n-simplices must have the same faces.

Proposition 9.1.12 The relation to be homotopic is reﬂexive.

There is no way to prove neither symmetry nor transitivity. We need to assume some extra condition on the simplicial set K.

Deﬁnition 9.1.13 A simplicial set K is said to satisfy the Kan extension condition (or, more simply, to be a Kan simplicial set if it has the extension property from any i-th horn to an n-simplex.

Equivalently, for any collection of (n+1) n-simplices x0, ··· , xi−1, xi+1, ··· , xn+1 satisfying the compatibility condition ∂jxk = ∂kxj for j < k and j, k 6= i, there is an (n + 1)-simplex y ∈ Kn+1 such that ∂jy = xj for all j 6= i.

Example 9.1.14 For any topological space X, the singular simplicial set S(X) is Kan.

Proposition 9.1.15 In a Kan simplicial set, the relation to be homotopic is an equivalence relation.

Deﬁnition 9.1.16 Let K be a Kan simplicial set and choose a vertex ν ∈ K0. We denote by ν also the subcomplex generated by ν and consider the simplices that have ν as boundary, i.e.

K˜n = {x ∈ Kn | ∂ix = ν, 0 ≤ i ≤ n}.

We define the n-homotopy groups of K as the quotient of Kñ with respect to the homotopy relation just defined, i.e. K˜ π (K, ν) = n . n ∼ The product is defined using the Kan extension condition.

Proposition 9.1.17 The homotopy groups πn(K, ν) are groups, abelian whenever n ≥ 2.

Everything can be repeated relative to a sub simplicial set.

0 Definition 9.1.18 Let K be a simplicial set and a sub simplicial set L ⊆ K. Two n-simplices x, x ∈ Kn are 0 homotopic relative to L (x ∼ x rel L) if there is an (n + 1)-simplex y ∈ Kn+1 such that the two last faces are 0 0 0 x and x , i.e. ∂n(y) = x, ∂n+1y = x , the first one is a homotopy in L, ∂0y : ∂0x ∼ ∂0x , and all others are degeneracies. In particular, in order to be homotopic two n-simplices must have the same faces except perhaps the first one. 256

Proposition 9.1.19 If both K and L ⊆ K are Kan simplicial sets, the relation to be homotopic relative to L is an equivalence relation.

Definition 9.1.20 If both K and L ⊆ K are Kan simplicial sets and ν ∈ K0 is a vertex. We consider the simplices that all faces but the first one are in ν as boundary and the first one is in L, i.e.

K˜ (L)n = {x ∈ Kn | ∂ix = ν, 1 ≤ i ≤ n, ∂0x ∈ Ln−1}.

We define the n-homotopy groups of (K,L) as the quotient of K˜ (L)n with respect to the homotopy relation just defined, i.e. K˜ (L) π (K, L, ν) = n . n ∼ The product is defined using the Kan extension condition.

Proposition 9.1.21 The homotopy groups πn(K, L, ν) are groups, abelian whenever n ≥ 3.

Theorem 9.1.22 The homotopy groups of a pair of Kan simplicial sets πn(K, L, ν) are functors to the category of groups (deﬁned if general for n ≥ 2 and also for n = 1 when L = ∅) that are abelian if n ≥ 3 and also for n = 2 when L = ∅. There is also a natural transformation of functors giving a natural long exact sequence

∂ · · · → πn(L, ν) → πn(K, ν) → πn(K, L, ν) −→ πn−1(L, ν) → · · · π1(L, ν) → π1(K, ν) → π1(K, L, ν).

The homotopy groups of a simplicial set may also be deﬁned via homotopy relation among the appropriate maps.

Deﬁnition 9.1.23 Let us consider a pair of simplicial maps f, g ∈ Simp(K,L). A homotopy from f to g (h : f ∼ g) is given by maps

hi : Kq → Lq+1, for 0 ≤ i ≤ q and q ∈ N satisfying

∂0h0 =f

∂ihj =hj−1∂i i < j

∂jhj =∂j+1hj

∂ihj =hj∂i−1 i > j + 1

∂q+1hq =g and

sihj =hj+1si i ≤ j

sihj =hjsi−1 i > j the homotopy is relative to a sub simplicial set K0 ⊆ K if all maps are constant on K0.

Proposition 9.1.24 The relation being homotopic is reﬂexive.

Proposition 9.1.25 The relation being homotopic is preserved by composition.

Proposition 9.1.26 Two homotopic maps induce the same morphism on homology. 257

Proposition 9.1.27 If L is a Kan complex, to be homotopic is an equivalence relation.

Proposition 9.1.28 Two n-simplices in a simplicial set K are homotopic if the associated simplicial maps of Proposition 9.1.6 x, y : ∆[n] → K are homotopic.

Remark 9.1.29 To interpret the homotopy groups of a simplicial set in terms of homotopy of maps, it remains to see which kind of simplicial maps correspond to the simplices in K˜ and K˜ (L).

Proposition 9.1.30 Elements of K˜n correspond via Proposition 9.1.6 to simplicial maps of pairs (∆[n], ∆[˙ n]) → (K, ν) and a homotopy of simplices of K˜n rel ν, correspond to homotopy of pairs of simplicial maps rel ∆[˙ n]).

0 Proposition 9.1.31 Elements of K˜ (L)n correspond via Proposition 9.1.6 to simplicial maps of triples (∆[n], ∆[˙ n], Λ [n]) → (K, L, ν) and a homotopy of simplices of K˜ (L)n rel ν, correspond to homotopy of triples of simplicial maps rel (∆[˙ n]), Λ0[n]).

weak equivalences

9.1.3 Realisation of simplicial sets.

We are going to deﬁne a left adjoint of the singular simplicial set functor

S : Top → Simp.

Deﬁnition 9.1.32 For any simplicial set K : ∆op → C, its realization |K| is the quotient space F K × ∆n |K| = n ≡ where the equivalence relation is generated by (∂ix, u) ≡ (x, δiu) and (six, u) ≡ (x, σiu) where x ∈ Kn and u ∈ ∆n.

Remark 9.1.33 The realization of a simplicial set |K| can also be interpreted either as a coend. Z n |K| = Kn × ∆ n or as a colimit ([93])

Proposition 9.1.34 The realization of a simplicial set is a CW-complex having an n-cell for each nondegenerate n-simplex.

Remark 9.1.35 Thus each point of the realization of a simplicial set |K| is an equivalence class | x, u | with n x ∈ Kn and u ∈ ∆ and it has a unique representative where x is a non-degenerate simplex.

We proceed to prove that the realization functor | | is left adjoint to the singular simplicial set S.

Deﬁnition 9.1.36 For each simplicial set K and topological space X, we deﬁne the map

Ψ: Top(|K|,X) → Simp(K,S(X)) 258 such that for any continuous map g : |K| → X, the simplicial map

Ψ(g): K → S(X) is given in dimension n by

Ψn(g)(x)(u) = g(| x, u |),

n for any n-simplex x ∈ Kn and point u ∈ ∆ .

Proposition 9.1.37 The realization functor | | is adjoint to the singular simplicial set S

Proof The map Ψ deﬁnes a natural transformation, we need to prove that it is bijective. The inverse of Ψ is given by sending a simplicial map f to the continuous map deﬁned by mapping any class

| xn, u | to fn(xn)(u). 2

Deﬁnition 9.1.38 Both maps produce natural units

−1 Ψ (1X ):| S(X) |→ X given by | x, u | 7→ x(u) corresponding to the identity map 1 : S(X) → S(X) and

Ψ(1K ): K → S(|K|) given by Ψ(1)(x): u 7→| x, u | corresponding to the identity map 1 : |K| → |K|.

Let us state the main homological and homotopical properties of these maps.

−1 Theorem 9.1.39 Let us consider the unit maps Ψ(1K ) and Ψ (1X ). They satisfy

1. Both unit maps induce homology isomorphisms.

2. If K is a Kan complex, Ψ(1K ) induces isomorphism of the fundamental group.

−1 3. For any X, Ψ (1X ) is a weak homotopy equivalence.

The realisation of a simplicial Kan ﬁbration is a Serre ﬁbration [157], and in fact has the covering homotopy property with respect to maps of all compactly generated spaces [163]

9.1.4 Simplicial sets as a closed category

The object of this subsection is to give a structure of cartesian closed category to Simp the category of simplicial sets and morphisms. We consider ﬁrst the product in the category of simplicial sets.

Deﬁnition 9.1.40 Let K,L be simplicial sets. Their product K × L is the simplicial set deﬁned on objects

(K × L)n = Kn × Ln and the obvious action on faces and degeneracies. 259

Proposition 9.1.41 It is clearly a simplicial set that is the product of K and L in Simp

Now we deﬁne the internal morphism structure for Simp.

Deﬁnition 9.1.42 Let K,L be simplicial sets. The simplicial set SIMP(K,L) is the one deﬁned on objects as

SIMP(K,L)n = Simp(K × ∆[n],L) and on faces and degeneracies is given by the obvious composition.

It gives Simp a structure of cartesian closed category. (Simplicial n-homotopies and SIMP(K,L)n)

Theorem 9.1.43 They are adjoint to each other, i.e. there is a natural equivalence

Simp(K, SIMP(L, M)) ≡ Simp(K × L, M).

Proof To a simplicial map f : K × L → M corresponds the simplicial map fˆ : K → SIMP(L, M) ˆ where for every x ∈ Kn, the map f(x) ∈ Simp(L × ∆[n],M) is the composite

f L × ∆[n] 1L−→×x L × K −→ M where x is the map corresponding to x seen in Proposition 9.1.6. Thus fˆ(x) is the simplicial map that sends

(y, ιn) to f(x, y). Its inverse is easier to describe. To any simplicial map

g : K → SIMP(L, M) we associate gˆ : K × L → M deﬁned byg ˆn(x, y) = g(x)(y, ιn). 2

Remark 9.1.44 As always when the product is associative, this natural equivalence in Sets extends to one in Simp SIMP(K, SIMP(L, M)) ≡ SIMP(K × L, M).

Now, we study the behaviour of the realisation functor with respect to the product and internal morphisms. First the product

Proposition 9.1.45 Realization preserves product (but no by means as a cellular map)

And now, using the cartesian closed structure, the internal morphisms.

Proposition 9.1.46 There is a weak equivalence

|SIMP(L, M)| → TOP(|L|, |M|) natural for all simplicial sets L, M with M Kan. 260

Proof This statement follows from a more general result in a pattern that we are going to use later, for instance in Proposition 9.2.8 and Corollary 9.2.9. For any simplicial set K there is a natural bijection.

Top(|K|, |SIMP(L, M)|) =∼ Simp(K, SIMP(L, M)) =∼ Simp(K × L, M) by adjointness =∼ Top(|K × L|, |M|) since M is Kan =∼ Top(|K| × |L|, |M|) by the previous property =∼ Top(|K|, TOP(|L|, |M|) by adjointness.

This natural bijection is given by composition with a map φ : |SIMP(L, M)| → TOP(|L|, |M|). By choosing the appropriate |K|, it is easy to see that φ induces isomorphisms on all homotopy groups. Moreover, it is a homotopy equivalence when |L| is a ﬁnite CW-complex. 2

9.1.5 Simplicial sets and ﬁltered topological spaces

The category FTop is a natural home for the realisation functor | | from the category of simplicial sets to Top. That is, any simplicial set K has a ﬁltration by skeleta, and this induces the skeletal ﬁltration of the realisation

For a filtered space X∗ there is a filtered singular complex RX∗ which in dimension n consists of the filtered n maps ∆∗ → X∗. This RX∗ has the structure of simplicial set, and it is easily verified that if Simp is the category of simplicial sets, then R : FTop → Simp is right adjoint to

| | : Simp → FTop.

9.2 The classifying space of a crossed complex

With the preliminaries on simplicial sets dealt with, we may deﬁne the classifying space of a crossed complex and its main properties.

9.2.1 The nerve of a crossed complex

Deﬁnition 9.2.1 The fundamental crossed complex of ﬁltered spaces functor Π : FTop → Crs of Chapter 7 composes with the realisation functor | | : Simp → FTop given before to give a functor, also written Π: Simp → Crs called the fundamental crossed complex of a simplicial set (Π(K) = Π(|K|)). 261

Proposition 9.2.2 The fundamental crossed complex ΠK of a simplicial set K can also be given as a coend Z n ΠK = Kn × Π∆ . n

Proof This follows from the van Kampen Theorem for the fundamental crossed complex of a CW -complex covered by a family of subcomplexes (Corollary 7.4.5). 2

Proposition 9.2.3 Let K and L be simplicial sets. Then there is a natural homotopy equivalence of crossed complexes ΠK ⊗ ΠL → Π(K × L) extending the inclusions ΠK ⊗ L0 → Π(K × L), K0 ⊗ ΠL → Π(K × L).

Proof This is proved in the Section 10.4 on acyclic models. 2

Remark 9.2.4 This last result is one place where cubical methods are more eﬃcient, since for cubical sets K and L we have a cellular isomorphism |K ⊗ L| =∼ |K| × |L|.

Deﬁnition 9.2.5 The nerve of a crossed complex functor N : Crs → Simp is deﬁned by

n (NC)n = Crs(Π∆∗ ,C)

n where ∆∗ , is the standard n-simplex with its standard cell structure and cellular ﬁltration, and NC has the n simplicial structure induced by the usual maps of ∆∗ , n ≥ 0.

n n The crossed complex Π∆∗ is a free crossed complex on the cells of ∆ , and the boundaries are determined by the universal example, which is itself given by the Homotopy Addition Lemma 8.7.4. n For n ≥ 2, the crossed complex Π∆∗ involves nonabelian groups in dimension 2, and a groupoid in dimension 1 which acts on the crossed complex. The Homotopy Addition Lemma (which says, intuitively, that the boundary of a simplex is the sum of its faces) therefore needs to be stated with care. n A morphism f :Π∆∗ → C for a crossed complex C can now be described as given by a family of elements in varying dimensions of C, indexed by the cells of ∆n, and related by the Homotopy Addition Lemma. Blakers in [15] uses such families to construct for a reduced crossed complex C (there called a group system) a simplicial set associated to C which is essentially our nerve of C, and this construction is also used in [6]. We can prove a result on adjointness that comes easily from standard manipulations with ends and coends.

Theorem 9.2.6 The functor Π: Simp → Crs is left adjoint to the nerve functor N : Crs → Simp.

Proof Let K be a simplicial set and let C be a crossed complex. Then we have natural bijections Z ∼ n Simp(K,NC) = Sets(Kn, Crs(Π∆∗ ,C)) Zn ∼ n n = Crs(Kn × Π∆∗ ,C) treating Kn × Π∆∗ as a crossed complex n Z ∼ n = Crs( (Kn × Π∆∗ ),C) because Crs commutes with coends n =∼ Crs(ΠK,C). 262 giving the adjointness. 2

Remark 9.2.7 The fact that the functor Π : Simp → Crs is a left adjoint implies that it preserves all colimits. However, the generalised van Kampen Theorem of Chapter 7 is not an immediate consequence of this fact since it is a theorem about the functor Π from ﬁltered spaces to crossed complexes, and one of the conditions for λ ∼ λ λ Π(colimλU∗ ) = colimλΠU∗ is that each ﬁltered space U∗ should be connected. It would be interesting to know whether this generalised van Kampen Theorem can be deduced from the fact that Π : Simp → Crs preserves all colimits.

It is useful to know that NC is a Kan complex, so that homotopy of simplicial maps K → NC is an equivalence relation. Actually, NC has the structure of Kan complex in a strong way, i.e. any horn has a unique filler. n n Define an element f :Π∆∗ → C of (NC)n to be thin if f maps the top dimensional element of Π∆∗ to zero in C. The thin elements satisfy Dakin’s axioms: degenerate elements are thin; any horn has a unique thin filler; if all faces but one of a thin element are thin, then so also is the last face. Thus NC has the structure of a T-complex defined by Dakin in [72], and in fact N yields an equivalence between crossed complexes and simplicial T -complexes (proved by Ashley in [6], reproved in a general setting by Nan Tie in [148, 149]).

Proposition 9.2.8 For each simplicial sets K,L and crossed complex C, there is a natural equivalence

Simp(K, SIMP(L, NC)) =∼ Simp(K,N(CRS(ΠL, C))).

Proof Let K and L be simplicial sets. Then we have natural maps

Simp(K, SIMP(L, NC)) =∼ Simp(K × L, NC) because Simp is a cartesian closed category =∼ Crs(Π(K × L),C) since N, Π are adjoint (Theorem 9.2.6) À Crs(ΠK ⊗ ΠL, C) by Proposition 9.2.3 =∼ Crs(ΠK, CRS(ΠL, C)) because Crs is a monoidal closed category =∼ Simp(K,N(CRS(ΠL, C))) again by Theorem 9.2.6. thus illustrating the power of generalities, in particular such of closed categories. 2

Corollary 9.2.9 The adjunction between the nerve functor N : Crs → Simp and the fundamental crossed module of a simplicial set functor Π: Simp → Crs extends to a natural equivalence

SIMP(L, NC) À N(CRS(ΠL, C)).

Proof The proof is standard in representable functors, i.e. we only have to choose the appropriate K to get the maps in the Corollary. 2

Corollary 9.2.10 The nerve functor N : Crs → Simp is a homotopy functor. More precisely, a homotopy f0 ' f1 : C → D in Crs induces a homotopy Nf0 ' Nf1 in Simp. 263

Proof Now let f0 ' f1 : C → D. Because the tensor product of crossed complexes is symmetric (see Section 8.2.3), we may view such a 1-fold homotopy either as a left homotopy I ⊗ C → D or as a right homotopy C ⊗ I → D. Choosing the latter form, we obtain a morphism C → CRS(I,D) and hence simplicial maps

NC → N(CRS(I,D)) → SIMP(I∗,ND). This gives a homotopy (NC) × I∗ → ND with end maps Nf0, Nf1. 2

9.2.2 The classifying space of a crossed complex

Deﬁnition 9.2.11 We deﬁne the classifying space BC of the crossed complex C by

BC = |NC|, the geometric realisation of the nerve of C. This deﬁnes the classifying space functor

B : Crs → Top.

Note that a crossed complex C is filtered by its skeleta C[n] = skntrn(C), for n ≥ 0, where C[n] agrees with C in dimensions ≤ n, and is trivial in dimensions > n. Applying the classifying space of crossed complexes to this filtration, we get a filtered space BC∗. It is proven by Ashley in [6] (following the analogous cubical result in Chapter 15) that there is a natural isomorphism

∼ ΠBC∗ = C.

We now describe the homotopy groups of BC.

Proposition 9.2.12 For any crossed complex C, there are natural isomorphisms

∼ ∼ ∼ π0BC = π0C, π1(BC,C0) = π1C, πn(BC, p) = Hn(C, p), and the two latter isomorphisms preserve the actions.

Proof The homotopy groups πn(BC, x) are isomorphic to those deﬁned combinatorially in the Kan complex NC. Thus an element of πn(NC, x) is represented by an n-simplex f ∈ (NC)n all of whose faces are degeneracies of the base point x. n n Now, f is a morphism Π∆∗ → C. Letc ˆn be a top dimensional generator of Π∆∗ . Then f(ˆcn) = cn ∈ Cn say. The Homotopy Addition Lemma now yields that in C, δcn = 0.

The deﬁnition of πn(NC, x) also shows thatc ˆn represents the trivial element if and only if there is an ˆ ˆ ˆ element d ∈ (NC)n+1 such that ∂nd =c ˆn and ∂id is a degeneracy of the base point x for 0 ≤ i ≤ n. Again, the Homotopy Addition Lemma shows that. 2

This result shows that the spaces BC should be regarded as generalising the classical Eilenberg-Mac Lane spaces. If En(G) is a crossed complex which is trivial except in dimension n, where is the group G (abelian if n ≥ 2), then NEn(G) is isomorphic to the classical K(G, n). We can now show that the classifying space functor B : Crs → Top is also a homotopy functor. This will lead to the proof of Theorem 9.2.14. 264

Corollary 9.2.13 The classifying space functor B : Crs → Top is a homotopy functor. More precisely, a homotopy f0 ' f1 : C → D in Crs induces a homotopy Bf0 ' Bf1 : BC → BD in Top.

Proof By Corollary 9.2.10, we get a homotopy (NC)×I∗ → ND with end maps Nf0, Nf1, and so a homotopy Bf0 ' Bf1 : BC → BD. 2

We are now able to prove our main result, giving the weak homotopy type of the function space TOP(X,BC). In the case BC is an Eilenberg- Mac Lane space K(G, n) for Abelian G, this result is essentially a theorem of Thom [166] (see also results by Brown [20], and Hansen [101]). It also includes a result of Gottlieb [95] for the case BC is a space K(G, 1) for any group G (proved as Proposition 9.3.7).

Theorem 9.2.14 If X is a CW -complex, and C is a crossed complex, then there is a weak homotopy equivalence

η : B(CRS(ΠX∗,C)) → TOP(X,BC), and a bijection of sets of homotopy classes ∼ [ΠX∗,C] = [X,BC] which is natural with respect to morphisms of C and cellular maps of X.

Proof It is suﬃcient to assume that X is the realisation |L| of a simplicial set L. As we have seen in Corollary 9.2.9, there is a homotopy equivalence of simplicial sets

N(CRS(ΠL, C)) → SIMP(L, NC).

Also, we have seen in Proposition 9.1.46 that there is a weak equivalence

|SIMP(L, M)| → TOP(|L|, |M|) for any simplicial sets L and M. They add together to give the map η and to prove that it is a homotopy equivalence. From this it is easy to deduce the ﬁnal result since

π0TOP(X,BC) = [X,BC], and π0N(CRS(ΠX∗,C)) = [ΠX∗,C], gives the bijection of homotopy classes and the naturality follows from the fact that the homotopy class of η is natural. 2

9.2.3 The pointed case

We consider brieﬂy the notions of tensor product and homotopy in the category Crs∗ of pointed crossed complexes, before getting Theorem 9.2.16, a version for pointed spaces of Theorem 9.2.14.

The pointed crossed complex category Crs∗ has objects the crossed complexes having a distinguished element ∗ in dimension 0 and only morphisms preserving this basepoint are included.

In the pointed category Crs∗ we deﬁne an m-fold pointed left homotopy C → D to be an m-fold left homotopy (H, f) satisfying f(∗) = ∗ and h(∗) = 0∗ ∈ Dm. The collection of all these is a sub-crossed complex Crs∗(C,D) of Crs(C,D) with basepoint the zero morphism c 7→ 0∗. 265

A pointed bimorphism θ :(C,D) → E is a bimorphism satisfying ( θ(c, ∗) = 0 for c ∈ C, (9.2.1) ∗ θ(∗, d) = 0∗ for d ∈ D, and C ⊗∗ D is the pointed crossed complex generated by all c ⊗∗ d with deﬁning relations (8.2.13) and ( c ⊗ ∗ = 0 for c ∈ C, (9.2.2) ∗ ∗ ∗ ⊗∗ d = 0∗ for d ∈ D.

∼ ∼ The symmetry C ⊗ D = D ⊗ C preserves the relations (9.2.2) and so gives a symmetry C ⊗∗ D = D ⊗∗ C.

Theorem 9.2.15 The pointed tensor products and hom functors described above deﬁne a symmetric monoidal closed structure on the pointed category Crs∗. 2

We denote by [X,Y ]∗ the set of pointed homotopy classes of pointed maps X → Y of pointed spaces X, Y . Similarly, for pointed crossed complexes D, C, we denote by [D,C]∗ the set of pointed homotopy classes of pointed morphisms D → C. If C is a pointed crossed complex, then BC is naturally a pointed space.

Theorem 9.2.16 If X is a pointed CW -complex and C is a pointed crossed complex, there is a bijection of sets of pointed homotopy classes which is natural with respect to pointed morphisms of C and pointed, cellular maps of X, and which ﬁts into a commutative diagram

∼= [X,BC]∗ / [ΠX∗,C]∗ SSS kk SSS kkk SSS kkk SSS kkk SS) ukkk Hom(π1(X, ∗), π1(C, ∗)),

in which we identify π1(BC, ∗) with π1(C, ∗), π1(X, ∗) with π1(X∗, ∗).

Proof The proof of the existence of the isomorphism of sets of pointed homotopy classes is similar to the proof of Theorem 9.2.14, but using the pointed constructions ⊗∗ and CRS∗ described before. We omit further details.

The slanting maps are induced by the functor π1(−, ∗) and the given identiﬁcations. To prove commutativity, it is suﬃcient to assume that X = |L| for some Kan simplicial set L. One then has to check that maps transformed by the following arrows induce the same map of fundamental groups:

Top(|L|,BC) ←− Simp(L, NC) → Crs(ΠL, C).

But this is clear on checking the values of these maps on 1-dimensional elements. 2

9.3 Applications

The ﬁrst set of applications of Theorem 9.2.14 give conditions about when is possible to realise some homotopy n-types as BC for some crossed complex C. 266

Theorem 9.3.1 Let n ≥ 1, and let X be a reduced CW -complex with πiX = 0, 1 < i < n. (Notice that this condition is vacuous if n = 1, 2.) Then there is a crossed complex C with Ci = 0, for all i > n together with a map f : X → BC inducing an isomorphism of homotopy groups f∗ : πiX → πiBC for 1 ≤ i ≤ n.

Proof Let X∗ be the skeletal ﬁltration of X, let X0 = {x}, and let D = ΠX∗. We deﬁne C be the crossed complex such that   Di 0 ≤ i < n Ci = Coker ∂n+1 i = n  0 i > n Then there is a unique morphism g : D → C which is the identity in dimensions < n and is the quotient morphism in dimension n. Clearly, this morphism g induces an isomorphism of fundamental groupoids, and of homology groups Hi(D, x) → Hi(C, x) for 2 ≤ i ≤ n. By Theorem 9.2.16 there is a pointed morphism

f : X → BC whose homotopy class corresponds to g :ΠX∗ → C. Without loss of generality we may assume f is cellular. Then for all i ≥ 1, the following diagram is commutative, where Si = e0 ∪ ei is the i-sphere:

f i ∗ / i [S ,X]∗ [S ,BC]∗ OOO OOO∼= ∼= OOO OO' (Πf)∗ [ΠSi , ΠX ] / [ΠSi , ΠBC ] / [ΠSi ,C] ∗ ∗ ∗ ∗ ∗ ∗ ∼= ∗ ∗

∼= ∼= ∼= Hi(ΠX∗, x) / Hi(C, x) Hig

i i The assumptions on X imply that the map [S ,X]∗ → [ΠS∗, ΠX∗]∗ is bijective for 1 ≤ i ≤ n. So the result on Πi follows. 2

Remark 9.3.2 (i) This Theorem shows that if πiX = 0, 1 < i < n, then the n-type of X is described completely by a crossed complex. For n = 1, this is well known (the Eilenberg-Mac Lane spaces do this), and for n = 2 it is essentially due to Mac Lane and Whitehead [134]. Indeed, they prove that the 2-type (for which they use the 1 1 term 3-type) of a reduced CW -complex X is described by the crossed module π2(X,X ) → π1X , which is the same crossed module as arises for n = 2 in the proof of Theorem 9.3.1.

(ii) The crossed complex C constructed in the proof of Theorem 9.3.1 has the property that Hi(C) = 0, ∼ n+1 1 < i < n; Hj(C) = πj(X), j = 1, n; Ci = 0, i > n. It is known that H (π1X, πnX) can be represented by equivalence classes of such complexes ([114], [109], [132], [127]). In particular, the equivalence class of C can be regarded as giving the ﬁrst k-invariant of X.

The following result is sometimes useful for giving an explicit presentation of a crossed module representing the 2-type of a space. It is due to Loday [125], with a diﬀerent proof. 267

Proposition 9.3.3 Let X be a reduced CW -complex and let P be a group such that there is a map f : BP → X which is surjective on fundamental groups. Let F (f) be the homotopy ﬁbre of f and let M = π1F (f), so that we have a crossed module M → P . Then there is a map X → B(M → P ) inducing an isomorphism of π1 and π2.

Proof Let f : BP → X be a cellular map which is surjective on fundamental groups. Let Y be the reduced mapping cylinder M(f) of f, and let j : BP → Y be the inclusion. Then the crossed module π2(Y,BP ) → π1BP is isomorphic to µ : M → P . Also j is surjective on fundamental groups, and it follows that the inclusion X1 → Y is deformable by a homotopy to a map g0, say, with image in BP . This homotopy extends to a homotopy of the inclusion X → Y to a map g : X → Y extending g0.

Let Y∗ be the ﬁltered space in which Y0 is the base point of Y , Y1 = BP , Yi = Y for i ≥ 2. Then C = ΠY∗ is the trivial extension by zeros of the crossed module M → P . The map g : X → Y induces a morphism g∗ :ΠX∗ → ΠY∗ which is realised by a map X → B(M → P ) inducing an isomorphism of π1 and π2. 2

Example 9.3.4 We now give an application of the last Proposition which uses the GvKT for crossed modules. Let X be a CW -complex which is the union of connected subcomplexes Y and Z such that A = Y ∩ Z is a K(P, 1), i.e. is a space BP . Suppose that the inclusions of A into Y and Z induce isomorphisms of fundamental groups. Then, as in Proposition 9.3.3, the 2-types of Y and Z may be described by crossed modules M → P and N → P respectively, say. By results of [39] which are applied in [25] to this situation, the crossed module describing the 2-type of X is the coproduct M ◦ N → P of the crossed P -modules M and N.

We now give another application to the homotopy classiﬁcation of maps. It is also addressed to n-aspherical spaces and says that the homotopy classes of maps from a CW -complex of dimension ≤ n to an n-aspherical space are classiﬁed by the homotopy classes of morphisms of their fundamental crossed complexes.

Proposition 9.3.5 For any CW -complex Y with skeletal ﬁltration Y∗, there is a homotopy ﬁbration

F → Y → BΠY∗

: Thus if πi(Y, y) = 0 for 1 < i < n, then the ﬁbre F is n-connected.

Proof Results of [6] give a Kan ﬁbration

RY∗ → NΠY∗

(since in the terminology of [6], NΠY∗ is the underlying simplicial set of the simplicial T-complex ρY∗).

Also for a CW -complex Y∗ the inclusion of RY∗ into the singular complex of Y is a homotopy equivalence. So when realising, we have a homotopy ﬁbration sequence

F → Y → BΠY∗.

The results on n-connectedness come from the homotopy exact sequence of a ﬁbration. (compare the cubical results in Section 15.6) 2 268

Corollary 9.3.6 If Y is a connected CW -complex such that πiY = 0 for 1 < i < n, and X is a CW -complex with dim X ≤ n, then there is a natural bijection of homotopy classes

∼ [X,Y ] = [ΠX∗, ΠY∗].

Proof The assumptions imply that the ﬁbration

Y → BΠY∗ induces a bijection [X,Y ] → [X,BΠY∗].

The fact that the map [X,BΠY∗] → [ΠX∗, ΠY∗] is a bijection follows from Theorem 9.2.14. 2

This Corollary may also be obtained as a concatenation of results proved by J.H.C.Whitehead in [175]. It is also proved in general circumstances by Baues in his book [10].

As another type of applications, we consider a couple of particular cases when we can identify the set Crs(C,A) appearing in the Theorem.

First recall that, for a groupoid G, E1(G) denote the crossed complex which consists of G in dimension 1 and which is trivial elsewhere. Then Crs(C,E1(G)) = Gpds(C1,G). Also recall that if G and H are groupoids, then GPDS(G, H) denotes the internal hom object in the category of groupoids, and [G, H] denotes the set π0 GPDS(G, H) of components of GPDS(G, H).

Proposition 9.3.7 If C is a crossed complex and G is a groupoid, then there is a homotopy equivalence of crossed complexes

CRS(BC,E1(G)) ' E1(GPDS(π1C,G)).

Proof Let Z be a crossed complex. Then there are natural bijections

∼ [Z, CRS(C,E1(G))] = [Z ⊗ BC,E1(G)] because Crs is a closed category ∼ = [π1(Z ⊗ C),G] because E1(G) is nontrivial only in dimension 1 ∼ = [π1Z × π1C,G] because π1 preserves products ∼ = [π1Z, GPDS(π1C,G)] because Gpds is a closed category ∼ = [Z,E1(GPDS(π1C,G))] as before.

The result follows by the same method used in Corollary 9.2.9. 2

If G is connected, x ∈ G0, and f : G → H is a morphism, then the vertex group GPDS(G, H)(f) is isomorphic to the centraliser of f(G(x)) in H(fx). So the previous result with Theorem 9.2.14 yields a result of Gottlieb [95] on the fundamental group of spaces of maps into an Eilenberg-Mac Lane space K(H, 1). In the pointed case the Proposition gives an even simpler result.

Proposition 9.3.8 If C is a pointed, connected crossed complex and G is a pointed groupoid, then the crossed complex CRS∗(C,E1(G)) has set of components bijective with Gpds(π1(C, ∗),G(∗)) the set of morphisms of groups π1(C, ∗) → G(∗), and all components of CRS∗(C,E1(G)) have trivial π1 and Hi for i ≥ 2. 269

Proof An argument similar to that in the proof of the previous proposition yields

∼ [∗, CRS∗(C,E1(G))] = [∗, GPDS∗(π1C,G)], which gives the ﬁrst result. The second result follows since for any pointed crossed complex Z and morphism f : C → E1(G) that we shall take as base point, we have ∼ [(Z, ∗), (CRS∗(C,E1(G), f)] = [Z ⊗ C,E1(G)]# ∼ = [π1Z × π1C,G]# ∼ ∼ = [π1C,G|f] = ∗. where the sets of homotopy classes marked # are of maps satisfying that restricted to some space are the appropriate ones, i.e. the conditions that come from duality, |1 ⊗ f : ∗ ⊗ C → E1(G), ∗ : Z ⊗ ∗ → E1(G) in the ﬁrst case and |1 × f : ∗ × π1C → G, ∗ : π1Z × ∗ → G in the second one. 2

There is another interesting special case of the homotopy classiﬁcation. Let n ≥ 2. Let M be an abelian group and let Aut M be the group of automorphisms of M. Let χ(M, n) be the pointed crossed complex which is: Aut M in dimension 1; M in dimension n; which has the given action of Aut M on M; and has trivial boundaries, i.e.

χ(M, n) = ··· / 0 / M / 0 / ······ / 0 / Aut M.

Let C be a crossed complex; in useful cases, C will be of free type. We suppose C reduced and pointed.

Let α : π1(C, ∗) → Aut M be a morphism. The set of pointed homotopy classes of morphisms C → χ(M, n) α which induce α on fundamental groups is written [C, χ(M, n)]∗ . This set is easily seen to have an Abelian group structure, induced by the addition on operator morphisms Cn → M over α. So we obtain the homotopy classiﬁcation:

Proposition 9.3.9 If X is a pointed reduced CW -complex, and α : π1(X, ∗) → Aut M, then there is a natural bijection α ∼ α [X, Bχ(M, n)]∗ = [ΠX∗, χ(M, n)]∗ where the former set of homotopy classes denotes the set of pointed homotopy classes of maps inducing α on fundamental groups.

Proof The proof is immediate from Theorem 9.2.16. 2

This result can be related to the case of local coeﬃcients (see Section 13.6), in such a way that Proposition 9.3.9 is essentially a result of Gitler in [92]. (See also the papers of McClendon [144] and Siegel [161]).

9.4 Fibrations of crossed complexes

We recall a deﬁnition due to Howie in [111].

Deﬁnition 9.4.1 A morphism p : E → D of crossed complexes is a ﬁbration if

(i) the morphism p1 : E1 → D1 is a ﬁbration of groupoids; 270

(ii) for each n ≥ 2 and x ∈ E0, the morphism of groups pn : En(x) → Dn(px) is surjective. The morphism p is a trivial ﬁbration if it is a ﬁbration, and also a weak equivalence, by which is meant that p induces a bijection on π0 and isomorphisms π1(E, x) → π1(D, px), Hn(E, x) → Hn(D, px) for all x ∈ E0 and n ≥ 2.

One of the ways in which the techniques of crossed complexes differ from those of chain complexes is the more elaborate exact sequences we get in comparison with those that arise from an exact sequence of chain complexes. The exact sequences in our context better reflect the geometry of fibrations of spaces than do those of chain complexes. Howie shows in [111] that a fibration of crossed complexes leads to a family of exact sequences involving the −1 Hn, π1 and π0, as follows. Let x ∈ E0 and let F = p (px) be the sub crossed complex of E of all elements of E0 which map by p to x and otherwise all elements of some En which map down by p to the identity at px.

Theorem 9.4.2 There is an exact sequence

in pn ∂n · · · → Hn(F, x) −→ Hn(E, x) −→ Hn(D, px) −→· · ·

p1 ∂1 · · · → π1(E, x) −→ π1(D, px) −→ π0(F ) → π0(E) → π0(D). Here the terms of the sequence are all groups, except the last three which are sets with base points the classes xF , xE, xD of x, x, px respectively.

(i) There is an operation of the group π1(E, x) on the group π1(F, x) making the morphism π1(F, x) → π1(E, x) into a crossed module.

∂1 (ii) There is an operation of the group π1(D, px) on the set π0(F ) such that the boundary π1(D, px) −→ π0(F ) is given by ∂1(α) = α · xF .

Proof The proof of this theorem is a development of the part of the theorem which deals with ﬁbrations of groupoids and that is given by Brown in the book [28]. We leave to work out the details as an exercise. 2

Remark 9.4.3 We want to remark the two big diﬀerences between this situation and the habitual one in homology. First we do not get one sequence but a whole family that varies with the base point x. Second, this our exact sequences are less precise in low dimensions since they are exact sequences of sets. Nevertheless, they contain a good amount of information, to be used later at several places.

We now consider coﬁbrations, following methods of [156] which were developed for crossed complexes in [31].

Deﬁnition 9.4.4 Consider the following diagram. / A > E i C / D.

If given i the dotted completion exists for all morphisms p in a class F, then we say that i has the left lifting property (LLP) with respect to F. We say a morphism i : A → C is a cofibration if it has the LLP with respect to all trivial fibrations. We say a crossed complex C is cofibrant if the inclusion ∅ → C is a cofibration. We shall also need the definition that p has the right lifting property (RLP) with respect to a class C if in the above diagram, given p, then the dotted completion exists for all i in the class C. 271

Now we can characterise ﬁbrations of crossed complexes in terms of the RLP.

Proposition 9.4.5 Let p : E → D be a morphism of crossed complexes. Then the following conditions are equivalent: (i) p is a ﬁbration; (ii) (covering homotopy property) p has the RLP with respect to the inclusion C ⊗ 1 → C ⊗ F(m) for all coﬁbrant crossed complexes C and m ≥ 1,; (iii) the covering property (ii) holds for m = 1.

Proof i)⇒(ii) We follow the method of [156]. We verify the covering property by constructing a lifting in the left hand of the following diagrams, where 1 → F(m) is the inclusion. Let p0 in the right hand

C ⊗ 1 / E ∅ / CRS(F(m),E)

p p0 / C ⊗ F(m) D C / CRS(1,E) ×D CRS(F(m),D) diagram be induced by p and the inclusion 1 → F(m). Then a lifting in the left hand diagram is equivalent to a lifting in the right hand diagram. Since C is cofibrant, such a lifting exists if p0 is a trivial fibration. But by the exponential law, for this it is sufficient to show that p has the RLP with respect to the inclusion

S(n) ⊗ F(m) ∪ F(n + 1) ⊗ 1 → F(n + 1) ⊗ F(m).

For n = −1, this corresponds precisely to the ﬁbration property of p. In general, a lifting of the image of the top basis element of F(n + 1) ⊗ F(m) is chosen, and the value of the lifting on the remaining basis element of

F(n + 1) ⊗ F(m), namely cn+1 ⊗ δcm if m ≥ 2, cn ⊗ 0 if m = 1, is determined by the boundary formula for cn ⊗ cm and the values on δcn+1 ⊗ cm if n ≥ 1 and 0 ⊗ cm and 1 ⊗ cm if n = 0. ii)⇒(iii) is immediate (iii)⇒(i) This is easily proved on taking C to be the crossed complex of free type on one generator of dimension n. 2

This gives a similar characterisation using coﬁbrant complexes.

Proposition 9.4.6 Let p : E → D be a morphism of crossed complexes. Then p is a fibration if and only if for any cofibrant crossed complex C then the induced morphism p : CRS(C,E) → CRS(C,D) is a fibration.

Proof It is clear that if p is a fibration and C is cofibrant, the induced morphism p : CRS(C,E) → CRS(C,D) is a fibration. To prove it in the other direction, one takes again C to be the crossed complex of free type on one generator of dimension n. 2

Finally, there is a relation between a map of crossed complexes being a fibration and its nerve being a Kan fibration. As a consequence the classifying spaces form a Serre fibration.

Proposition 9.4.7 Let p : E → D be a morphism of crossed complexes. Then p is a ﬁbration; if and only if the induced map of nerves Np : NE → ND is a Kan ﬁbration. 272

n n+1 n+1 Proof Let Λi for i = 1, . . . , n be the simplicial subset of ∆ generated by all faces ∂jc except for j = i. To say that Np : NE → ND is a Kan ﬁbration is equivalent [138] to saying that any diagram

m / Λi NE;

Np ∆n+1 / ND has a regular completion given by the dotted arrow. By adjointness, this is equivalent to the existence of a regular completion in Crs of the following diagram:

m k / Π(Λi ) ; E g (*) j p k0 Π(∆n+1) / D

If n = 0, this existence is equivalent to E1 → D1 being a ﬁbration of groupoids.

If n ≥ 1, let us see that this existence is equivalent to each En+1(x) → Dn+1(px) being surjective. To see this, note that if these maps are surjective, and v is the usual base point of ∆n+1, then we can choose 0 n+1 n+1 a ∈ En+1(pv) such that pa = k c . If we now deﬁne g(c ) = a and g(x) = k(x) for each non-degenerate n n+1 element x of Λ1 , then there is a unique value for g(∂ic ), determined by the homotopy addition lemma, which deﬁnes a morphism g : Π(∆n+1) → E. This g is a regular completion of (*).

On the other hand, suppose each diagram (**) has a regular completion. Let b ∈ Dn+1(px). Deﬁne n 0 n+1 0 n+1 k : Π(Λ1 ) → E to be the trivial morphism with value 0x. Deﬁne k : Π(∆ ) → D by k (c ) = b, 0 n 0 n+1 0 n+1 k (Λ1 ) = 0px and k (∂1c ) = δb . Then pk = k j. Let g be a regular completion. Then pg(c ) = b. 2

−1 Corollary 9.4.8 Let p : E → D be a ﬁbration of crossed complexes and let x ∈ D0. Let F = p (x). Then the sequence of classifying spaces BF → BE → BD is a ﬁbration sequence.

Proof By a fibration of spaces we will mean a map which has the covering homotopy property with respect to all maps of compactly generated spaces. It is known that the realisation of a simplicial Kan fibration is a Serre fibration [157], and in fact has the covering homotopy property with respect to maps of all compactly generated spaces [163]. Thus all we have to check is that the fibre of NE → ND over x is precisely NF . This follows n from the formula (NF )n = Crs(Π∆ ,F ) given in Section 2. 2

We now give some applications of BC to the case where the crossed complex C is essentially a crossed module. Similar results are proved in [125] using a classifying space of a crossed module deﬁned using the equivalence of crossed modules and 1-cat-groups. We use the same notation µ : M → P for a crossed module and for the crossed complex obtained from it by trivial extension. Hence the above crossed module has a classifying space which we write B(M → P ). We will use the fact that the identity crossed module M → M has contractible classifying space B(M → M). This can be proved either by noting that it has all homotopy groups zero, or by realising a contracting homotopy of the crossed complex extending M → M. 273

Example 9.4.9 Let µ : M → P be an inclusion of a normal subgroup. Then we have an exact sequence of crossed modules

M / M / 1

µ M / P / P/M

It follows from the ﬁbration sequence of classifying spaces that the induced map B(M → P ) → B(P/M) is a homotopy equivalence.

Example 9.4.10 Let µ : M → P be a crossed module. Then we have a short exact sequence of crossed modules

µ0 Ker µ / M / Im µ

µ 1 / P / P in which µ0 is the restriction of µ and the unlabelled maps are inclusions. This exact sequence yields a ﬁbration sequence which up to homotopy is

K(Ker µ, 2) → B(M → P ) → B(Coker µ).

Example 9.4.11 Again let µ : M → P be a crossed module. Then we have short exact sequences of crossed modules

1 1 1 / M / M M / M / 1

i µ 1 M / M o P / P M / M o P / P j τ i σ in which i : m 7→ (m, 1), j : m 7→ (m, (µm)p), σ :(m, p) 7→ p, τ :(m, p) 7→ (µm)p. It follows that we have a homotopy ﬁbration B(M) → B(P ) → B(M → P ).

This shows as in [125] that a morphism µ : M → P of groups can arise from a homotopy ﬁbration B(M) → B(P ) → X if and only if µ can be given the structure of a crossed module.

Example 9.4.12 Let A be a crossed complex, and let n ≥ 1. We write A(n) for the crossed complex D where   Ai i ≤ n Di = δ(An+1) i = n + 1  0 i > n + 1 with the boundary Dn+1 → Dn being inclusion, and other structures induced by that of A. The natural map

A → A(n) 274

(n) is a fibration and induces isomorphisms of π0 and π1 and of H for i ≤ n. Further Hi(A ) = 0 for i > n. Thus the induced map BA → BA(n) is a Postnikov fibration. Let A[n] = Ker(A → A(n)). Suppose A is reduced. Then A[n] may be regarded as a chain complex with a groupoid operating (see Chapter 12). It is clear that NA[n] is a simplicial Abelian group and so BA[n] may be given the structure of a topological Abelian group. If n = 1, then BA is a K(G, 1) space. Thus, as first pointed out by Loday (private communication), the homotopy types of the form BA for A a crossed complex are restricted. In particular, all Whitehead products in BA are zero. One should think of these homotopy types as giving a first approximation to homotopy theory. This idea is developed in the tower of homotopy theories due to H.J. Baues [10], in which crossed complexes (reduced, of free type) give the first level of the tower, and in a sense represent the linear approximation to homotopy theory. However, the use of classifying spaces in conjunction with crossed complexes A with some kind of algebra structure A ⊗ A → A does allow some deeper levels of homotopy information to be obtained (see [11], [13]). Such objects could be called crossed differential graded algebras. Chapter 10

Resolutions.

In this section, we give an introduction to the notion of resolution from the point of view of crossed complexes. As we saw in Chapter 3 of Part I, a presentation hX | Ri of a group G gives a free crossed module

φ C(R) −→∂ F (X) −→ G → 1.

By methods close to those of traditional homological algebra, this crossed module can be extended to give a free crossed complex ∂ · · · → Fn → Fn−1 → · · · F2 = C(R) −→ F1 = F (X) whose homology is trivial for n ≥ 1 and isomorphic to G in dimension 1. We call this a crossed complex resolution of G In this Chapter we study the existence, properties and examples of these resolutions and the closely related technique of acyclic models. In Chapter 11 we shall give a computational procedure to get resolutions (some of the examples of this Chapter are produced this way) and in Chapter 13 we establish some relations with the cohomology of groups. To relate this approach to the traditional one, we need the relations between crossed complexes and chain complexes with operators, which is dealt with in another chapter.

10.1 The notion of syzygy and of resolution

A central problem in mathematics is the representation of infinite objects in manipulable, and preferably finite, terms. One method of doing this is by what is called a resolution. There is not a formal definition of this, but we will see several examples. This notion first arose in the 19th century study of invariants.

Invariant theory deals with subalgebras of polynomial algebras Λ = k[x1, . . . , xn], where k is a ring. Consider for example, the subalgebra of A of Z[a, b, c, d] generated by a2 + b2, c2 + d2, ac + bd, ad − bc.

It is called an invariant subalgebra since it is invariant under the action of Z2 which switches the variables a, b and c, d. These generators satisfy the relation (ac + bd)2 + (ad − bc)2 = (a2 + b2)(c2 + d2)

275 276 which is classically called a syzygy, and the algebra A of invariant polynomials turns out to be the homomorphic image of the polynomial algebra in four variables given by the quotient algebra

Z[x, y, z, w]/(z2 + w2 − xy).

In particular, the algebra is finitely generated by four explicit polynomials, and the ideal of relations is finitely generated by a single explicit relation. It was natural then to turn to chains of syzygies, studying relations among the generating set of relations and so on. More precisely, this work involved the sequence of finitely generated k[y1, . . . , yn]-modules / / / / 0 J1 F1 B1 0 / / / / 0 J2 F2 J1 0

0 / Jq / Fq / Jq−1 / 0, where the Fi are free with rank equal to the minimal number of generators of the i’th syzygy Ji. Hilbert’s main theorem on the chains of syzygies says that if k is a ﬁeld then Jq = 0 if q > n. In eﬀect, this launched the theory of homological dimension of rings.

It was also natural to splice the morphisms Fq → Jq−1 → Fq−1 together to get a sequence

∂q+1 ∂q ∂q−1 ∂2 ∂1 ··· −→ Fq −→ Fq−1 −→ · · · −→ F1 −→ B1 which was exact in the sense that

Ker ∂q = Im ∂q+1 for all q. This sequence was called a free resolution of the module B1. A natural question was the independence of this sequence on the choices made. It was found that given 0 0 any two such free resolutions F∗ → B1,F∗ → B1, then there was a morphism F∗ → F∗ and any two such morphisms were homotopic. It was also later found that the condition ‘free’ could conveniently be replaced by the condition projective. Another source for homological algebra was the homology and cohomology theory of groups. As pointed out in [131], the starting point for this was the 1942 paper of Hopf [110]. Let X be an aspherical space (i.e. connected and with πiX = 0 for i > 1) , and let

1 → R → F → π1X → 1 be an exact sequence of groups with F free. Hopf proved the formula ∼ H2X = (R ∩ [F,F ])/[F,R].

Later work of Eilenberg-Mac Lane [80] found an algebraic formula for HnX, n > 2 as follows. Produce sequences of ZG-modules / / / / 0 J1 F1 Z 0 / / / / 0 J2 F2 J1 0

0 / Jq / Fq / Jq−1 / 0, in which Z is the trivial ZG-module, and each Fn is a free ZG-module. Splice these together to give a free resolution of Z:

F∗ : ... → Fn → Fn−1 → ... → F2 → F1 → Z. 277

∼ Form the chain complex C = F ⊗ZG Z. Then HnC = HnX. Using particular choices of the Fn, the Hopf formula may be deduced see p.46 in K.S.Brown’s book [19]. Thus we see an input from the homotopy and homology theory of spaces into the development of homological algebra. The use of homological methods across vast areas of mathematics is a feature of 20th century mathematics. It seems the solution of Fermat’s last theorem depended on it, but it has also been applied in diﬀerential equations, coding theory and theoretical physics. In its 20th century form, homological algebra is primarily an abelian theory. There is considerable work on nonabelian homological algebra, but this is only beginning to link with work in homotopical algebra, diﬀerential topology, and related areas. This book has an aim of showing one kind of start to a more systematic background to such an area. As we saw in Example 5.5.6, the Hopf formula follows from our van Kampen Theorem for crossed modules. Thus we see the advantage for homotopy theory of having a 2-dimensional algebraic model of homotopy types.

Theorem 10.1.1 For any groupoid G there is an aspherical free crossed complex

∂n+1 / ∂n / ∂n−1 / ∂3 / ∂2 / F (G) = ( ··· Fn+1 Fn ······ F2 F1 ) ∼ and an isomorphism φπ1(F (C)) = G.

Proof We ﬁrst choose a presentation P = hX | Ri of the groupoid G and so we get a free crossed module over a free groupoid φ C(R) −→∂2 F (X) −→ G → 1. Now A = Ker ∂ is a G-module and we proceed as in classical homological algebra as outlined above. 2

This is a traditional method of constructing complexes, either CW -complexes or forms of resolutions, by ‘killing kernels’. It must be said that this is not the most economical way of constructing such crossed complexes.

Notice that at each stage we are producing a free F1-module Fn and an epimorphism Fn → Ker δn−1. This always exists (in the worst case take Fn the F1-module generated by all Ker δn−1) but can be enormous. We will see in Chapter 11 that the use of homotopies in an inductive manner does allow a more algorithmic procedure. But ﬁrst, we study some properties of the free crossed complexes like the one in Theorem 10.1.1, i.e. resolutions of G.

10.2 Free crossed resolutions of groups and groupoids

In this Section we introduce the concept of free crossed resolution of a groupoid G, prove that any two resolutions of the same groupoid are homotopy equivalent and give some examples.

Deﬁnition 10.2.1 A crossed complex D is called aspherical if for all n > 2 and p ∈ D0, we have Hn(D, p) = 0. It is acyclic if it is aspherical, connected and in addition π1(D, p) = 0 for all p ∈ D0. A free, aspherical crossed complex F together with an isomorphism of groupoids φ : π1F → G where G is a groupoid, is called a free crossed resolution of the groupoid G.

A main technical tool in the use of free crossed resolutions is the following two theorems, which imply that free crossed resolutions of a groupoid are determined up to homotopy. 278

Theorem 10.2.2 Let C,D be crossed complexes such that C is free and D is aspherical. Let α : π1C → π1D be a morphism of groupoids. Then there is a morphism f : C → D of crossed complexes such that π1(f) = α, and any two such are homotopic. Such a morphism f is said to be a lift of α.

Proof We consider the diagram

δn / / / δ2 / φ / ··· Cn Cn−1 ··· C2 C 1 π1C f1 α / / / / / ··· Dn Dn−1 ··· D2 D1 π1D δn δ2 ψ in which φ, ψ are the quotient morphisms.

Let the free basis of C be denoted by X∗, where X0 = C0, and we assume Xn is a subgraph of Cn.

For x ∈ X1 we choose f1(x) ∈ D1 such that ψf1(x) = αφ(x). This is possible because ψ is surjective. Since X1 is a free basis of C1, this choice extends to a morphism

f1 : C1 → D1.

Since ψf1 = αφ on the generating set X1, it follows that

ψf1 = αφ on C1. Note also that

ψf1δ2 = αφδ2 = 0.

Since Ker ψ = Im δ2, it follows that Im f1 ⊆ Im δ2. Now we proceed in the same way. Suppose

fn−1 : Cn−1 → Dn−1 has been deﬁned so that

δn−1fn−1 = fn−2δn−1. Then

δn−1fn−1δn = fn−2δn−1δn = 0.

By asphericity of D, Im (fn−1δn) ⊆ Im δn. So for all x in the free basis Xn, there is an fn(x) ∈ Dn such that δnfn(x) = fn−1δn(x). This deﬁnes a morphism

fn : Cn → Dn such that δnfn = fn−1δn. 2

Theorem 10.2.3 Let C,D be crossed complexes such that C is free and D is aspherical. Let α : π1C → π1D be 0 1 0 1 a morphism of groupoids and f , f : C → D morphisms of crossed complexes such that π1(f ) = π1(f ) = α. Then there is a homotopy h : f 0 ' f 1. 279

Proof We proceed as before to deﬁne the homotopy (see Deﬁnition 8.2.6) starting for

h0 : C0 → D1.

0 1 0 1 Since π1(f ) = π1(f ) = α, we have ψf1 = αφ = ψf1 . We set h0(c) = 1βc ∈ D1, for c ∈ C0. Let us deﬁne

h0 : C1 → D2.

If c ∈ C1, then the formula for a homotopy gives that h1 should satisfy

0 0 1 1 −1 f1 (c) = (h0δ c)(f1 c)(δ2h1c)(h0δ c) which because of our deﬁnition of h0 reduces to

0 1 f1 (c) = f1 (c)(δ2h1c) or 1 −1 0 δ2h1c = (f1 (c)) f1 (c). But 1 −1 0 ψ((f1 (c)) f1 (c)) = 1. 1 −1 0 1 Hence for each x ∈ X1 we can choose an h1(x) such that δ2h1x = (f1 (x)) f1 (x). This extends to an f1 derivation h1 : C1 → D2, as explained in Remark 8.2.2. 1 1 0 0 At the next level, for c ∈ C2, we note that f1 δ2 = δ2f2 , f1 δ2 = δ2f2 and we require h2 such that

0 1 f2 (c) = f2 (c)(h1δ2(c))((δ3h2(c)).(∗)

But −1 1 −1 0 1 −1 0 −1 1 −1 0 δ2((h1δ2(c)) f2 (c) f2 (c)) = ((f1 (δ2c)) f1 (δ2c)) δ2f2 (c) δ2f2 (c) = 1. 1 So again, we can choose h2(x) for x ∈ X2 so that (*) holds for c = x. This extends to an f1 -derivation h2 : C1 → D2 as required. We now look at the situation around dimension n.

δ δ / n+1 / δn / n−1 / / Cn+1 Cn Cn−1 Cn−2 r r q r rrr qqq r r rr qq f 0 f 1 h f 0 f 1 h f 0 f 1 h f 0 f 1 n+1 n+1 r n n n r n−1 n−1 n−1 qq n−2 n−2 n−2 r rrr qq yr yrr xqq / / / / / Dn+1 Dn Dn−1 Dn−2 δn+1 δn δn−1

0 1 We suppose given the morphisms f , f and also the hn−2, hn−1 such that

0 1 fn−1 = fn−1 + hn−2δn−1 + δnhn−1.

But for c ∈ Cn

0 1 0 1 δn(fnc − fnc − hn−1δnc) = fn−1δnc − fn−1δnc − δnhn−1δnc 0 1 0 1 = fn−1δnc − fn−1δnc − (fn−1δnc − fn−1δnc − hn−2δn−1δnc)

= 0 since δn−1δn = 0. 280

By asphericity of D, for each x in the basis Xn we can ﬁnd an hnx in Dn+1 such that

0 1 hnx = fnx − fnx − hn−1δnx.

This extends to an operator morphism hn : Cn → Dn+1 with the required properties for the next stage of the induction. 2

This proof is typical of the method of constructing homotopies, and is important in later sections.

Corollary 10.2.4 Any two free crossed resolutions of a group G are homotopy equivalent.

Remark 10.2.5 A reﬁnement of Theorems 10.2.2 and 10.2.2 is to assume that we have two morphisms α0, α1 : G → H, that the morphisms f 0, f 1 lift α0, α1 respectively and that η is a homotopy (or natural transformation) α0 ' α1. Then we assert that under the same conditions of freeness and asphericity, η lifts to a homotopy 0 1 h : f ' f . Let us assume φ, ψ are the identity on objects. Here η0 assigns to each p in C0 an element η(p) ∈ H(α0p, α1p) such that the usual naturality condition holds: if g ∈ G(p, q) then α0(g)η(q) = η(p)α1(g).

For each p ∈ C0 choose an h0(p) ∈ D1(p, q) such that ψ(h0(p)) = η(p). Now we repeat the arguments of the proof of Theorems 10.2.2 and 10.2.2 but using the more complicated formulae for homotopies which involve h0. We leave the details as an exercise for the reader. 2

The uniqueness up to homotopy of the free crossed resolution leaves open the important problem of the construction of a free crossed resolutions of a group or groupoid G that can be used in practice. We give some examples of resolutions leaving in some cases the proof for a later Section (11.4). Most of them can be checked directly, but the proof of exactness is a calculation in modules over a group ring. In general, it is quite hard to prove asphericity of a crossed complex, particularly at dimension 2 where the problem involves non commutative, and possibly inﬁnitely generated, groups. Part of the problem is to calculate a set of generators for the kernel of the boundary in a free crossed module δ : C(R) → F (X). We will give later some ways of dealing with this problem. Let us ﬁnish this Section by giving some examples of resolutions even if we leave the proof that some are actually resolutions to a Chapter 11. First a very simple case, when G is itself free.

Example 10.2.6 If G is a free groupoid F (X1), then G has a free crossed resolution which is F (X1) in dimension 1 and is trivial in higher dimensions.

Now a small free crossed resolution of the ﬁnite cyclic group.

Example 10.2.7 A cyclic group Cq of order q has a free crossed resolution F = F (Cq) as follows:

δ4 δ3 δ2 φ F (Cq) = · · · → Z[Cq] −→ Z[Cq] −→ Z[Cq] −→ C∞ −→ Cq with C∞ the inﬁnite cyclic group with free generator x1, with φ(x1) = c; Z[Cq] is the free Cq module with free generator xn for n > 2; and   q  x1 if n = 2 ; δ (x ) = x (1 − c) if n is odd; n n  n−1  2 q−1 xn−1 (1 + c + c + ··· + c ) otherwise.

This resolution is due to Brown and Wensley in [60]. In Section 11.4 we prove that it is indeed a resolution. 281

Exercise 10.2.8 1. Prove directly that the preceding example gives a free crossed resolution of Cq. r 2. Let α : Cq → Cqr be the morphism which sends a generator c of Cq to c1, where c1 generates Cqr. Find a morphism of free crossed resolutions F (Cq) → F (Cqr) which lifts α.

Now a free crossed resolution for any group, called the standard crossed resolution.

st Example 10.2.9 There is a free crossed F∗ (G) resolution of a group G

δ st / st δn / st n−1 / δ3 / st δ2 / st φ / F (G) = ··· Fn (G) Fn−1(G) ······ F2 (G) F1 (G) G

st n in which Fn (G) is free on the set G with generators written [a1, a2, . . . , an], ai ∈ G and boundary

st st δn : Fn (G) → Fn−1(G) given by −1 δ2[a, b] = [a][b][ab] , −1 −1 [a]−1 δ3[a, b, c] = [a, bc][ab, c] [a, b] [b, c] , and for n > 4 nX−1 a−1 i n δn[a1, a2, . . . , an] = [a2, . . . , an] 1 + (−1) [a1, a2, . . . , ai−1, aiai+1, ai+2, . . . , an]+ +(−1) [a1, a2, . . . , an−1]. i=1

Some introduction to the next two examples

Example 10.2.10 Let hX | Ri be a one relator presentation of a group G, that is R consists of a single element r ∈ F (X), and suppose r is not a proper power. It is a theorem that the kernel of C(r) → F (X) is trivial, so that this crossed module itself is in essence a free crossed resolution of G. However, the proof of this triviality is by no means easy.

Later we will prove by topological means the following theorem, which, combined with the fact that the tensor product of free crossed complexes is free, gives one method of making new free crossed resolutions from pairs of given ones.

Theorem 10.2.11 If C, D are aspherical free crossed complexes, then their tensor product C ⊗ D is also aspherical.

Proof ?? 2

Example 10.2.12 Let PA = hXA | RAi, PB = hXB | RBi be presentations of groups A, B respectively, and let F(PA), F(PB) be the corresponding free crossed modules, regarded as 2-truncated crossed complexes. The tensor product T = F(PA) ⊗ F(PB) is 4-truncated and is given as follows (where we now use additive notation in dimensions 3, 4 and multiplicative notation in dimensions 1, 2):

• T1 is the free group on generating set XA t XB;

• T2 is the free crossed T1-module on RA t (XA ⊗ XB) t RB with the boundaries on RA,RB as given before and −1 −1 δ2(a ⊗ b) = b a ba for all a ∈ XA, b ∈ XB ; 282

• T3 is the free (A × B)-module on generators r ⊗ b, a ⊗ s, r ∈ RA, s ∈ RB with boundaries

−1 b −1 −1 a δ3(r ⊗ b) = r r (δ2r ⊗ b), δ3(a ⊗ s) = (a ⊗ δ2s) s s ;

• T4 is the free (A × B)-module on generators r ⊗ s, with boundaries

δ4(r ⊗ s) = (δ2r ⊗ s) + (r ⊗ δ2s) .

The important point is that we can if necessary calculate with these formulae, because elements such as

δ2r ⊗ b may be expanded using the rules for the tensor product. Alternatively, the forms δ2r ⊗ b, a ⊗ δ2s may be left as they are since they naturally represent subdivided cylinders.

10.3 Free products with amalgamation and HNN-extensions

This requires two deep results We illustrate the use of crossed complexes of groupoids with the construction of a free crossed resolution of a free product with amalgamation, given free crossed resolutions of the individual groups, and a similar result for HNN-extensions. These are special cases of results on graphs of groups which are given in [147, 52], but these cases nicely show the advantage of the method and in particular the necessary use of groupoids. Suppose the group G is given as a free product with amalgamation

G = A ∗C B, which we can alternatively describe as a pushout of groups

j C / B

i i0 A / G. j0

We are assuming the maps i, j are injective so that, by standard results, i0, j0 are injective. Suppose we are given free crossed resolutions A = F(A), B = E(B), C = F(C). The morphisms i, j may then be lifted (non uniquely) to morphisms i00 : C → A, j00 : C → B. However we cannot expect that the pushout of these morphisms in the category Crs gives a free crossed resolution of G. To see this, suppose that these crossed resolutions are realised by CW -ﬁltrations K(Q) for Q ∈ {A, B, C}, and that i00, j00 are realised by cellular maps K(i): K(C) → K(A),K(j): K(C) → K(B), as in Proposition ?? This requires Baues result. However, the pushout in topological spaces of cellular maps does not in general yield a CW -complex — for this it is required that one of the maps is an inclusion of a subcomplex, and there is no reason why this should be true in this case. The standard construction instead is to take the double mapping cylinder M(i, j) given by the homotopy pushout

K(j) K(C) / K(B)

K(i) ' K(A) / M(i, j) 283 where M(i, j) is obtained from K(A) t (I × K(C)) t K(B) by identifying (0, x) ∼ K(i)(x), (1, x) ∼ K(j)(x) for 1 x ∈ K(C). This ensures that M(i, j) is a CW -complex containing K(A),K(B) and { 2 }×K(C) as subcomplexes and that the composite maps K(C) → M(i, j) given by the two ways round the square are homotopic cellular maps. It follows that the appropriate construction for crossed complexes is obtained by applying Π to this homotopy pushout: this yields a homotopy pushout in Crs

j00 C / B

i00 ' A / F(i, j) .

Since M(i, j) is aspherical we know that F(i, j) is aspherical and so is a free crossed resolution. Of course F(i, j) has two vertices 0, 1. Thus it is not a free crossed resolution of G but is a free crossed resolution of the homotopy pushout in the category Gpds j C / B

i ' A / G(i, j) which is obtained from the disjoint union of the groupoids A, B, I × C by adding the relations (0, c) ∼ i(c), (1, c) ∼ j(c) for c ∈ C. The groupoid G(i, j) has two objects 0, 1 and each of its object groups is isomorphic to the amalgamated product group G, but we need to keep its two object groups distinct. This idea of forming a fundamental groupoid is due to Higgins in the case of a graph of groups [106], where it is shown that it leads to convenient normal forms for elements of this fundamental groupoid. This view is pursued in [147], from which this section is largely taken. The two crossed complexes of groups F(i, j)(0), F(i, j)(1), which are the parts of F(i, j) lying over 0, 1 respectively, are free crossed resolutions of the groups G(i, j)(0),G(i, j)(1). From the formulae for the tensor product of crossed complexes we can identify free generators for F(i, j) : in dimension n we get

• free generators an at 0 where an runs through the free generators of An ;

• free generators bn at 1 where bn runs through the free generators of Bn ;

• free generators ι ⊗ cn−1 at 1 where cn−1 runs through the free generators of Cn−1 .

Example 10.3.1 Let A, B, C be inﬁnite cyclic groups, written multiplicatively. The trefoil group T can be presented as a free product with amalgamation A ∗C B where the morphisms C → A, C → B have cokernels of orders 3 and 2 respectively. The resulting homotopy pushout we call the trefoil groupoid. We immediately get a free crossed resolution of length 2 for the trefoil groupoid, whence we can by a retraction argument deduce 3 −2 the free crossed resolution F(T ) of the trefoil group T with presentation PT = ha, b | a b i. We show in the last section that there is a free crossed resolution of T of the form

φ2 φ1 3 −2 F(T ): ··· / 1 / C(r) / F {a, b} ___ / T where φ2 r = a b .

Hence a 2-cocycle on T with values in K can also be speciﬁed totally by elements s ∈ K, c, d ∈ Aut(K) such that ∂(s) = c3d−2, which is a ﬁnite description. It is also easy to specify equivalence of cocycles. 284

More elaborate examples and discussion are given in Emma Moore’s PhD thesis [147] and her forthcoming paper with Brown and Wensley [52]. Now we consider HNN-extensions. Let A, B be subgroups of a group G and let k : A → B be an isomorphism. Then we can form a pushout of groupoids

(k0, k1) {0, 1} × A / G

i j I × A / ∗ G f k where

k0(0, a) = ka, k1(1, a) = a, and i is the inclusion.

In this case of course ∗k G is a group, known as the HNN-extension. It can also be described as the factor group

(Z ∗ G) / {z−1a−1z (ka) | a ∈ A} of the free product, where Z is the inﬁnite cyclic group generated by z. Now suppose we have chosen free crossed resolutions A, B, G of A, B, G respectively. Then we may lift k to 00 a crossed complex morphism k : A → B and k0, k1 to

00 00 k0 , k1 : {0, 1} × A → G . Next we form the pushout in the category of crossed complexes:

00 00 (k0 , k1 ) {0, 1} ⊗ A / G

i00 j00 / I ⊗ A ⊗k00 G f 00

Theorem 10.3.2 The crossed complex ⊗k00 G is a free crossed resolution of the group ∗k G .

The proof will be given in a future paper of Brown, Moore and Wensley [52] as a special case of a theorem on the resolutions of the fundamental groupoid of a graph of groups This requires BMW result. Here we show that Theorem 10.3.2 gives a means of calculation. Part of the reason for this is that we do not need to know in detail the deﬁnition of free crossed resolution and of tensor products, we just need free generators, boundary maps, values of morphisms on free generators, and how to calculate in the tensor product with I using the rules given previously.

−1 −1 −1 Example 10.3.3 The Klein Bottle group K has a presentation gph a, z | z a z a i. Thus K = ∗k A where A = hai is inﬁnite cyclic and ka = a−1. This yields a free crossed resolution

φ2 φ1 K : ··· / 1 / C(r) / F {a, z} ___ / K

−1 −1 −1 where φ2 r = z a z a . Of course this was already known since K is a surface group, and also a one relator group whose relator is not a proper power, and so is aspherical. 2 285

q −1 −1 −1 Example 10.3.4 Developing the previous example, let L = gph c, z | c , z c z c i. Then L = ∗k Cq −1 where Cq is the cyclic group of order q generated by c and k : Cq → Cq is the isomorphism c 7→ c . A small free crossed resolution of Cq is given in [60] as

χn χ2 χ1 Cq : ··· / Z[Cq] / Z[Cq] / ··· / Z[Cq] / A ___ / Cq with a free generator a of A in dimension 1 ; with χ1 a = c ; free generators cn in dimension n ≥ 2 ; and   q  a if n = 2 , χ c = c (1 − c) if n is odd, n n  n−1  2 q−1 cn−1 (1 + c + c + ··· + c ) otherwise.

00 The isomorphism k lifts to a morphism k : Cq → Cq which is also inversion in each dimension. Hence L has a free crossed resolution

L = (L−, λ−) = ⊗k00 Cq .

This has free generators a, z in dimension 1; generators c2, z ⊗ a in dimension 2; and generators cn, z ⊗ cn−1 in dimension n ≥ 3 . The extra boundary rules are

−1 −1 −1 λ2(z ⊗ a) = z a z a , q −1 −1 −1 z λ3(z ⊗ c2) = (z ⊗ a ) c2 (c2 ) , z λn+1(z ⊗ cn) = −(z ⊗ χncn) − cn − cn for n ≥ 3 .

In particular, the identities among relations for this presentation of L are generated by

−1 −1 −1 z c2 and λ3(z ⊗ c2) = (z ⊗ χ2c2) c2 (c2 ) .

Similarly, relations for the module of identities are generated by

z c3 and λ4(z ⊗ c3) = − (z ⊗ c2(1 − c)) − c3 − c3 .

Of course we can expand expressions such as (z ⊗χncn) using the rules for the cylinder given in Example 8.2.19. Further examples are developed in [147]. 2

10.4 Acyclic models

The theory of acyclic models is in the traditional methods of homology very useful for comparing diﬀerent representations of homology by chain complexes. It has also been useful for comparing cohomology theories of algebraic structures. The same sort of technique works for crossed complexes, but with some technical diﬀerences. The methods of proof are closely related to those of Theorem 10.2.2. Suppose given a category C and a functor G : C → Crs. Let

M = {Mi}i∈I ⊆ Ob(C) be a set of elements called models (if necessary we shall call I the model indices). We shall examine two conditions of G with respect to the set of models M. The ﬁrst about the homology of G(Mi).

Deﬁnition 10.4.1 We say that G is aspherical relative to M if G(Mi) is aspherical for each i ∈ I, i.e. Hn(G(Mi)) = 0 for all n ≥ 2. It is acyclic if G(Mi) is aspherical and π1(G(Mi) is a contractible groupoid for each i ∈ M. 286

The crucial deﬁnition is the existence of a basis for each dimension that can be obtained through morphism in (some) models. This shall permit deﬁnition by freeness.

Deﬁnition 10.4.2 For each n > 0 and i ∈ I, let J(n, i) be an indexing set and suppose given a family

c = {c(j) | j ∈ J(n, i), i ∈ I, n ≥ 0} of distinct elements where c(j) ∈ G(Mi)n. We say that G is free on M, c if for all X ∈ Ob(C) the crossed complex G(X) is free on the (assumed distinct) elements

G(f)(c(j)) ∈ G(X)n, f ∈ C(Mi,X), n ≥ 0.

Notice that freeness in dimension 0 means that each element of G(X)0 has a unique representation as G(f)(c(j)) for some model Mi, morphism f : Mi → X and c(j) ∈ G(Mi)0. Let us see a couple of examples that we are going to use later. First the fundamental crossed complex of the geometric realization of the simplicial singular complex of a space.

Example 10.4.3 Let ∆ n M = {∆ }n∈N be the set models. Let Π∆ : Top → Crs be the functor Π ◦ |S∆|, where |S∆(X)| is the geometric realisation of the simplicial singular complex of X. It is well known that Π∆ is acyclic relative to the models M ∆, which follows from showing they have trivial fundamental group and trivial homology. Let ∆ ∆ n c (n) ∈ Π(|S (∆ )|)n be the class of the characteristic map of the top dimensional cell of ∆n. Then Π∆ is clearly free on M∆, c∆.

And now the cubical counterpart.

Example 10.4.4 A similar result is obtained if in the previous example we replace the n-simplex ∆n by the n-cube In so that S∆ becomes S2, G∆ becomes G2, and c∆ becomes c2. Here the acyclicity is a consequence of the contractibility of the spaces |S2(In)|, which follows from showing they have trivial fundamental group and trivial homology. For the latter calculation, see the book on cubical singular homology by Massey [137].

The acyclic models are useful for comparing functors. This comparison is obtained using the following theorem.

Theorem 10.4.5 Let C be a category, and let M be a sets of models. Let G, G0 : C → Crs be functors and suppose that G is free relative to a given M, J, c and G0 is acyclic on M. Then the following holds:

(a) there is a natural transformation T : G → G0, and

(b) given any two such natural transformations, there is a natural crossed complex homotopy between them. 287

Proof The ideas for this are closely related to those in the proof of Theorem 10.2.2 and use both the concept of models and of free crossed complex. For each object X of C, we have to construct a morphism T X : G(X) → G0(X) of crossed complexes and prove that the one we construct is natural in X. So we first define the natural transformation on the elements c(j) using G0(M) are acyclic, by naturality we define the values on a basis of G(X) and extend using freeness. It only remains to check that this definition is natural. Let us apply this process inductively on dimension.

In dimension 0, we know that G(X)0 consists of distinct elements G(f)(c(j)) for all morphisms f : M → X 0 0 of models, where c(j) ∈ G(M)0. For each such c(j), choose an element c (j) ∈ G (M)0 and deﬁne

X 0 0 T0 (G(f)(c(j)) = G (f)(c (j)).

We have to prove that this gives a natural transformation. Let φ : X → Y be a morphism of C. We have to prove that the following diagram commutes:

G(φ) G(X)0 / G(Y )0

X Y T0 T0 0 0 G (X)0 / G (Y )0 G0(φ)

In this dimension we are dealing with a set of elements G(f)(c(j)). So we calculate

Y Y T0 G(φ)(G(f)(c(j))) = T0 G(φf)(c(j)) = G0(φf)(c0(j)) = G0(φ)G0(f)(c0(j)) 0 X = G (φ)(T0 (G(f)(c(j)))).

Thus naturality follows from the form of the deﬁnition, and so we will omit proofs of naturality in higher dimensions.

We now move to dimension 1. The groupoid G(X)1 is free on the graph deﬁned by elements G(f)(c(j)) for 0 morphisms f : Mi → X. Since G (Mi)1 is a connected groupoid, we can choose an element in that groupoid

0 Mi 0 Mi 1 c (j): T0 (δ c(j)) → T0 (δ c(j)).

We then deﬁne similarly to the case n = 0

X 0 0 T1 (G(f)(c(j)) = G (f)(c (j)),

X 0 and extend this by freeness to a groupoid morphism T1 : G(X)1 → G (X)1. Again, this gives a natural transformation. In this and higher dimensions, note that it is suﬃcient to verify commutativity on the given set of free generators, so that the proof follows exactly the line given in dimension 0. X 0 For the general inductive step, we assume given Tr : G(X)r → G (X)r, for r < n giving a natural morphism of (n − 1)-truncated crossed complexes. For each map from a model f : M → X and c(j) ∈ G(M)n, we note that

0 M M δn−1Tn−1δnc(j) = Tn−2δn−1δnc(j) = 0. 288

0 0 By acyclicity, there is a c (j) ∈ G (M)n such that

0 0 M δnc (j) = Tn−1δnc(j) and we deﬁne X 0 0 Tn (G(f)(c(j)) = G (f)(c (j)).

Thus the pattern of the argument is to use the models to show how to construct morphisms and homotopies, with the veriﬁcation of the required relations taking place in the models, and then transferred to the other objects. The mode of deﬁnition ensures naturality. 2

Corollary 10.4.6 Let C be a category, and let M be a sets of models. Let G, G0 : C → Crs be functors and suppose that G is free relative to a given M, J, c and G0 is free on M0,J 0, c0. Moreover both G and G0 are acyclic on both M and M0. Then the natural transformation T of Theorem 10.4.5 is a natural homotopy equivalence.

Proof We already have the natural transformation T : G → G0. The theorem also applies (using M0,J 0, c0) to give a natural transformation T 0 : G0 → G. Now we consider the natural transformation T 0 ◦ T : G → G. Again by the theorem, using M, J, c, we get a 0 0 0 natural homotopy η : T ◦ T ' 1G. Similarly, we get a natural homotopy η : T ◦ T ' 1G0 . This completes the proof. 2

We are now going to use this corollary to get equivalences in several interesting situations. First the functors given in Examples 10.4.3,10.4.4.

Theorem 10.4.7 [The equivalence of the simplicial and cubical homologies] There is a natural homotopy equivalence Π∆ ' Π2 : Top → Crs between the functors Π∆ = Π◦|S∆|, where |S∆(X)| is the geometric realisation of the simplicial singular complex of X and Π2 = Π ◦ |S2|, where |S2(X)| is the geometric realisation of the simplicial singular complex of X.

Proof We have seen that Π∆ is acyclic on the models M∆ and M2 and is free on the models M∆, c∆ in Example 10.4.3. Also Π2 is acyclic on the models M2and M∆ and is free on the models M2, c2 in Example 10.4.4. So the Theorem follows from Corollary 10.4.6. 2

∆ 2 Corollary 10.4.8 Let ηX : |S X| → |S X| be any natural transformation. Then for all X, ηX is a homotopy equivalence.

Proof The assumptions imply that Π◦η is a natural transformation of crossed complex functors. By Corollary

10.4.7, ηX induces isomorphisms on homotopy groups and so is a homotopy equivalence. 2

Recall that the geometric realisation functor | | : Simp → Top is adjoint to the singular functor S∆ : Top → Simp, so that there is a natural bijection

Simp(K,S∆X) =∼ Top(|K|,X) 289 for simplicial sets K and topological spaces X. This adjoint pair determines a unit natural transformation

ε : K → S∆|K| for a simplicial set K.

Theorem 10.4.9 The morphism induced by ε,

Π|K| → Π|S∆|K||, is a natural homotopy equivalence.

Proof [OK I hope!] We set up the models and prove acyclicity and freeness...... 2

and as last consequence a version of the Eilenberg-Zilber Theorem

Theorem 10.4.10 For simplicial sets K,L, there is a natural homotopy equivalence

η :Π|K| ⊗ Π|L| → Π|K × L|.

Proof We consider a model system based on M = N×N, with M(p, q) = (∆p, ∆q), where C = Simp×Simp, and Simp is the category of simplicial sets. Our two functors G, G0 are given, of course, by G(K,L) = Π|K| ⊗ Π|L|, G0(K,L) = Π|K × L|. Both G, G0 are acyclic on the models given by M. [More details here.] Now we have to assign the indexing sets J, J 0 and families c, c0. We let J(n, p, q) consist of a single element c(n, p, q) where p+q = n and c(n, p, q) is the class of ep ⊗eq where p p 0 e is a generator in dimension of Π(|∆ |)p. Let J (n, p, q) be indexed by the family c(p, q, α) of top dimensional p q p q cells in |∆ × ∆ | and let c(p, q, α) be the class in (Π|∆ × ∆ |)n represented by such cells. Then G is free on M, J, c and G0 is free on M 0,J 0, c0. We now apply Corollary 10.4.6 to deduce the result. 2

Remark 10.4.11 The above result is useful but it is sometimes even more useful to have explicit formulae for the equivalence and homotopies and to give their properties. In fact the reason why models are aspherical is often because they are contractible, and a contracting homotopy gives the required lifts. For the Eilenberg- Zilber theorem for crossed complexes, explicit formulae have been given by A. Tonks in his thesis [168], and published as [169], and their properties established. These results are used in an equivariant theory of crossed complexes in [32, 33]. 2

Example 10.4.12 Let M = N × N, let C = Top × Top and let M(m, n) = (∆m, ∆n). Let G, H : C → Crs, be the two functors G(X,Y ) = ΠS∆(X × Y ),H(X,Y ) = ΠS∆(X) ⊗ ΠS∆(Y ). It is a consequence of the Eilenberg-Zilber theorem that there is a natural homotopy equivalence from G to H. 290 Chapter 11

Construction of free crossed resolutions of groupoids

In this Chapter we are going to address the problem of getting a resolution for a groupoid G associated to a presentation hX | Ri. As we have seen Theorem 10.1.1 in the previous Chapter gives a theoretical solution. First we construct the free crossed module

φ C(R) −→∂ F (X) −→ G → 1.

Now we take a free resolution of the G-module A = Ker ∂ by the ‘usual way’. That means that at each step we have to get a free G-module mapping surjectively to a G-module

F (Hn) → An−1 and get a set of generators Xn of the kernel.

How can we proceed? Theoretically the answer is easy. If in trouble, just take Xn = An. Obviously this has no practical application (An could be inﬁnite), so we would like to get a way of constructing smaller resolutions. In the chain complex of R-modules case there is a way of constructing at the same time the resolution and the contracting homotopy (in chain complexes resolutions are contractible) that can be easily described. We shall use earlier homotopy information to construct Cn+1, ∂n+1 and hn from free generators of Cn Assume that we have constructed both the resolution and the contracting homotopy up to dimension n. Thus, we have the diagram

∂n / ∂n−1 / / ∂3 / ∂2 / Cn Cn−1 ··· C3 C2 C1 y } } yy }} }} fn yy fn−1 }} f2 }} f1 y n−1 } } |yy h ~}} h2 ~}} h1 / / / / / Cn Cn−1 ··· C3 C2 C1 ∂n ∂n−1 ∂3 ∂2 where the squares commute and

∂i+1hi + hi−1∂i = 1 for i ≤ n − 1 and Ci is free on say Xi.

We want a free G-module Cn+1 and a morphism hn : Cn → Cn+1 satisfying ∂n+1hn + hn−1∂n−1 = 1. We construct the Cn+1 with just enough room to deﬁne the hn. So, if Xn is a free generating set of Cn, we deﬁne

291 292

Cn+1 to be the free R-module over Xn, Cn+1 = F (Xn). We deﬁne

∂n+1 : Cn+1 → Cn as the only morphism extending

∂n+1(x) = x − hn−1∂n−1(x) for all x ∈ Xn and hn : Cn → Cn+1 to be the only morphism extending the identity.

Clearly this produces an homotopy and it is not difficult to check that ∂n∂n+1 = 0. By iterating we get a free chain complex and a contracting homotopy, so the resulting chain complex is a resolution. The trouble with repeating this process for crossed resolutions is that they are not contractible (since the first homology group is isomorphic to G), and that they contain some nonabelian part. Essentially, the universal covering of the ‘resolution’ is going to be a contractible free crossed complex and apply we can apply the same procedure as before. An important point is that in passing to the universal cover, we produce a non-symmetrical situation. The process deals with Cayley graphs, a standard tool in combinatorial group theory. The theory well reflects the geometry of covering spaces and extends the notion of Cayley graph to include higher dimensional information. First we develop the machinery of coverings of crossed complexes in order to prove that the cover of a free crossed complex is free. Only then we can set properly the computational procedure and we end the Chapter using it to get some crossed resolutions of groups that we have already used.

11.1 Covering morphisms of groupoids and crossed complexes

For the convenience of readers, and to ﬁx the notation, we recall here the basic facts on covering morphisms of groupoids.

Let G be a groupoid. For each object a of G the star of a in G, denoted by StG a, is the union of the sets G(a, b) for all objects b of G, i.e. StGa = {g ∈ G | sg = a}. A morphism p : Ge → G of groupoids is a covering morphism if for each object ea of Ge the restriction of p

StGe ea → StG pea is bijective. In this case Ge is called a covering groupoid of G. A basic result for covering groupoids is unique path lifting. That is, let p : Ge → G be a covering morphism of groupoids, and let (g1, g2, . . . , gn) be a sequence of composable elements of G. Leta ˜ ∈ Ob(Ge) be such that pa˜ is the starting point of g1. Then there is a unique composable sequence (˜g1, g˜2,..., g˜n) of elements of Ge such thatg ˜1 starts ata ˜ and pg˜i = gi, i = 1, . . . , n. If G is a groupoid, the slice category GpdsCov/G of coverings of G has as objects the covering morphisms p : H → G and has as arrows (morphisms) the commutative diagrams of morphisms of groupoids, where p and q are covering morphisms, f / H @ K @@ ~~ @@ ~~ p @ ~ q @ ~~ G 293

By a result of [28], f also is a covering morphism. It is convenient to write such a diagram as a triple (f, p, q). The composition in GpdsCov/G is then given as usual by

(g, q, r)(f, p, q) = (gf, p, r).

It is a standard result (see for example [105, 23]) that the category GpdsCov/G is equivalent to the functor op category SetsG . Thus if X : Gop → Sets is a functor, then Ge = X o G has object set the disjoint union of the sets X(a) for a ∈ Ob(G) and arrows x → y the pairs (x, g) such that x ∈ X(sg) and X(g)(x) = y. This ‘semidirect product’ or ‘Grothendieck construction’ is useful for constructing covering morphisms of the groupoid G. For example, if a is an object of the transitive groupoid G, and A is a subgroup of the object group G(a), then the groupoid G operates on the family of cosets {Ah | h ∈ StG a}, by (Ah).g = Ahg whenever hg is deﬁned, and the associated covering morphism Ge → G deﬁnes the cover Ge of the groupoid G determined by the subgroup A. When A is trivial this gives the universal cover at a of the groupoid G. In particular, this gives the universal covering groupoid of a group. We now explain the generalisation of this notion to crossed complexes due to Howie [111].

Deﬁnition 11.1.1 A morphism p : Ce → C of crossed complexes is a covering morphism if

(i) the morphism p1 : Ce1 → C1 is a covering morphism of groupoids;

(ii) for each n ≥ 2 and xe ∈ Ce0, the morphism of groups pn : Cen(xe) → Cn(pxe) is an isomorphism.

In such case we call Ce a covering crossed complex of C. This definition may also be expressed in terms of the unique covering homotopy property similar to the one given for fibrations (Section 9.4. Actually, coverings are fibrations with discrete fibre. so we can use the long exact sequence of a fibration Theorem 9.4.2.

Proposition 11.1.2 Let p : Ce → C be a covering morphism of crossed complexes and let a˜ ∈ Ob(Ce). Let −1 e e a = pa,˜ and let K = p0 (a) ⊆ Ob(C). Then p induces isomorphisms πn(C, a˜) → πn(C, a) for n > 2 and a sequence

1 → π1(C,e a˜) → π1(C, a) → K → π0(Ce) → π0(C) which is exact in the sense of the exact sequence of a ﬁbration of groupoids.

The comment about exactness has to do with operations on the pointed sets, see Theorem 9.4.2. The following result gives a basic geometric example of a covering morphism of crossed complexes.

Theorem 11.1.3 Let X∗ and Y∗ be ﬁltered spaces and let

f : X → Y

−1 be a covering map of spaces such that for each n ≥ 0, fn : Xn → Yn is also a covering map with Xn = f (Yn). Then

Πf :ΠX∗ → ΠY∗ is a covering morphism of crossed complexes.

−1 Proof By a result of [28], Πf : π1X1 → π1Y1 is a covering morphism of groupoids. Since X0 = f (Y0), the restriction of π1f to π1(X1,X0) → π1(Y1,Y0) is also a covering morphism of groupoids. Now for each n ≥ 2 294

and for each x0 ∈ X0, f∗ : πn(Xn,Xn−1, x0) → πn(Yn,Yn−1, p(x0)) is an isomorphism (see for example, [112]). 2

Proposition 11.1.4 Let p : Ce → C be a covering morphism of crossed complexes. Then the induced morphism

π1(p): π1Ce → π1C is a covering morphism of groupoids.

Proof Letx ˜ ∈ Ce . We will show that p0 : St x˜ → St px˜ is bijective. Let [a] ∈ St px˜, where 0 x˜ π1Ce π1C π1C 0 a ∈ StC px˜. Since p is a covering morphism, there exists a uniquea ˜ of StCex˜ such that pa˜ = a. So px˜[˜a] = [a] 0 and thus px˜ is surjective. 0 0 ˜ ˜ −1 ˜ −1 Now suppose that px˜[˜a] = px˜[b]. Then (pb) pa˜ ∈ δC2(px˜) which implies that (pb) (pa˜) = δpc˜ for a unique ˜ −1 c˜ ∈ Ce2(˜x). Because p is a covering morphism, we need only show that (b) a˜ = δc˜. This follows by star 0 injectivity. Therefore px˜ is injective and so is bijective. Hence π1(p) is a covering morphism of groupoids. 2

Here is an important method of constructing new covering morphisms. Let C be a crossed complex. We write CrsCov/C for the full subcategory of the slice category Crs/C whose objects are the covering morphisms of C.

Proposition 11.1.5 Suppose given a pullback diagram of crossed complexes

f Ce / Ee

q q C / E f in which q is a covering morphism. Then q is a covering morphism.

Proof 2 The groupoid case is done in [9.7.6] of Brown’s book [28]. See also the paper by Brown, Heath and Kamps [36] for uses of pullbacks of covering morphisms of groupoids. Our next result is the analogue for covering morphisms of crossed complexes of a classical result for covering maps of spaces (see, for example, [9.6.1] of Brown’s book [28]).

Theorem 11.1.6 If C is a crossed complex, then the functor π1 : Crs → Gpds induces an equivalence of categories 0 π1 : CrsCov/C → GpdsCov/(π1C).

Proof If p : Ce → C is a covering morphism of crossed complexes, then π1p : π1Ce → π1C is a covering 0 morphism of groupoids, by Proposition 11.1.4. Since π1 is a functor, we also obtain the functor π1. To prove 0 π1 is an equivalence of categories, we construct a functor ρ : GpdsCov/(π1C) → CrsCov/C and prove that there 0 0 are equivalences of functors 1 ' ρπ1 and 1 ' π1ρ.

Let C be a crossed complex, and let q : D → π1C be a covering morphism of groupoids. Let Ce be given by 295 the pullback diagram in the category of crossed complexes:

φ Ce / iD

q q / C iπ1C φ

By proposition 11.1.5, q : Ce → C is a covering morphism of crossed complexes. We deﬁne the functor ρ by ρ(q) = q, and extend ρ in the obvious way to morphisms. 0 The natural transformation π1ρ ' 1 is deﬁned on a covering morphism q : D → π1C to be the composite morphism π1(φ) ∼ λ : π1(Ce) −→ π1(iD) = D where φ : Ce → iD is given in the pullback diagram. The proof that λ is an isomorphism is simple and is left to the reader. 0 To prove that 1 ' ρπ1, we show that the following diagram is a pullback:

e φ / Ce iπ1Ce

0 q q =iπ1(q) / C iπ1C φ

This is clear in dimension 0 and in dimensions > 2. For the case of dimension 1, let c : x → y in C, and 0 e 0 [˜c] ∈ (π1Ce)(˜x, y˜) be such that q[˜c] = φ(c). Then there exists a uniquec ˜ :x ˜ → y˜ such that φ(˜c ) = [˜c] and 0 q(˜c ) = c. Now, q(˜cδCe2(˜x)) = φ(c) = cδC2(x). This implies that (qc˜)δC2(x) = cδC2(x). So q(˜c) = c(δc2) for −1 some c2 ∈ C2(x). Therefore there exists a uniquec ˜2 ∈ Ce2(˜x) covering c2, and q(˜c(δc˜2) ) = c. So the above 0 diagram is a pullback and thus we have proved that 1 ' ρπ1. This proves the equivalence of the two categories. 2

11.2 Coverings of free crossed complexes

Recall that the importance of a free crossed complex is because if C is a free crossed complex on X∗, then a morphism f : C → D can be constructed inductively provided one is given the values fnx ∈ Dn, x ∈ Xn, n > 0 α α provided the following geometric conditions are satisﬁed: (i) δ f1x = f0δ x, x ∈ X1, α = 0, 1; (ii) βfn(x) = f0(βx), x ∈ Xn, n > 2; (iii) δnfn(x) = fn−1δn(x), x ∈ Xn, n > 2.

Notice that in (iii), fn−1 has to be deﬁned on all of Cn−1 before this condition can be veriﬁed. We now show that freeness can be lifted to covering crossed complexes, using the following result of Howie ([111, Theorem 5.1]).

Theorem 11.2.1 Let p : A → B be a morphism of crossed complexes. Then p is a ﬁbration if and only if the pullback functor p∗ : Crs/B → Crs/A has a right adjoint.

As a consequence we get the following. 296

Corollary 11.2.2 If p : A → B is a covering morphism of crossed complexes, then p∗ : Crs/B → Crs/A preserves all colimits.

We shall use this last result to prove that coverings of free crossed complexes are free.

Theorem 11.2.3 Suppose given a pullback square of crossed complexes

j Ae / Ce

p0 p A / C j in which p is a covering morphism and j : A → C is relatively free. Then j : Ae → Ce is relatively free.

Proof We suppose given the sequence of pushout diagrams ` S(m − 1) / n−1 λ∈Λn λ C

` E(m ) / n λ∈Λn λ C . deﬁning C as relatively free. Let Cˆn = p−1(Cn). By corollary 11.2.2, the following diagram is also a pushout:

¡` ¢ p∗ S(m − 1) / ˆn−1 λ∈Λn λ C

¡` ¢ p∗ E(m ) // ˆn λ∈Λn λ C .

¡` ¢ ` Since p is a covering morphism, we can write p∗ E(m ) as E(m ) for a suitable Λe . This λ∈Λn λ λ∈Λen λ n completes the proof. 2

Corollary 11.2.4 Let p : Ce → C be a covering morphism of crossed complexes. If C is free on X∗, then Ce is −1 free on p (X∗).

A similar result to Corollary 11.2.4 applies in the m-truncated case. The signiﬁcance of these results is as follows. We start with an m-truncated free crossed resolution C of a group G, so that we are given φ : C1 → G, and C is free on X∗, where Xn is deﬁned only for n 6 m. Our extension process of section 11.3 will start by constructing the universal cover p : Ce → C of C; this is the covering crossed complex corresponding to the universal covering groupoid p0 : Ge → G. By the results above, −1 Ce is the free crossed complex on p (X∗). It also follows from Proposition 11.1.2 that the induced morphism φe : Ce → Ge makes Ce a free crossed resolution of the contractible groupoid Ge. Hence Ce is an acyclic and hence, since it is free, also a contractible crossed complex. 297

11.3 A computational procedure

The initial motivation for this work was to determine in an algorithmic mode generators and relations for the G-module π(P) of identities among relations for a presentation P = hX|ωi of a groupoid G. Here ω : R → F (X) is a map where we regard R as a set disjoint from F (X). Recall from Chapter 3 that a presentation P gives the beginning of a crossed resolution

φ (11.3.1) C(R) −→δ2 F (X) −→ G where δ2 is the free crossed module associated to ω. Then π(P) is Ker δ2. The advantages of using the function ω are (i) to allow for the possibility of repeated relations, and (ii) to distinguish between an element r ∈ R and the corresponding element w(r) ∈ F (X). The elements of C(R) are ‘formal consequences’ Yn εi ui c = (ri ) i=1

ε u −1 ε where n > 0, ri ∈ R, εi = ±1, ui ∈ F (X), δ2(r ) = u (ωr) u, subject to the crossed module rule ab = baδ2b, a, b ∈ C(R).

Remark 11.3.1 It follows from the van Kampen Theorem for crossed modules that π(P) is given geometrically as π2(K(P)), the second homotopy group of the cell complex of the presentation. This result is not necessary for the work of this chapter, but it does emphasise the topological importance of our methods.

Actually 11.3.1 is equivalent to the more general situation

F2 → F1 → G where F1 is a free groupoid, F2 is a free crossed module over F1, G is a groupoid and φ induces an isomorphism ∼ Coker δ2 = G because the free generators of F1 have to map in a set of generators X of G so F1 = F (X) and F2 has to be free on some map ω : R → F1 = F (X). Thus, we want to extend 11.3.1 to a crossed resolution of G. To do this we require algebraic analogues of methods of covering spaces developed in the preceding Sections. We are following the method outlined in the introduction. 1.- The beginning of the process. First we construct a covering of part of diagram 11.3.1 getting

φe F (Xe) / Ge

(11.3.2) p1 p0 C(R) / F (X) / G δ2 φ where

1. The morphism p0 : Ge → G is the universal covering groupoid of the group G. The objects of Ge are the elements of G, and an arrow of Ge is a pair (g, g0) ∈ G × G with source g = δ0(g, g0) and target 0 1 0 0 0 gg = δ (g, g ). The projection morphism p0 is given by (g, g ) 7→ g . 298

2. The space Xˆ is the Cayley graph of the pair (G, X). Its objects are the elements of G and its arrows are pairs (g, x) ∈ G × X with source g = δ0(g, x) and target g(φx) = δ1(g, x), also written β(g, x). Notice that we shall use Xˆ because in general is not the universal covering of X. 3. The space F (Xˆ) is the free groupoid on Xˆ. Its objects are the elements of G and its arrows are pairs (g, u) ∈ G × F (X) with source g and target g(φu). We also write β(g, u) = g(φu). The multiplication is given by (g, u)(g(φu), v) = (g, uv). ˆ 4. The morphism φ is given by (g, u) 7→ (g, φu). The morphism p1 is given by (g, u) 7→ u. It maps the object group F (Xˆ)(1) isomorphically to N = Ker φ.

As we have seen in Section 11.1, Ge → G is the covering morphism corresponding to the trivial subgroup of G, and F (Xˆ) → F (X) is the covering morphism corresponding to the subgroup N of F (X). 2a.- Covering up to dimension 2. Next step is to complete diagram 11.3.2 to give ˆ ˆ δ2 φ C(Rˆ) / F (Xˆ) / Gˆ

(11.3.3) p2 p1 p0 C(R) / F (X) / G δ2 φ ˆ where Rˆ = G×R and δ2 : C(Rˆ) → F (Xˆ) is the free crossed F (Xˆ)-module onω ˆ : Rˆ → F (Xˆ), (g, r) 7→ (g, ω(r)).

Then C(Rˆ) is the disjoint union of groups C(Rˆ)(g), g ∈ G, all mapped by p2 isomorphically to C(R). Elements of C(Rˆ)(g) are pairs (g, c) ∈ {g} × C(R), with multiplication (g, c)(g, c0) = (g, cc0). The action of (g,u) u ˆ F (Xˆ) is given by (g, c) = (g(φu), c ). The boundary δ2 is given by (g, c) 7→ (g, δ2c). The morphism p2 : C(Rˆ) → C(R) is given by (g, c) 7→ c. The elements of C(Rˆ)(g) are also all ‘formal consequences’ Yn Yn Yn εi (gi,ui) εi ui εi ui (g, c) = ((gi, ri) ) = (g, (ri ) ) = (g, (ri ) ) i=1 i=1 i=1 where n > 0, ri ∈ R, εi = ±1, ui ∈ F (X), gi ∈ G, gi(φui) = g, subject to the crossed module rule ab = ˆ baδ2b, a, b ∈ C(Rˆ). Here the first form of the product is useful geometrically, and the last is useful computationally. In effect, we are giving first a presentation hXˆ|ωî of the groupoid Ge [104], and second the free crossed module corresponding to this presentation. The proof that the construction does give the free crossed module as claimed is given in theorem 11.2.3. 2b. Contractibility of the covering up to dimension 1. So we have started the construction of a crossed complex that we want to be acyclic. To prove it we construct a contracting homotopy at the same time we are constructing the crossed complex. Let us consider the contracting homotopy up to this point.

δˆ φˆ ˆ 2 / ˆ / C(R) F (X) v G ss vv ss vv (11.3.4) 1 ss 1 vv 1 ss vv ys h1 {v h0 C(Rˆ) / F (Xˆ) / G ˆ ˆ δ2 φ 299

−1 To get h0, choose a section σ : G → F (X) of φ such that σ(1) = 1, and write σ(g) = σ(g) . Then σ determines a function

h0 : G → F (Xˆ) by g 7→ (g, σg). Thus h0(g) is a path g → 1 in the Cayley graph Xˆ.

Remark 11.3.2 Computationally, such a choice σg is a ‘normal form’ for the element g of G; even for a finite presentation, σ cannot always be found by a finite algorithm, and the usual way of finding it is by a ‘rewriting’ process, which may not complete in finite time.

The choice of h0 is often, but not always, made by choosing a maximal tree in the graph Xˆ – such a choice is equivalent to a choice of Schreier transversal for the subgroup N = Ker φ of F (X).

g g0 = g(φx) ¡ (g, x) ¡ ¡ Z Z Z h0(g)

X XXX 1

−1 Now we deﬁne h1. For each arrow (g, x) of Xˆ the element ρ(g, x) = (h0g) (g, x)h0(g(φx)) is a loop at 1 in ˆ F (Xˆ) and so is in the image of δ2. For each arrow (g, x) of Xˆ choose an element h1(g, x) ∈ C(Rˆ)(1) such that ˆ δ2(h1(g, x)) = ρ(g, x).

Then h1 extends uniquely to a morphism

h1 : F (Xˆ) → C(Rˆ)(1) such that ˆ −1 δ2(h1(g, u)) = (h0g) (g, u)h0(g(φu)) for all arrows (g, u) of F (Xˆ). ˆ It follows that δ2h1(h0(g)) = (1, 1) for all g ∈ G. Further, if h0 is determined by a choice of maximal tree T in the Cayley graph, then for each (g, x) in T we may choose h1(g, x) = (1, 1).

Remark 11.3.3 The choice of h1 is equivalent to choosing a representation as a consequence of the relations R for each element of N, given as a word in the elements of X. There is no algorithm for such a choice. It will be shown in [103] how a ‘logged Knuth-Bendix procedure’ will give such a choice when the monoid rewrite system determined by R may be completed, and that this allows for an implementation of the determination of h1.

g (g, x) g0 = g(φx) ¡ A ¡ A ¡ h1(g, x) A Z Z 0 h0(g)Z h0(g )

¢ X ¢ XXX ¢ 1 300

This gives all maps represented in diagram 11.3.3 that give a contracting homotopy up to dimension 2. We now extend everything to dimension 3. 3a. Covering up to dimension 3. Let I be a set in one-to-one correspondence with G × R with elements written [g, r], g ∈ G, r ∈ R. Let

C3(I) = FG(I) be the free G-module on I. For any [g, r] ∈ I we deﬁne

¡ −1¢ σg δ3[g, r] = p2 (h1(g, w(r))) r that extends uniquely to

δ3 : C3(I) → C(R).

It follows from equation the deﬁnition of δ2 that δ2δ3[g, r] = 1, and so the given values δ3[g, r] lie in the G-module π(P). So we have got a truncated crossed complex

δ3 δ2 φ (11.3.5) C3(I) / C(R) / F (X) / G

Let us see to the covering. We deﬁne ˆ ˆ C3(I) = FGe(I) to be the free Ge-module on Iˆ = G × I. Then C3(Iˆ) is the disjoint union of abelian groups C(Iˆ)(g), g ∈ G, all mapped by p3 isomorphically to C3(I). Elements of C3(Iˆ)(g) are pairs (g, i) ∈ {g} × C3(I) with addition 0 0 0 0 0 (g, i) + (g, i ) = (g, i + i ). The action of Ge on C3(Iˆ) is given by (g, i).(g, g ) = (gg , i.g ). ˆ ˆ Let δ3 : C3(Iˆ) → C(Rˆ) be the Ge-morphism given by δ3(g, d) = (g, δ3d), d ∈ C3(I). With these deﬁnitions we have got the diagram

δˆ δˆ φˆ ˆ 3 / ˆ 2 / ˆ / C3(I) C(R) F (X) Ge

(11.3.6) p3 p2 p1 p0 C3(I) / C(R) / F (X) / G δ3 δ2 φ

where the upper row is acyclic up to dimension 1. 3b. Contractibility of the covering up to dimension 2.

To go a dimension higher we define h2 : C(Rˆ) → C3(Iˆ)(1) be the groupoid morphism killing the operation of F (Xˆ) i.e. it is defined (g,u) h2((g, c) ) = h2(g, c) for all (g, c) ∈ C(Rˆ), u ∈ F (X)) and satisfies (g, r) 7→ (1, [g, r]), (g, r) ∈ G × R. ˆ Then from the definition of δ3 we deduce that

ˆ ˆ −1 σg δ3h2(g, c) = (h1δ2(g, c)) (1, c ) for all g ∈ G, c ∈ C(R) and we have got a contracting homotopy up to dimension 2 301

δˆ δˆ φˆ ˆ 3 / ˆ 2 / ˆ / C3(I) C(R) F (X) v G ss ss vv ss ss vv (11.3.7) 1 ss 1 ss 1 vv 1 ss ss vv ys h2 ys h1 {v h0 C (Iˆ) / C(Rˆ) / F (Xˆ) / G 3 ˆ ˆ ˆ δ3 δ2 φ

This level of the construction allows us to extract some consequences on the presentation. In particular

Theorem 11.3.4 The module π(P) is generated as G-module by elements

−1 (σg)−1 δ3[g, r] = (k1(g, ωr)) r for all g ∈ G, r ∈ R, where (i) σ : G → F (X) is a section of the quotient mapping φ : F (X) → G, (ii) k1 is a morphism F (Xˆ) → C(R) from the free groupoid on Xˆ, the Cayley graph of the presentation, to the free crossed −1 module of the presentation, such that δ2k1(g, x) = (σg)x(σ(g(φx))) , for all x ∈ X, g ∈ G.

ˆ ˆ ˆ Proof Since h2 and h1 give a contracting homotopy, we have δ2δ3 = 0, and so the elements p2(δ3h2(g, c)) do ˆ give identities. On the other hand, if c ∈ C(R) and δ2c = 1, then (1, c) = δ3h2(1, c), and so c = δ3(d) for some d. Theorem follows just taking k1 = p2h1. 2

The identities δ3[g, r] may be seen as separation elements in the geometry of the Cayley graph with relators. It is easy to see from properties (i), (ii) and the ﬁrst crossed module rule that these elements all lie in π(P). Thus the main feature of the theorem is that these elements generate this module. For each g ∈ G, r ∈ R we have a 2-cell (g, r) in the Cayley graph with relations, where the boundary of (g, r) in F (Xˆ) is (g, ωr).

aa aa D (g, r) (g, y)D gD (g, x) g0 = g(φx)

Now we put all the choices for all edges of (g, r) to give with (g, r) a ‘separation element’, representing a

2-sphere, with one face (g, r) and the other faces all the h1(g, x), as partially shown in the following picture:

aa aa g0(φz) D (g, r) aa D (g, y) aa D aa g g0

eh1(g, y) ¡ (g, x) A e ¡ A h (g0, z) 1 e ¡ h1(g, x) A ¥ Z ¥ Z 0 ¥ h0(g)Z h0(g ) ¥ ¢ Q Q X ¢ Q XXX ¢ Q Q 1 Q 302

4.a Dimension 4 and higher

However some of the elements of δ3(I) may be trivial, and others may depend ZG-linearly on a smaller subset. That is, there may be a proper subset J of I such that δ3(J) also generates the module π(P). Then for each element i ∈ I \ J there is a formula expressing δ3i as a ZG-linear combination of the elements of δ3(J). These formulae determine a ZG-retraction r : C3(I) → C3(J) such that for all d ∈ C3(I), δ3(rd) = δ3(d). So we replace I in the above diagram by J, replacing the boundaries by their restrictions. Further, and this is the 0 0 0 ˆ ˆ crucial step, we replace h2 by h2 = r h2 where r : C3(I)(1) → C3(J)(1) is mapped by p3 to r. 0 ˆ ˆ This h2 : C(R) → C3(J)(1) is now used to continue the above construction. We deﬁne C4(J) to be the free G-module on elements written [g, d] ∈ J = G × J, with

0 −1 δ4[g, d] = −p3(h2(g, δ3d)) + d.g .

These boundary elements give generators for the relations among the generators δ3(J) of π(P). This can also be applied to presentations of groupoids.

Theorem 11.3.5 A G-module generating set of relations among these generators δ3(J) of π(P) is given by

−1 δ4[g, γ] = −k2(g, δ3γ) + γ.g ˆ for all g ∈ G, γ ∈ J, where k2 : C(Rˆ) → C3(J) is a morphism from the free crossed F (Xˆ)-module on δ2 : G × R → F (Xˆ) such that k2 kills the operation of F (Xˆ) and is determined by a choice of writing the generators δ3[g, r] ∈ δ3(I) for π(P) in terms of the elements of δ3(J).

Proof This is a similar argument to the proof of Theorem 11.3.4, using the deﬁnition of δ4 and setting 0 k2 = p3h2. 2

From here onwards we proceed as in the Introduction.

Remark 11.3.6 In the above we have defined morphisms and homotopies by their values on certain generators, 0 and so it is important for this that the structures be free. For example, h2 is defined by its values on the ˆ 0 u v elements (g, r) ∈ G × R. So, noting that h2 kills the operation of F (X), we calculate for example h2(g, r s ) = 0 −1 0 −1 h2(g(φu) , r) + h2(g(φv) , s). In this way the formulae reflect the choices made at different parts of the Cayley graph in order to obtain a contraction.

Remark 11.3.7 The determination of minimal subsets J of I such that δ3J also generates π(P) is again not straightforward. Some dependencies are easy to find, and others are not. A basic result due to Whitehead [175] is that the abelianisation map C(R) → (ZG)R maps π(P) isomorphically to the kernel of the Whitehead-Fox derivative (∂r/∂x):(ZG)R → (ZG)X . Hence we can test for dependency among identities by passing to the free ZG-module (ZG)R, and we use this in the next section. For bigger examples, this testing can be a formidable task by hand. An implementation of Gröbnerbasis procedures for finding minimal subsets which still generate is described in [102].

11.4 Examples

11.4.1 The standard crossed resolution of a group

The standard crossed resolution of a group was deﬁned by Huebschmann in [114] and applied also in for example [55, 168]. Here we show how this resolution arises from our procedure. 303

We start with a group G and let C1 = F (G), the free group on the set G, with generators written [a], a ∈ G. Let φ : C1 → G be the canonical morphism. This has a section σ : G → F (G), a 7→ [a], a 6= 1, 1 7→ 1. This −1 deﬁnes h0 : G → Ce1, a 7→ (a, [a] ). −1 −1 The Cayley graph of this presentation has arrows (a, [b]) : a → ab so that h0(a) (a, [b])h0(ab) = (1, [a][b][ab] ). So we may take C2 to be the free crossed C1-module on elements [a, b] and deﬁne δ2 : C2 → C1 by

−1 δ2[a, b] = [a][b][ab] .

Then in the universal cover we can deﬁne h1 : Ce1 → Ce2(1) by (a, [b]) 7→ (1, [a, b]).

Theorem 11.4.1 There is a free crossed C∗(G) resolution of a group G in which Cn(G) is free on the set n G with generators written [a1, a2, . . . , an], ai ∈ G, with contracting homotopy of the universal cover given by (a, [a1, a2, . . . , an]) 7→ (1, [a, a1, a2, . . . , an]), and boundary δn : Cn(G) → Cn−1(G) given by δ2 as above,

−1 −1 [a]−1 δ3[a, b, c] = [a, bc][ab, c] [a, b] [b, c] , and for n > 4

nX−1 a−1 i (11.4.1) δn[a1, a2, . . . , an] = [a2, . . . , an] 1 + (−1) [a1, a2, . . . , ai−1, aiai+1, ai+2, . . . , an]+ i=1 n + (−1) [a1, a2, . . . , an−1].

Proof We ﬁrst verify

ˆ ˆ −1 (a,[a]−1) δ3h2(a, [b, c]) = h1δ2(a, [b, c]) (a, [b, c]) −1 −1 [a]−1 = h1(1, a, [b][c][bc] ) (1, [b, c] ) −1 −1 [a]−1 = h1((a, [b])(ab, [c])(abc, [bc] )) (1, [b, c] ) −1 −1 [a]−1 = h1((a, [b])(ab, [c])(a, [bc]) ) (1, [b, c] ) −1 = (1, [a, bc][ab, c]−1[a, b]−1[b, c][a] ).

In order to have a contracting homotopy we require for n > 3

ˆ δn+1hn(a1, [a2, . . . , an+1]) −1 ˆ a1 = −hn−1δn(a1, [a2, . . . , an+1]) + (1, [a2, . . . , an+1] ) Xn a−1 i n+1 = (1, [a2, . . . , an+1] 1 + (−1) [a1, . . . , aiai+1, . . . , an+1] + (−1) [a1, a2, . . . , an]). i=1

This completes the proof that the family hn give a contracting homotopy and so that C∗(G) is a resolution. 2

11.4.2 A small crossed resolution of ﬁnite cyclic groups

This is the resolution ﬁrst given by Brown and Wensley in [60]. Here we shall prove that it is a resolution describing its universal cover and a contracting homotopy. 304

We write C∞ for the (multiplicative) inﬁnite cyclic group with generator x, and Cr for the ﬁnite cyclic group of order r with generator t. Let φ : C∞ → Cr be the morphism sending x to t. We show how the inductive procedure given earlier recovers the small free crossed resolution of Cr together with a contracting homotopy of the universal cover. e ˆ Let p0 : Cr → Cr be the universal covering morphism, and let p1 : C∞ → C∞ be the induced cover of C∞. ˆ Then C∞ is the free groupoid on the Cayley graph Xˆ pictured as follows:

(tn−1,x)

t 1 / t / t2 / / tn−2 / tn−1 (1,x) (t,x) (t2,x) (tn−2,x) A section

σ : Cr → C of φ is given by ti 7→ xi, i = 0, . . . , r − 1, and this deﬁnes

h0 : Cr → Fe1 by ti 7→ (ti, x−i). It follows that for i = 0, . . . , r − 1 we have ( i −1 i i+1 (1, 1) if i 6= r − 1, h0(t ) (t , x)h0(t ) = (1, xr) if i = r − 1.

So we take a new generator x2 for F2 with δ2x2 = xr and set ( i (1, 1) if i 6= r − 1, h1(t , x) = (1, x2) if i = r − 1.

Then for all i = 0, . . . , r − 1 we have

e i i −1 i i+1 δ2h1(t , x) = h0(t ) (t , x)h0(t ), and it follows that

i r i i+1 i+r−1 h1(t , x ) = h1((t , x)(t , x) ... (t , x))

(11.4.2) = (1, x2).

Hence

e i i −i −i −h1δ2(t , x2) + (t , x2).x = (1, −x2) + (1, x2.t ) r−i = (1, x2.(t − 1))

i−1 r−i This gives us 0 for i = 0, and (1, x2.(t − 1)) for i = r − 1. Let N(i) = 1 + t + ··· + t , so that t − 1 = (t − 1)N(r − i) for i = 1, . . . , r − 1. Hence we can take a new generator x3 for F3 with δ3x3 = x2.(t − 1) and deﬁne ( i (1, 0) if i = 0, h2(t , x2) = (1, x3.N(r − i)) if 0 < i 6 r − 1. Now we ﬁnd that if we evaluate

e i −i i−1 i −i −h2δ2(t , x3) + (1, x3.t ) = −h2((t , x2).t + (t , x2)) + (1, x3.t ) 305 we obtain for i = 0 r−1 −h2(t , x2) + (1, x3) = (1, 0), for i = 1 r−1 r−1 0 + h2(t, x2) + (1, x3.t ) = (1, x3.(N(r − 1) + t )) = (1, x3.N(r)) and otherwise r−i (1, x3(−N(r − i + 1) + N(r − i) + t )) = (1, 0).

Thus we take a new generator x4 for F4 with δ4x4 = x3.N(r) and ( i (1, x4) if i = 1, h3(t , x3) = (1, 0) otherwise.

Then

e i −i i −i −h3δ4(t , x4) + (1, x4.t ) = −h3(t , x3.N(r)) + (1, x4.t ) −i −i = −h3(1, x3.N(r).t ) + (1, x4.t ) r−i = (1, x4.(t − 1)).

Thus we are now in a periodic situation and we have the theorem:

Theorem 11.4.2 A free crossed resolution F∗ of Cr may be taken to have single free generators xn in dimension n > 1 with φ(x1) = t, and   r x1 if n = 2, δ (x ) = x .(t − 1) if n > 1, n odd, n n  n−1  xn−1.N(r) if n > 2, n even. 306 Chapter 12

Crossed complexes and chain complexes with operators

Chain complexes with a group of operators are a well known tool in algebraic topology, where they arise naturally as the chain complex C∗X˜ of cellular chains of the universal cover X˜ of a reduced CW -complex X. The group of operators here is the fundamental group of X.

J. H. C. Whitehead in his classical but little read paper [175] showed that the chain complex C∗(X˜) is useful for the homotopy classiﬁcation of maps between non-simply connected spaces (see below). His methods must have seemed at the time to be circuitous. In modern parlance, he introduced the categories CW of CW - complexes, HS of homotopy systems, and FCC of free chain complexes with operators, together with functors

ρ CW −→ HS −→C FCC.

In each of these categories he introduced notions of homotopy and he proved that C induces an equivalence of the homotopy category of HS with a subcategory of the homotopy category of FCC. He also showed that if X and Y are reduced CW -complexes such that dimX ≤ n and πiY = 0 for 2 ≤ i ≤ n − 1, then ρ induces a bijection of homotopy classes [X,Y ] → [ρX, ρY ]. Further, CρX is isomorphic to the chain complex C∗X˜ of cellular chains of the universal cover of X, so that under these circumstances there is a bijection of sets of homotopy classes

[X,Y ] → [C∗X,C˜ ∗Y˜ ]. This result can be interpreted as an operator version of the Hopf classification theorem. It is surprisingly little known. It includes results of Olum [150] and Schellenberg [160] published later, and it enables quite useful calculations to be done easily, such as the homotopy classification of maps from a surface to the projective plane done by Ellis [83]. A recent account of Whitehead’s work is in Baues book [10]. However, our aim is somewhat different. In the first place we eliminate from Whitehead’s account both the freeness assumptions, and the assumptions of only one vertex. So we consider categories FTop of filtered spaces, Crs of crossed complexes, and Chn of chain complexes with a groupoid as operators. We construct functors

FTop −→Π Crs −→∆ Chn and show that for a class of ﬁltered spaces X, ∆ΠX does give the chain complex of universal covers of X. We also show that ∆ has a right adjoint Θ (which it does not have in the case of only one vertex); these functors

307 308 are related to some well known tools in the homology of groups, such as relation modules, Alexander modules, and derived modules. Notions of homotopy and higher homotopy have been analysed for the categories FTop and Crs in Section ??. We give a similar analysis for the category Chn and discuss the homotopy-preserving properties of ∆. This enables us to give a more general version of Whitehead’s results. Thus our results and methods give a clearer picture of the relation between Whitehead’s work and standard methods of homological algebra. One advantage of our approach is that, because crossed complexes appear in a variety of algebraic situations (cf. Lue article [127], and the survey article by Brown [26]) one can expect analogues of these methods to form part of the general machinery of non-Abelian homological and homotopical algebra. In fact, applications of these methods have been found by Porter both in commutative algebra [154] and, in collaboration with Korkes, in the theory of proﬁnite groups [121].

12.1 Modules and chain complexes over groupoids

We have seen that the symmetric monoidal closed structure on the category Crs of crossed complexes, constructed in Section 8.2 from tensor products and homotopies, relies crucially on the consideration of crossed complexes over groupoids as well as over groups. The same is true for chain complexes with operators that we study in this section as an introduction. There are well known deﬁnitions of tensor product and internal hom functor for chain complexes of Abelian groups (without operators). If one allows operators from arbitrary groups the tensor product is easily generalised (the tensor product of a G-module and an H-module being a (G × H)-module) but the adjoint construction of internal hom functor does not exist, basically because the group morphisms from G to H do not form a group. To rectify this situation we allow operators from arbitrary groupoids and we start with a discussion of the monoidal closed category structure of Mod the category of modules over groupoids given in Deﬁnition 7.2.7. As is customary, we write M for the G-module (M,G) when the operating groupoid G is clear from the context. Also, to simplify notation, we will assume throughout this chapter that the Abelian groups M(p) for p ∈ G0 are all disjoint; any G-module is isomorphic to one of this type. Perhaps it is more natural in this case to start describing the internal hom functor in Mod.

Deﬁnition 12.1.1 Let (M,G), (N,H) be modules, to construct the internal hom MOD((M,G), (N,H)) we consider the set of morphisms of modules

Mod((M,G), (N,H)) = {(θ, f):(M,G) → (N; H) | (θ, f) is a morphism of modules} and we have to give this set the structure of module over a groupoid. We shall use the internal hom groupoid GPDS(G, H) described in the Appendix B. Recall that its objects are functors f : G → H and its morphisms are natural transformations φ : f → f 0. For a ﬁxed functor f : G → H, we deﬁne the set of morphisms of groupoids over f, i.e.

Modf ((M,G), (N,H)) = {(θ, f):(M,G) → (N; H) | (θ, f) is a morphism of modules} ⊆ Mod((M,G), (N,H)).

It is easy to see that the morphisms Modf ((M,G), (N,H)) form an Abelian group under element-wise addition, so all morphisms Mod((M,G), (N,H)) form a family of Abelian groups indexed by the set of objects of the groupoid GPDS(G, H). [ MOD((M,G), (N,H)) = {Modf ((M,G), (N,H))}f∈Gpds(G,H). 309

It remains to describe the action, i.e. for each f, g ∈ Gpds(G, H) we need an action

Modf ((M,G), (N,H)) × GPDS(G, H)(f, g) → Modg((M,G), (N,H))

So for each θ such that (θ, f) is a morphism of groupoids and each natural transformation φ : f → g. This groupoid, acts on the morphisms M → N we deﬁne

(θ, f)φ = (θφ, g)

φ where θp : Mp → Ng(p) is the composition of θp : Mp → Nf(p) and φ(p): Nf(p) → Ng(p). It is not diﬃcult to prove that this deﬁnition gives an action giving a structure of module

MOD(M,N) = (Mod((M,G), (N,H)), GPDS(G, H)) that is the internal hom functor in Mod.

It is not diﬃcult to see that, as in the group case, we can characterize the elements of this internal hom functor in terms of ‘bilinear’ maps

Deﬁnition 12.1.2 A bilinear map of modules (M,G) × (N,H) → (P,K) is given by a couple of maps (θ, f) where f : G × H → K is a map of groupoids and θ : M × N → P is given by a family of bilinear maps

M(p) × N(q) → Pf(p,q) and they preserve actions, i.e.

θ(xa, yb) = θ(x, y)f(a,b).

Proposition 12.1.3 There is a natural bijection between bilinear maps M ×N → P and morphisms of modules from M to MOD(N,P )).

Proof Let us consider an element (θ, f) ∈ Mod(M, MOD(N,P )) then we can deﬁne

fˆ(p, q) = f(p)(q) and θˆ(x, y) = θ(x)(y).

It is easy to see that (θ,ˆ fˆ) is a bilinear map ant that this assignation is a natural bijection. 2

Now the tensor product is just deﬁned as the one that transforms these bilinear maps in morphisms of modules.

Deﬁnition 12.1.4 The tensor product in Mod of modules (M,G), (N,H) is the module

(M ⊗ N,G × H) where, for p ∈ G0, q ∈ H0,M ⊗ N(p, q) = M(p) ⊗Z N(q) and the action is given by

(x ⊗ y)(a,b) = xa ⊗ yb.

Proposition 12.1.5 There is a natural bijection between bilinear maps M ×N → P and morphisms of modules from M ⊗ N to P .

Proof Let us consider a bilinear map (θ, f): M × N → P then we can deﬁne

fˆ(p, q) = f(p)(q) and θˆ(x ⊗ y) = θ(x, y). 310

It is easy to see that (θ,ˆ fˆ) is a morphism of modules ant that this assignation is a natural bijection. 2

Proposition 12.1.6 There is a natural bijection

MOD(L ⊗ M,N) =∼ MOD(L, MOD(M,N)).

Proof It is straightforward to verify the natural bijection

Mod(L ⊗ M,N) ' Mod(L, MOD(M,N)), where L is a G-module. These families of groups are modules over GPDS(G×H,K) =∼ GPDS(G, GPDS(H,K)) and the actions agree, giving a natural isomorphism of modules

MOD(L ⊗ M,N) =∼ MOD(L, MOD(M,N)).

These ideas can be extended with little extra trouble to chain complexes over groupoids.

Deﬁnition 12.1.7 A chain complex C over G is a sequence

∂ ∂ ∂ ∂ ∂ C = ··· −→ Cn −→ Cn−1 −→ · · · −→ C1 −→ C0 of G-modules and G-morphisms satisfying ∂∂ = 0. A morphism f :(C,G) → (D,H) is a family of morphisms fn :(Cn,G) → (Dn,H) (over some f : G → H, independent of n) satisfying ∂fn = fn−1∂. These form a category Chn and, for a ﬁxed groupoid G, we have a subcategory ChnG of chain complexes over G.

Deﬁnition 12.1.8 The tensor product of chain complexes C, D over groupoids G, H respectively is the chain complex C ⊗ D over G × H where

(C ⊗ D)n = ⊕i+j=n(Ci ⊗ Dj).

Here, the direct sum of modules over a groupoid G is deﬁned by taking the direct sum of the Abelian groups at each object of G. The boundary map

∂ :(C ⊗ D)n → (C ⊗ D)n−1 is deﬁned on the generators a ⊗ b of (C ⊗ D)n by

∂(a ⊗ b) = ∂a ⊗ b + (−1)ia ⊗ ∂b, where a ∈ Ci, b ∈ Dj, i + j = n. 311

This tensor product clearly gives a symmetric monoidal structure to the category Chn, with unit object the complex C(Z, 0) = · · · → 0 → · · · → 0 → Z over the trivial group. The symmetry map C ⊗ D → D ⊗ C is given by

x ⊗ y 7→ (−1)ijy ⊗ x for x ∈ Ci, y ∈ Dj.

Deﬁnition 12.1.9 The internal hom functor CHN(−, −) is deﬁned as follows. Let C, D be chain complexes over the groupoids G, H respectively. As in the case of morphisms of modules, it is easy to see that the morphisms of chain complexes Chn(M,N) form an GPDS(G, H)-module. We write

S0 = Chn(M,N) for this module and take it as the 0-dimensional part of the chain complex S = CHN(C,D). The higher-dimensional elements of S are chain homotopies of various degrees. An i-fold chain homotopy

(i ≥ 1) from C to D is a pair (s, f) where s : C → D is a map of degree i (that is, a family of maps s : Cn → Dn+i for all n ≥ 0) which in each dimension is a morphism of modules over f : G → H . Again the i-fold homotopies

Si = {s : C → D | i-fold homotopies} have a structure of an GPDS(G, H)-module and we deﬁne the boundary map

∂ : Si → Si−1 (i ≥ 1) by (∂s)(x) = ∂(s(x)) + (−1)i+1s(∂x), the morphism f : G → H being the same for ∂s as for s. We observe that ∂s is of degree i − 1. Also ∂s commutes or anticommutes with ∂, namely

∂((∂s)(x)) = (−1)i+1(∂s)(∂x).

It follows that ∂∂ : Si → Si−2 is 0 for i ≥ 2. We deﬁne CHN(C,D) to be the chain complex

∂ CHN(C,D) = · · · −→ Si −→ Si−1 −→ · · · −→ S0 over F = GPDS(G, H).

Proposition 12.1.10 The functors ⊗ and CHN give Chn the structure of symmetric monoidal closed category.

Proof Again, if L is a chain complex over G, there is a natural bijection

Chn(L ⊗ C,D) =∼ Chn(L, CHN(C,D)) which extends to a natural isomorphism of chain complexes

CHN(L ⊗ C,D) =∼ CHN(L, CHN(C,D)) over GPDS(G × H,K) =∼ GPDS(G, GPDS(H,K)). 2 312

12.2 Augmentation modules

As we have seen in Section 7.2, a crossed complex is a type of non-Abelian chain complex with operators, the non-Abelian features being conﬁned to dimensions ≤ 2. We have seen also that the category Crs of crossed complexes is monoidal closed using the tensor product and the internal hom deﬁnes in Section 8.2. Our aim now is to construct ∆ : Crs → Chn which relates the two monoidal closed structures. That means ‘abelianising’ the crossed module part of a crossed complex. The basic constructions used to linearise the theory of groups in homological algebra are the group ring

ZG and augmentation module IG of a group G, and the derived module Dφ of a group morphism φ : H → G (usually appearing in the form Dφ = IH ⊗H ZG). These are the ingredients of ∆ and one advantage of working with the category Mod (which includes modules over all groups) is that one can exploit the formal properties of these functorial constructions. We ﬁrst extend them to the case of groupoids.

Deﬁnition 12.2.1 Let G be a groupoid. For q ∈ G0, we deﬁne

ZG(q) = Fab{g ∈ G | t(g) = q}, the free Abelian group on the elements of G with target q. Thus an element has the form of a ﬁnite sum Σnigi with ni ∈ Z and gi ∈ G with t(gi) = q. Clearly, ZG is a (right) G-module under the action

(a, g) 7→ ag of G on basis elements. Thus

ZG = {ZG(q)}q∈G0 is a G-module. it is called the groupoid ring of G. Indicate in which sense is ‘free on G’, i.e. it is freely generated as G-module by G0 (embedded in ZG as the set of identities of G),

There is also a generalization of the augmentation map and the augmentation ideal.

Deﬁnition 12.2.2 We deﬁne the G-groupoid that is going to play the role the integers play in homological algebra. Let Z be the (right) G-module consisting of the constant family Z(p) = Z for p ∈ G0, with trivial action of G. The augmentation map ε : ZG → Z, given by Σnigi 7→ Σni is a morphism of G-modules and its kernel IG is the (right) augmentation module of G. The group IG(q) has Z-basis consisting of all g − 1q, where g is a non-identity element of G with target q. Any morphism of groupoids φ : H → G induces a module morphism ZH → ZG over φ which maps IH to IG.

These constructions deﬁne functors Z(−),I : Gpds → Mod. We are going to prove that both of them preserve colimits by getting right adjoints for each of them. 313

Deﬁnition 12.2.3 Given a module (M,H), the semidirect product H n M of H and M is the groupoid with the same set of objects as H, and (H n M)(p, q) = H(p, q) × M(q.) The composition is given by (x, m)(y, n) = (xy, my + n), when x ∈ H(p, q), m ∈ M(q) and y ∈ H(q, r), n ∈ M(r).

Proposition 12.2.4 The functor (M,H) 7→ H n M is a right adjoint of I : Gpds → Mod. Hence I preserves colimits.

Proof Let us begin by studying Gpds(G, H n M) for a groupoid G. A morphism

G → H n M is of the form x 7→ (ψx, fx) where ψ : G → H is a morphism of groupoids and f : G → M is a ψ-derivation, that is, f maps G(p, q) to M(ψq) and satisﬁes

f(xy) = (fx)ψy + fy whenever xy is deﬁned in G. (In particular, all sections G → H n M are x 7→ (x, fx) where f : is a derivation.)

It is easy to prove that the map κ : G → IG, given by κ(x) = x−1q for x ∈ G(p, q), is a universal derivation, i.e. for every derivation f : G → N for a G-module N, there is a unique G-morphism

fˆ : IG → N such that f = fκˆ . (see [94] for the corresponding fact for categories). On the other hand, if ψ : G → H is a morphism of groupoids and M is an H-module, then any ψ-derivation f : G → M is uniquely of the form f = fκˆ where fˆ : IG → M is a morphism of modules over ψ. Thus we have a natural bijection Mod((IG, G), (M,H)) =∼ Gpds(G, H n M). 2

Deﬁnition 12.2.5 Let us recall a construction in the category of groupoids. Let G be a groupoid and f : X → ∗ G0 a map. The pullback groupoid f G is deﬁned as having X as set of objects and f ∗G = {(x, g, y) | x, y ∈ X, g ∈ G(fx, fy)} and composition (x, g, y)(y, h, z) = (x, gh, z) 314

It satisﬁes the obvious universal property. Let us use this construction in relation to the ring groupoid.

Deﬁnition 12.2.6 On the other hand, given a module (M,H), we consider M as set and the target map t : M → Ob H. We may therefore form the pull-back groupoid

P (M,H) = t∗H = {(m, h, n) | m, n ∈ UM, h ∈ H(βm, βn)} with Ob (β∗H) = UM and multiplication

(m, h, n)(n, k, p) = (m, hk, p).

(see Ehresmann [78], p. 245, Mackenzie [135], p. 11).

This groupoid, with its canonical morphism to H,(m, h, n) 7→ h, is universal for morphisms ψ : G → H of groupoids such that Ob ψ factors through maps β : M → H0.

Proposition 12.2.7 The functor P is a right adjoint of Z(−): Gpds → Mod. Hence Z(−) preserves colimits.

Proof By the deﬁnition of P (M,H), the groupoid morphisms G → P (M,H) are naturally bijective with pairs (α, ψ) where α : Ob G → UM is a map, ψ : G → H is a morphism and Ob ψ = β ◦ α.

However, since ZG is freely generated as G-module by G0 (embedded in ZG as the set of identities of G), such pairs (α, ψ) are naturally bijective with morphisms of modules (γ, ψ):(ZG, G) → (M,H). 2

Both adjointness are related as follows

Proposition 12.2.8 The natural transformation IG,→ ZG is conjugate to the natural transformation

θ = θ(M,H) : P (M,H) → H n M

h given by θ(m, h, n) = (h, m − n). For each module (M,H), this θ(M,H) is a covering morphism.

Proof Any commutative triangle

(IG, G) i / (ZG, G) JJ s JJ ss JJ ss JJ ss (α,ψ) J% yss (γ,ψ) (M,H) in Mod corresponds to a commutative triangle

θ H n M o P (M,H) cFF v: FF vv FF vv ξ FF vv η F vv G in Gpds, where θ is natural and, if g ∈ G(p, q), then

ξg = (ψg, α(g − 1q)) and ηg = (γ1p, ψg, γ1q). 315

Given (m, h, n) ∈ P (M,H), we may take G = H, ψ = id, and choose γ so that γ1p = m, γ1q = n. Then

θ(m, h, n) = ξh = (ψh, α(h − 1q))

= (h, γ(h − 1q))

= (h, γ(1ph) − γ1q) = (h, mh − n).

Finally, let (h, x) ∈ H n M, with h ∈ H(p, q) and x ∈ M(q), and let m ∈ M(p) be an object of P (M,H) lying over the source p of (h, x). Then there is a unique n ∈ M(q) such that mh − n = x. Hence there is a unique arrow (m, h, n) over (h, x) with source n. 2

Surely it is worth commenting that if one restricts attention to groups, and modules over groups, the restricted functor Z(−) does not have a right adjoint since, for example, it converts the initial object 1 in the category of groups to the module (Z, 1) which is not initial in the category of modules over groups. However, the functor I, when restricted to groups does have a right adjoint given by the split extension as above.

12.3 Derived modules

Deﬁnition 12.3.1 Let φ : H → G be a morphism of groupoids. Its derived module Dφ is the G-module constructed as follows: Let

F (q) = FG({x ∈ H | tφ(x) = q} , the free G-module on the family of sets of elements x of H such that φ(x) has target q. Then F (q) has an additive basis of pairs (x, g) such that φ(x)g is deﬁned in G, and the action of G is given by

g1 (x, g) = (x, gg1) when gg1 is deﬁned in G. There is a natural map i : H → F, given by i(x) = (x, 1q), where φ(x) has target q. Now, we impose on F the relations i(xy) = i(x)φ(y) + i(y) whenever xy is deﬁned in H we obtain a quotient G-module Dφ, a quotient morphism s : F → Dφ and a φ-derivation h = si : H → Dφ.

2 φ This produces a functor D : Gpds → Mod given by D(H −→ G) = (DφG).

This deﬁnition is an extension to groupoids of Crowell’s deﬁnition for groups [69].

Proposition 12.3.2 The φ-derivation h : H → Dφ is a universal φ-derivation. That is, every φ-derivation 0 0 f : H → M is uniquely of the form f = f h, where f : Dφ → M is a G-morphism.

Remark 12.3.3 Alternatively, regarding the category of G-modules as the functor category (Ab)G, any functor

M : H → Ab has a left Kan extension φ∗M : G → Ab along φ : H → G. Then the derived module Dφ is canonically isomorphic to φ∗(IH), the G-module induced from IH by φ : H → G. In the case of a group morphism φ, this induced module is just IH ⊗H ZG, where ZG is viewed as a left H-module via φ and left multiplication; however the construction is a little more subtle in the case of groupoids. 316

The adjointness property of the derived module follows from the universality.

Proposition 12.3.4 The functor D has a right adjoint Mod → Gpds2 given by

(M,K) 7→ (K n M −→π1 K).

Proof This is an immediate consequence of the adjointness of I and n seen in proposition (12.2.4) and the formula Dφ = φ∗(IH). 2

Compute some derived modules for further use

Example 12.3.5 1. The augmentation ideal is the derived module of the identity. 2. The derived module of the 0 : G → 0 map is the abelianisation Gab.

12.4 The functor ∆0 : Crs → Chn

We are now able to turn to the construction of the main functors used in the chapter. We want to associate a chain complex with a groupoid of operators to each crossed complex in such a way that we lose as little information as possible. Things are quite clear in dimensions higher than 2 since both deﬁnitions agree. Thus we shall leave everything as it is. We are left with δ2 δ2 C3 −→ C2 −→ C1 where δ2 is a crossed module with cokernel G and we want to change it to get

∂2 δ2 C3 −→ L2 −→ L1

ab where L2 and L1 are G-modules. Since G-modules are abelian, the bare minimum is L2 = C2 . Let us see the appropriate deﬁnition for L1.

Deﬁnition 12.4.1 Let C be the crossed complex

δn / / / δ2 / C = ··· Cn Cn−1 ··· C2 C1 .

Then all the Cn (n ≥ 1) have object set C0, which is mapped identically by δn if n ≥ 2.

Since N = δ2C2 is a totally intransitive normal subgroupoid of C1, we may deﬁne G = π1(C) = C1/N (with Ob G = C0) and let φ : C1 → G be the quotient morphism.

For n ≥ 3, N acts trivially on Cn, so Cn is a G-module and δn+1 : Cn+1 → Cn is a G-morphism. δb ab Similarly, N acts on C2 by conjugation: a = −b + a + b for a, b ∈ C2(q), so N acts trivially on C2 , the ab ab family of Abelianised groups C2(p) . This makes C2 a G-module, and since δ3 : C3 → C2 is C1-equivariant, ab ab we have a G-morphism ∂3 = α2δ3 : C3 → C2 , where α2 is the Abelianisation map C2 → C2 . Then, we deﬁne

0 0 ∂n ab ∂2 ∂1 ∆ C = · · · −→ Cn −→ Cn−1 −→ · · · −→ C2 −→ Dφ −→ IG where the ﬁnal part is got using the next proposition. 317

Proposition 12.4.2 There are G-morphisms

0 ab ∂2 ∂1 C2 −→ Dφ −→ IG such that the diagram

/ δn / / / δ3 / δ2 / φ / ··· Cn Cn−1 ... C3 C2 C1 G

= = = α2 α1 α0 → / / / / / ab / / ··· Cn Cn−1 ··· C3 C Dφ IG ∂ ∂ 2 ∂ 0 n 3 2 ∂1 commutes and the lower line is a chain complex over G, where α1 is the universal φ-derivation, α0 is the G-derivation x 7→ x − 1q for x ∈ G(p, q) and ∂n = δn for n ≥ 4.

Proof The functor D : Gpds2 → Mod, applied to the sequence of morphisms

/ δ3 / δ2 / φ / ··· C3 C2 C1 G

ε3 ε2 φ = ... / 1 / 1 / G / G gives a sequence of module morphisms

... → (Dε3 , 1) → (Dε2 , 1) → (Dφ,G) → (IG, G).

Since a derivation Cn → M over a null map εn : Cn → 1 is just a morphism to an Abelian group, we may ab identify Dεn with Cn and its universal derivation with the Abelianisation map. Thus we obtain the stated commutative diagram in which the vertical maps are the corresponding universal derivations. 0 This establishes all the stated properties except the G-invariance of ∂2 and the relations ∂2∂3 = 0, ∂1∂2 = 0.

Clearly ∂2∂3 = α1δ2δ3 = 0. 0 0 Also ∂1∂2α2 = α0φδ2 = 0 and since α2 is surjective, this implies ∂1∂2 = 0. ab g Finally, if x ∈ C2 , g ∈ G and x is deﬁned, choose a ∈ C2, b ∈ C1 such that α2a = x, φb = g. Then

g b ∂2(x ) = α1δ2(a ) −1 = α1(b cb), where c = δ2a, −1 φc φb = [(α1(b )) + α1c] + α1b, since α1 is a φ-derivation, φb = (α1c) , since φc = 1, g = (∂2x) , as required.

Deﬁnition 12.4.3 For a chain complex L over a groupoid H,Θ0L = Θ0(L, H) is the crossed complex

0 / ∂ / / / ∂ / (0,∂)/ Θ L = ··· Ln Ln−1 ··· L3 L2 H n L1.

Here H n L1 acts on Ln (n ≥ 2) via the projection H n L1 → H, so that L1 acts trivially. 318

Proposition 12.4.4 The functor Θ0 is the right adjoint of ∆0 : Crs → Chn. Hence ∆0 preserves colimits.

Proof It follows easily from Propositions 12.2.4 and 12.3.4. 2

0 0 0 We note that Θ L is independent of L0; this reﬂects the fact that, in ∆ C, the boundary map ∂1 : Dφ → IG is an epimorphism. We are going to change that.

12.5 The functor ∆ : Crs → Chn

Deﬁnition 12.5.1 For any crossed complex C, ∆C is the chain complex

∂3 ab ∂2 ∂1 ∆C = · · · → Cn → Cn−1 → · · · → C3 −→ C2 −→ Dφ −→ ZG

0 over G in which ∂1 is the composite of ∂1 : Dφ → IG with the inclusion of IG in ZG.

The functor ∆ : Crs → Chn also has a right adjoint Θ, but now ΘL involves L0 in an essential way. Let us begin by deﬁning ΘL and checking that the deﬁnition works.

Deﬁnition 12.5.2 For any chain complex L, we consider the canonical covering morphism

θ : P (UL0,H) → H n L0 of Proposition 12.2.8. Then Θ(L) = θ∗Θ0L, the pull-back along θ of the crossed complex of Deﬁnition 12.4.3. We obtain a commutative diagram / / / / ··· E3 E2 E1 P (L0,H)

σ3 σ2 σ1 θ / / / / ··· L3 L2 H n L1 H n L0 (0,∂) (1,∂) in which each En is a groupoid over E0 = L0, the underlying set of the groupoid L0), and each σn is a covering morphism.

For n ≥ 2, the composite map Ln → H n L0 is 0 and , since Ker θ is discrete, it follows that En is just a family of groups each isomorphic to a group of Ln. There is also an action of E1 on En (n ≥ 2) induced by the action of H n L1 on Ln; for if e1 ∈ E1(x, y), where x ∈ L0(p), y ∈ L0(q), and if en ∈ En(x), then σ1e1 acts on σnen to give an element of Ln(q) which lifts uniquely to an element of En(y). 0 It is now easy to see that E = {En}n≥0 is a crossed complex and that the σi form a morphism σ : E → Θ L of crossed complexes.

An explicit description of E = Θ(L, H) can be extracted from the constructions given above. The set of objects of every En is L0.

An arrow of E1 from x to y, where x ∈ L0(p), y ∈ L0(q), p, q ∈ H0, is a triple (h, a, y), where h ∈ H(p, q), h a ∈ L1(q), and x = y + ∂a. Composition in E1 is given by

(h, a, y)(k, b, z) = (hk, ak + b, z) 319 whenever hk is deﬁned in H and yk = z + ∂b.

For n ≥ 2, En is a family of groups; the group at the object y ∈ L0(q) has arrows (a, y) where a ∈ Ln(q), with composition (a, y) + (b, y) = (a + b, y).

The boundary map δ : E2 → E1 is given by

δ(a, y) = (1q, ∂a, y) for a ∈ L2(q), y ∈ L0(q).

The boundary map δ : En → En−1 (n ≥ 3) is given by δ(a, y) = (∂a, y) and the action of E1 on En (n ≥ 2) is given by (a, y)(k,b,z) = (ak, z),

k where k ∈ H(q, r), a ∈ Ln(q), y ∈ L0(q) and y = z + ∂b.

Proposition 12.5.3 The functor Θ, is the right adjoint of ∆ : Crs → Chn . Hence ∆ preserves colimits.

Proof A morphism (β, ψ) : (∆C,G) → (L, H) in Chn is equivalent to a commutative diagram in Mod:

→ → / / ab / / i / ··· C3 C2 Dφ IG ZG }} β β β β0 } 3 2 1 0 }} } β0 ~}} / / / / ··· L3 L2 L1 L0 ∂ ∂ ∂

(over some morphism ψ : G → H) and hence, by Propositions 12.2.4, 12.3.4, to a commutative diagram in Gpds:

φ ··· / C / C / C / G 3 2 1 LLL LLL β γ1 ξ LL 3 β2 LL L& / / / / o ··· L3 L2 H n L1 H n L0 P (L0,H) ∂ (0,∂) (1,∂) θ where (. . . β3, β2, γ1) is a morphism of crossed complexes, and θ is the canonical covering morphism. This in turn is equivalent to a commutative diagram

/ / δ / ω / ··· C3 C2 C1 P (L0,H)

β3 β2 γ1 θ / / / / ··· L3 L2 H n L1 H n L0 ∂ because, in any such diagram, θωδ = 0 and θ is a covering morphism, so ωδ = 0, that is, ω factorises through

φ : C1 → G. This diagram is therefore equivalent to a morphism of crossed complexes C → E. Hence (β, ψ) is therefore equivalent to a morphism of crossed complexes C → E. This shows that the functor Θ : Chn → Crs is right adjoint to ∆. 2

Remark 12.5.4 This adjoint pair (∆, Θ) has been used in several places before it was assembled together by Brown and Higgins ([45]). The construction of ∆ is already in Whitehead’s paper [175], but his construction 320

requires C1 to be a free group. (If C1 is free on a set X, then Dφ is just the free G-module on X.) The general construction of (∆C)1 = Dφ was suggested by Crowell in [69]. The existence of an adjoint was suggested by results in MacPherson [145] that the Alexander module preserves colimits. Special cases of the groupoid E1 = (ΘL)1 appear in Crowell [68], Gruenberg and Roggenkamp [99] and Grothendieck [96].

12.6 Properties of ∆ : Crs → Chn

12.6.1 ∆ and colimits

The fact that ∆ : Crs → Chn preserves all colimits implies that the van Kampen theorem proved in section 7.4 for the fundamental crossed complex ΠX of a ﬁltered space X can be converted into a similar theorem for the chain complex CX = ∆ΠX. The interpretation of this result will be discussed in Section 12.7. The following simple example illustrates some of the interesting features that arise in computing colimits in Crs and Chn. Note that if all the crossed complexes in a diagram {Cλ} are reduced then the colimit of {Cλ} is reduced provided that the diagram is connected, in which case the colimit of {∆Cλ} can be computed in the category of chain complexes over groups instead of groupoids.

Example 12.6.1 Let M → P , N → P be crossed modules over a group P . Their coproduct in the category of crossed modules over P is given by the pushout in Crs:

(··· 0 → M → P → ∗) i4 VVV iiii VVVV iiii VVVV iii VVVV iiii VVV* (··· 0 → 0 → P ⇒ ∗) (··· 0 → M ./ N ⇒ P → ∗) UUU hh4 UUU hhhh UUUU hhhh UUU hhhh UUU* hhhh (··· 0 → N → P ⇒ ∗) where the group M ./ N is the Peiﬀer product described in [25], [91]. To ﬁnd the corresponding chain complexes let G = P/δM, H = P/δN and write φ, ψ for the quotient maps P → G, P → H. Then the corresponding derived modules are Dφ = IP ⊗P ZG and Dψ = IP ⊗P ZH and we wish to compute the pushout in Chn (or in chain complexes over groups) of

(··· 0 → M ab → IP ⊗ ZG → ZG, G) g3 P ggggg ggggg ggggg ggggg (··· 0 → 0 → IP → ZP,P ) WW WWWWW WWWWW WWWWW WWWW+ ab (··· 0 → N → IP ⊗P ZH → ZH,H) 321

To do this, we ﬁrst form the pushout K of G ~> ~~ ~~ ~~ P @@ @@ @@ @ H namely K = P/(δM.δN); this is the group acting on the pushout chain complex. Next we form the induced modules over K of each module in the diagram and then form pushouts of K-modules in each dimension. This gives the chain complex

ab ab (··· 0 → (M ⊗P ZK) ⊕ (N ⊗P ZK) → IP ⊗P ZK → ZK,K).

ab ab ab ab ab Since K = P/δMδN, and δM acts trivially on M , we have M ⊗P ZK = M /[M ,N]; similarly N ⊗P ZK = N ab/[N ab,M]. Thus the pushout in dimension 2 is

M ab/[M ab,N] ⊕ N ab/[N ab,M], which is easily identiﬁable as (M ./ N)ab, conﬁrming that ∆ preserves this pushout.

12.6.2 ∆ and the closed category structure

In Section ?? an internal hom functor Crs(−, −) has been deﬁned for crossed complexes similar to that deﬁned in Section 12.1 for chain complexes over groupoids. The relationship between the two monoidal closed structures is best described in terms of the adjoint functors ∆ and Θ.

Theorem 12.6.2 For crossed complexes B, C and chain complexes L there are natural isomorphisms

(i) CRS(C, ΘL) =∼ ΘCHN(∆C,L),

(ii) ∆(B ⊗ C) =∼ ∆B ⊗ ∆C.

Proof The two natural isomorphisms are equivalent because

CHN(∆(B ⊗ C),L) =∼ Crs(B ⊗ C, ΘL) =∼ Crs(B, CRS(C, ΘL)), while

Chn(∆B ⊗ ∆C,L) =∼ Chn(∆B, CHN(∆C,L)) =∼ Crs(B, ΘCHN(∆C,L)).

The isomorphism (i) is easier to verify than (ii) because we have explicit descriptions of the elements of both sides, whereas in (ii) we have only presentations. In dimension 0 we have on the left of (i) the set Crs(C, ΘL) of morphisms fˆ : C → ΘL; on the right we have ˜ the set Chn(∆C,L) of morphisms (f, ψ) : ∆C → L, where ψ is a morphism of groupoids from G = π1C to H, 322 the operator groupoid for L. These sets are in one-one correspondence, by adjointness, and their elements are also equivalent to pairs (f, ψ) where ψ : G → H and f is a family

δ0 δ / δ / / ··· C2 C1 / C0 δ1 f2 f1 f0 ∂ / ∂ / ∂ / ··· L2 L1 L0 such that

(i) f0(p) ∈ L0(ψ(p)) (p ∈ C0),

(ii) f1 is a ψφ-derivation, where φ is the quotient map C1 → G,

(iii) fn is a ψ-morphism for n ≥ 2,

(iv) ∂fn+1 = fnδ (n ≥ 1),

0 ψφx 1 (v) ∂f1(x) = (f0δ x) − (f0δx)(x ∈ C1).

Such a family will be called a ψ-derivation f : C → L. ˆ ˆ We recall from Deﬁnition 8.2.6 that an element in CRSi(C,E) is an i-fold homotopy (h, f): C → E, where fˆ is a morphism C → E and hˆ is a family of maps

/ / / ··· C2 C1 / C0

hˆ2 hˆ1 hˆ0 ··· Ei+2 Ei+1 Ei satisfying

ˆ ˆ (i) h0(p) ∈ Ei(f0(p)) (p ∈ C0); ˆ ˆ (ii) h1 is a f1-derivation; ˆ ˆ (iii) hn is a f1-morphism for n ≥ 2.

In the case E = ΘL, where L is a chain complex over H, it is easy to see that, if i ≥ 2, such a homotopy is equivalent to the following data: a morphism of groupoids ψ : G → H; a ψ-derivation f : C → L as in diagram (*); and a family h of maps / / / ··· C2 C1 / C0 δ1 h2 h1 h0 ··· Li+2 Li+1 Li satisfying

(i) h0(p) ∈ Li(ψp)(p ∈ C0);

(ii) h1 is a ψφ-derivation;

(iii) hj is a ψ-morphism for j ≥ 2. 323

ˆ The maps hj of diagram (**) are then given by

ˆ hj(x) = (hj(x), f0(q)) if x ∈ Cj(q), j ≥ 2, ˆ h1(x) = (h1(x), f0(q)) if x ∈ C1(p, q), ˆ h0(q) = (h0(x), f0(q)) if q ∈ C0.

In the case i = 1, because of the special form of E1, we also need a map τ : C0 → H satisfying

(iv) τ(q) ∈ H(ψ0(q), ψ(q)) for some ψ0(q) ∈ Ob H,

ˆ and in this case h0(q) = (τ(q), h0(q), f0(q)). It is now an easy matter to see that these data are equivalent to an element of dimension i in ΘCHN(∆C,L). In the case i = 1, the map τ deﬁnes a natural transformationτ ˜ : ψ0 → ψ, where ψ0(g) = τ(p)ψ(g)τ(q)−1 for g ∈ G(p, q). Thisτ ˜ is an element of the groupoid GPDS(G, H) (the operator groupoid for CHN(∆C,L)) and ˜ ˜ provides the ﬁrst component of the triple (˜τ, h, f) which is the required element of Θ1CHN(∆C,L); the other components are f˜ : ∆C → L, the morphism of chain complexes induced by f, and h˜, the 1-fold homotopy ˜ ˜ ∆C → L induced by h. Here h0(1p) = h0(p) and hnαn = hn for n ≥ 1, where the αi are as in the diagram in Proposition (12.4.2). The rest of the proof is straightforward. 2

12.6.3 ∆ and the free standard resolution of a group

The functor ∆ also relates the standard free crossed resolution of a group with the bar resolution.

Theorem 12.6.3 Let G be a group. The image by ∆ of F st(G) the standard free crossed resolution of a group is B(G) the bar resolution.

st st n Recall from Section 11.4.1 that F (G) is given by which Fn (G) is free on the set G with generators written [a1, a2, . . . , an], ai ∈ G, with boundary

δn : Cn(G) → Cn−1(G) given by −1 δ2[a, b] = [a][b][ab] .

, −1 −1 [a]−1 δ3[a, b, c] = [a, bc][ab, c] [a, b] [b, c] , and for n > 4

nX−1 a−1 i (12.6.1) δn[a1, a2, . . . , an] = [a2, . . . , an] 1 + (−1) [a1, a2, . . . , ai−1, aiai+1, ai+2, . . . , an]+ i=1 n + (−1) [a1, a2, . . . , an−1].

and B(G) is given (see Mac Lane [129]) 324

12.7 The chain complex of a ﬁltered space and of a CW -complex.

Our object in this section is to identify the chain complex ∆ΠX in terms of chains of universal covers for certain ﬁltered spaces X. All spaces which arise will now be assumed to be Hausdorﬀ and to have universal covers.

Let X∗ be a ﬁltered space. For v ∈ X0, let p : X˜(v) → X denote the universal cover of X and let Xˆ (v) denote the ﬁltered space consisting of X˜(v) and the family of subspaces

−1 Xˆi(v) = p (Xi) for all i ≥ 0.

Suppose X∗ is a connected ﬁltered space. This implies that Xˆi(v) is the universal cover of Xi based at v for i ≥ 2. (Proof

Proposition 12.7.1 If X∗ is a connected ﬁltered space, then ∆ΠX∗ has operating groupoid π1(X,X0) and has chain complex C(v) at v ∈ X0 given by

Ci(v) = Hi(Xˆi(v), Xˆi−1(v)) for all i ≥ 1.

Proof Let v ∈ X0 and let i ≥ 3. The pair (Xˆi(v), Xˆi−1(v)) is (i − 1)-connected, and so

∼ πi(Xi(v),Xi−1(v), v) = πi(Xî(v), Xî−1(v), v) since p is a covering, ∼ = Hi(Xî(v), Xî−1(v)) by the relative Hurewicz theorem,

since Xˆi(v) and Xˆi−1(v) are in fact the universal covers at v of Xi and Xi−1 respectively. If i = 2, a similar argument applies but in this case π1(Xˆ1, v) = δπ2(X2,X1, v). So the relative Hurewicz theorem now gives

∼ ab H2(Xˆ2(v), Xˆ1(v)) = π2(Xˆ2(v), Xˆ1(v), v) = C2(v).

The case i = 1 is essentially the result of [69], section 4. (Prove) 2

In view of the above we define for a filtered space X∗ the chain complex with operators CX∗ to have groupoid of operators π1(X,X0) and to have CiX∗(v) = Hi(Xî(v), Xî−1(v)). This defines the functor

C : FTop → Chn.

The result given above is that if X∗ is connected then CX∗ = ∆ΠX∗.

λ Corollary 12.7.2 Let X∗ be a filtered space and suppose that X is the union of a family U = {U }λ∈Λ of open λ sets such that U is closed under finite intersection. Let U∗ be the filtered space obtained from X∗ by intersection λ λ with U . Suppose that each U∗ is a connected filtered space. Then X∗ is connected and the natural morphism in Chn λ λ colim CU∗ → CX∗ is an isomorphism. 325

Proof This is a consequence of the Union Theorem (Theorem C) of [39] which gives a similar result for Π rather than C, and the fact that ∆ has a right adjoint and so preserves colimits. 2

We note that results such as this have been used by various workers ([126, 152]) in the case X∗ is the skeletal ﬁltration of a CW -complex and the family U is a family of subcomplexes, although usually in simple cases. The general form of this ‘van Kampen Theorem’ for CX∗ does not seem to have been noticed, and this is probably due to the unfamiliar form of colimits in the category Chn of chain complexes over varying groupoids. Even in the group case these colimits are not quite what might be expected (see Example 12.6.1). 326 Chapter 13

Cohomology of groups

The aim of this chapter is to show how the theory of crossed complexes sheds light on aspects of the well known cohomology theory of groups. The standard theory (see for example [KSBrown, Mac Lane, ...] deﬁnes abelian groups Hn(G, A) for a group G and G-module A, and n ≥ 0, which are functorial in G and in A. There are other interpretations of these groups of which the easiest are in low dimensions: H0(G, A) =∼ AG, the subgroup of A of invariant elements, i.e. elements a with ag = a for all g ∈ G; H1(G, A) is the group of derivations G → A modulo the subgroup of principal derivations, i.e. functions of the form f(g) = ag − a for a ﬁxed a ∈ A; H2(G, A) is the group of equivalence classes of central extensions of A by G, i.e. exact sequences

p 0 → A −→i E −→ G → 1, where i(A) is central in E, and two such extensions are equivalent if there is a commutative diagram

i p 0 / A / E / G / 1 f 0 / A / E0 / G / 1 i0 p0 in which case the 5-lemma implies that f is an isomorphism; H3(G, A) is the group of equivalence classes of crossed sequences, i.e. exact sequences

µ φ 0 → A −→i M −→ P −→ G → 1, where µ : M → P is a crossed module, iA is invariant under the action of P and the induced operation of G on i(A) is the given action. An elementary equivalence of such is a commutative diagram

i µ φ 0 / A / M / P / G / 1 f g 0 / A / M 0 / P 0 / G / 1 i0 µ0 φ0

327 328 and the equivalence of two crossed sequences is the equivalence relation generated by the elementary equivalences. (Note that in this case we cannot deduce that an elementary equivalence is an equivalence; we will give examples of this later.) There is an analogous interpretation of Hn+1(G, A) which we will discuss later. Notice that we have not explained at this stage the addition on the classes of extensions or crossed sequences which gives the addition in the cohomology groups, and to do so would take us too far afield. Notice that any crossed module M = (µ : M → P ) gives rise to a crossed sequence with A = Ker µ, G = 3 Coker µ. So one of the problems is to calculate the element of cohomology, usually written kM determined by such a crossed module, and in particular to say whether or not this so called k-invariant is 0. The traditional theory of the cohomology of groups takes it as a branch of homological algebra and is firmly based on the notions of free (or projective) resolution of G-modules. This has the great advantage of linking the theory with a broad theory of wide ramifications, namely that of homological algebra. However, this approach has certain disadvantages when it comes to some problems which are essentially nonabelian, such as the classification of nonabelian extensions of A by G

p 0 → A −→i E −→ G → 1, where A is a possibly nonabelian group, and so the extension is not central. (Note that the literature also describes such an extension as ‘of G by A’, which seems anomalous, since an extension of A should describe something bigger than A.) There is a description of Hn(G, A) based on n-cocycles f : Gn → A, where Gn is the n-fold product of G with itself, indeed this was the original Eilenberg-Mac Lane definition, and this gives a description of nonabelian extensions as above in terms of so-called factor sets k1 : G → AutA, k2 : G2 → A, but one comes across here the problem of calculation, since the group G may be infinite, and it may be difficult to specify such a factor set. Our intention in this chapter is to sketch an account of the cohomology of groups based firmly on the notion of crossed complex, and the homotopy theory of these. One great advantage of this is that allows it allows for more general coefficients, so that we have the notion of cohomology of a group G with coefficients in a crossed complex A. Indeed this generalises easily to the case where G is a groupoid, or even a CW -complex. This leads to concepts of nonabelian homological algebra, seen here as a special case of nonabelian algebraic topology. This reflects the historical process, where the homology of groups arose out of considerations of algebraic topology(see [Mac Lane - history]). In the traditional cohomology theory the notion of exact sequence plays a key rôle.This notion turns out to be more complicated in the nonabelian theory, in a way which reflects known complications in algebraic topology, in the homotopy theory of fibre spaces and of function spaces. Thus if p : E → B is a fibration of spaces, then there is an exact sequence for each x ∈ E

(13.0.1) · · · → πn(Fx, x) → πn(E, x) → πn(B, px) → πn−1(Fx, x) → · · ·

→ π1(B, px) → π0(Fx) → π0(E) → π0(B)

−1 where the last three terms are sets with base points, and Fx = p (px). That is, the standard situation in homotopy theory is to obtain a family of exact sequences. This is commonly avoided by taking spaces with base points, so that the exact sequence is determined. However we are also interested in the induced ﬁbre sequence of function spaces

p∗ : TOP(X,E) → TOP(X,B) and do not necessarily wish to take the constant function as base point in the function space TOP(X,E). 329

An analogous situation applies in the crossed complex theory. Taking a direct analogy with topology, we consider some ﬁbrations of crossed complexes p : E → B which arise simply in practice, and wish to consider exact sequences arising from

p∗ : CRS(C,E) → CRS(C,B) where C could be either ΠX∗ for a CW -space X∗ or some free crossed resolution of a group (or groupoid) G. We will find that the exact sequences arising from this fibration contain interesting information, and that a varying choice of base point is useful. For example, it will allow for cohomology with varying operations of a group G on an abelian group A, and will also describe in a convenient way the notion of cohomology of a CW -complex X with coefficients in varying local systems A. Other points in favour of this approach are in terms of calculation in the case of nonabelian problems. Thus we would like to calculate the equivalence classes of all possible nonabelian extensions of A by G; it is difficult to compute with the description by factor sets. We show how this traditional theory can be replaced by one working with any free crossed resolution of G, and this may be calculable even if G is infinite, for example in some cases when G is finitely presented. This theory brings in again the notion of identity among relations. A further point is that it makes it easy to present the theory of nonabelian extensions of the type of a crossed module µ : M → P , a theory due initially to Dedecker. This notion crops up in topological examples; for example in [Brown-Mucuk] it is used to discuss covering maps onto non connected topological groups. In effect, once crossed modules are allowed into the scene, the natural theory is that of crossed complexes. This is indeed a part of a more general nonabelian homological algebra, investigated earlier by [Gerstenhaber, Reinhart, Frohlich, Lue] and others, and currently of renewed interest for its application to geometric problems [Breen, .....]. Again the traditional theory is awkward for calculating the k-invariant of a crossed module µ. We show how to complete some calculations again by working with a ‘small’ free crossed resolution of G = Coker µ. We also need to relate this theory with the traditional theory which is based on chain complexes of modules and free (or projective) resolutions of modules. For this relation, we need the results of Chapter 12, on the relation between crossed complexes and chain complexes with a group(oid) of operators. That theory does lead naturally to the use of a groupoid of operators, and suggests that calculation in such a category may in some cases be more convenient than restricting to groups. However that topic requires further investigation. In this chapter we demonstrate the usefulness of free crossed complexes resolutions to analyse and generalise the classical cohomology of groups defined via free chain complex resolutions. In the first Section we prove how the usual definition can be stated in terms of homotopy classes of morphisms of a free crossed resolution C(G), getting

n H (G, A) = [C(G), En(A); φ].

This expression lends itself to generalization. In Section 13.4 we consider the second cohomology group with coeﬃcients in a crossed module M, deﬁning

2 H (G, M) = [C(G), En(M); φ] and we get that this group classiﬁes also in this case group extensions (now of type M). We can also generalise the theory of abstract kernels, getting obstructions to the existence of extensions of type M and, in the case they exist, the classiﬁcation of them. This is done in Section 13.5 following Mac Lane ([129], Ch.IV, Thm.8.7)). 330

13.1 Cohomology of a group

Recall that if G is a group, A is a G-module, and n ≥ 2, we write

En(M,G) = ··· / 0 / M / 0 / ······ / 0 / G. for the crossed complex which is G in dimension 1, A in dimension n, is trivial otherwise, and has zero boundary maps (this follows if n > 2). Using this notation, we deﬁne some cohomology groups of a group G in terms of homotopy classes of maps from a free crossed resolution C(G) of the group to this particular structure. The fact that any two free crossed resolutions of G are homotopy equivalent makes this deﬁnition essentially independent of the choices made in constructing a resolution, and allows us to choose the resolution according to convenience for the purposes at hand.

Since a resolution also includes an morphism φ : C1(G) → G inducing an isomorphism φ : π1C(G) → G, this information should be taken in account when getting the homotopy classes. In this direction, we introduce the notion of homotopy relative to a ﬁxed map.

Deﬁnition 13.1.1 Let C,D be crossed complexes, and let f : C → D be a morphism. Let i : A → C be a morphism. A homotopy h : f ' g rel i (or rel A) is a homotopy which satisﬁes hi is a constant homotopy, so that this implies fi = gi. We often employ this when A is a truncation say trnC of C and i is the inclusion, so that we are dealing with a homotopy relative to levels ≤ n. These set of homotopy classes we write

[C,D; i].

We will use this idea to deﬁne some notions of cohomology of a group G (or also of a groupoid) with coeﬃcients, which are related to the choice of the crossed complex D. The information on the group G is encapsulated by the choice of C as a free crossed resolution of the group G.

Definition 13.1.2 Let G be a group, and A a G-module. The nth cohomology of G with coefficients in A is defined to be the set of homotopy classes

n (13.1.1) H (G, A) = [C(G), En(A); φ] where C(G) is a free crossed resolution of the group G, and φ is the inclusion of the ﬁrst skeleton

φ : C1(X) → G.

First, let us prove that this agrees with the traditional deﬁnition For the proof we have to use the standard resolution, whose associated chain complex is the bar resolution Recall that the standard free crossed resolution of a group G is:

st δ3 / st δ2 / st φ / / F∗ (G)3 F∗ (G)2 F∗ (G)1 G 1

st n in which Fn (G) is free on the set G with generators written [a1, a2, . . . , an], ai ∈ G, and boundary given by

−1 δ2[a, b] = [a][b][ab] ,

−1 −1 [a]−1 δ3[a, b, c] = [a, bc][ab, c] [a, b] [b, c] , 331 and for n > 4

nX−1 a−1 i (13.1.2) δn[a1, a2, . . . , an] = [a2, . . . , an] 1 + (−1) [a1, a2, . . . , ai−1, aiai+1, ai+2, . . . , an]+ i=1 n + (−1) [a1, a2, . . . , an−1].

Remark 13.1.3 The formulae for the differential given above are different in detail from those given in [114, 43, 168]. This reflects only the different conventions used.

There are some advantages of using the free crossed complex approach. One is that allows for certain kinds of nonabelian coefficients, such as a group when n = 1 or crossed module when n = 2. Another one is computational. Traditionally, such cohomology is defined by functions kn on the product Gn given for say n = 1, 2 and satisfying certain conditions, which make the pair what is called a ‘factor set’ or ‘2-cocycle’. The problem with this approach is how to do any calculations, if G is described in some other way than as a finite group. For example, G may be defined by a presentation, and may be infinite. In such case it is not clear how such functions on G and G2 may be written down and the required conditions satisfied. We overcome this problem by using the standard free crossed resolution of G to model such factor sets by morphisms of crossed complexes, and then see that since we are dealing with homotopy classes, we can replace this standard free crossed resolution by any other, since they are all equivalent. In particular, if we can find a free crossed resolution finitely presented (as free crossed resolution) in the required dimensions, then we will be able to describe the factor set in finite terms. A further advantage is that equivalence of factor sets is then seen as homotopy of morphisms of the crossed complexes, so that this again puts what seems to be a special concept in a convenient general setting, and again allows for specific computations. We will give examples of that later.

13.2 Cohomology of groups as classes of crossed extensions

Let us start by introducing the idea of crossed n-fold extensions of a group G by a G-module A.

Deﬁnition 13.2.1 An crossed n-fold extension of A by G

∂n ∂n−1 ∂2 H = 0 −→ Cn = A −→ Cn−1 −→ · · · C2 −→ C1 is a crossed resolution of G, C such that Cn = A and Ci = 0 for all i ≥ n + 1.

Example 13.2.2 Let G be a group and C a crossed resolution of G. We denote by Cn the crossed n-fold extension n i ∂n−1 ∂2 C = 0 −→ Zn−1 = Ker ∂n−1 −→ Cn−1 −→ · · · C2 −→ C1.

Now, we relate the cohomology groups with classes of crossed n-fold extensions. The relation of similarity among crossed n-fold extensions of A by G is defined as the equivalence relation generated by two crossed n-fold extensions H and H0 being similar if there is a morphism of crossed n-fold extensions over (Id, Id). We denote the quotient set by OpExtn(G, A). It is a group with the ”Baer sum” whose definition may be found in [129]. Let us define a map n OpExt (G, A) → Hn(G, A). To start with, we indicate how to associate a cocycle to a crossed n-fold extension H. 332

Let C be a free crossed resolution of G. Applying 10.2.2 we get a morphism of crossed n-fold extensions n n {fk}k=1 : C −→ H such that f = Id. Since it is a morphism, φ = fn : Cn → A satisﬁes φ∂n = 0, i.e. it is a cocycle. So we have got φ by successive extensions as the diagram indicates

/ ∂n / ∂n−1/ ∂2 / / ··· Cn Cn−1 ··· C1 G

fn=φ fn−1 f1

∂0 ∂0 / / n−1/ 2 / 0 / 0 0 A Hn−1 ··· C1 G .

Theorem 13.2.3 The above construction induces a well deﬁned map OpExtn(G, A) → Hn(G, A) deﬁned by [H] 7→ [φ] ∈ Hn(G, A)

Proof The first step is to see that for a fixed crossed n-fold extension H, the cohomology class is independent n n of the extension φ chosen. This is so since two extensions {fk}k=1 and {gk}k=1 both inducing the Id : G → G are homotopic by 10.2.3 and the homotopy gives fn − gn = ∂nDn−1, i.e. their difference is a coboundary. 0 n 0 Next, if we consider two crossed n-fold extensions H and H related by a morphism {hk}k=1 : H −→ H n n 0 n n 0 over (Id, Id), we construct an extension {fk}k=1 : C −→ H and as extension {fk}k=1 : C −→ H we choose 0 0 0 the composite fk = hkfk. Then the maps fn = hnfn = fn so φ = φ. 2

Remark 13.2.4 It is going to be used later on a kind of partial inverse of the first part of the preceding theorem, 0 namely that all cocycles cohomologous to φ can be got from appropriate extensions. In fact if φ = φ + h∂n, 0 n 0 0 0 0 where h is a homomorphism Cn−1 → A we define {fk}k=1 as follows, fn = φ , fn−1 = fn−1 + ih and fi = fi. 0 n It is not difficult to prove that {fk}k=1 is also a morphism of crossed n-fold extensions inducing Id.

Now we want to deﬁne a map back

n Hn(G, A) → OpExt (G, A).

First let us give an alternative description of Hn(G, A). We consider C a crossed resolution of G. An n-cocycle is a homomorphism ψ : Cn → A such that ψ∂n+1 = 0, or, equivalently, a homomorphism ψ : Cn/∂n+1(Cn+1) → ∼ A. Considering that ∂n+1(Cn+1) = Ker ∂n and Cn/Ker ∂n = ∂n(Cn) = Ker ∂n−1 = Zn−1, we deduce that n ∼ Z (G, A) = Hom(Zn−1,A).

Under this isomorphism, the coboundaries, i.e. the homomorphisms of the type ψ∂n go to ψ|Zn , thus Hom(Z ,A) Hn(G, A) =∼ n−1 . Hom(Cn−1,A)

Thus let us start with a homomorphism φ : Zn−1 → A. Then we consider the crossed n-fold extension

n ∂n−1 ∂2 φ C = 0 −→ A −→ Cn−1φ −→ · · · C2 −→ C1 where Cn−1φ is the coequaliser of the maps Zn−1 → A × Cn−1 given by φ and the inclusion. This produces a map n HomG(Zn−1,A) → OpExt (G, A). Let us check that this map is onto 333

Lemma 13.2.5 For each equivalence class of crossed n-fold extensions there is a representative of the form n φC for some φ ∈ HomG(Zn−1,A).

Proof For any crossed n-fold extension H there is a morphism

n n {fk}k=1 : C −→ H such that f1 induces the identity (see 10.2.3.) By the universal property of the coequaliser, this gives a morphism of crossed n-fold extensions φCn −→ H where φ = fn. The induced maps on G and A are clearly the identity. 2

Since the map is onto, the group OpExtn(G, A) is abelian. Let us check that the map just deﬁned induces another one Hn(G, A) → OpExtn(G, A). n Since H (G, A) = HomG(Zn−1,A)/HomG(Cn−1,A) all we have to check is that the construction maps HomG(Cn−1,A) to 0.

˜ n Lemma 13.2.6 If the homomorphism φ extends to a G-morphism φ : Cn−1 → A, the extension φC represents the trivial class.

Proof In this case there is a section

Zn−2 → Cn−1φ splitting the short exact sequence

0 −→ A −→ Cn−1φ −→ Zn−2 −→ 0 producing a homomorphism α : Cn−1φ −→ A that gives a morphism of crossed n-fold extensions φCn → 0 where 0 = 0 −→ A −→ A −→ · · · 0 −→ G and this last extension represents the 0. 2

Now, we can state and prove the main theorem of [114].

Theorem 13.2.7 The map above is an isomorphism of abelian groups.

Proof All that remains to prove is that given a crossed resolution C and maps φ and φ0, φ − φ0 extends over

Cn−1. Let us assume that φCn =∼ φ0Cn, i.e. there are crossed n-fold extensions and morphisms between them

n α1 α2 α2m 0 φC −→ H1 ←− H2 ···H2m −→ φ C (if necessary, we repeat some H to get an even number). n We construct by 10.2.2 extensions of the identity φC → Hk, giving maps νk and βk. There is a decomposition

0 0 φ − φ = (φ − ν2) + (ν2 − ν4) + ··· (ν2m − φ ).

0 Since by 10.2.3 there are homotopies ν2k ∼ ν2k+2, the map φ − φ extend over Cn−1. 2 334

13.3 An interpretation of H3(G, A) and some examples

Now, we use this resolution to get a better inkling of the elements of H3(G, A). The idea of the relation of this group with crossed modules goes back to MacLane 1949. A crossed extension of G by A is an exact sequence of the form

µ 0 −→ A −→ M −→ P −→ G −→ 1 where µ : M → P is a crossed module and the action of P on M induces the given action of G on A. Notice that this is what has been previously deﬁned as a 1-fold crossed extension. Also, they are just crossed modules µ : M → P , since A = Ker µ and G = Coker µ. µ µ0 Two crossed extensions, 0 → A → M → P → G → 1 and 0 → A → M 0 → P 0 → G → 1 are equivalent when there is a commutative diagram

0 / A / M / P / G / 1

1 1 0 / A / M 0 / P 0 / G / 1 where the vertical arrows are compatible with the actions of P and P 0 on M and M 0. The equivalence classes of crossed extensions is denoted OpExt(G, A). As we have seen, there is a 1-1 correspondence between OpExt(G, A) and the elements of H3(G, A). Let us recall the deﬁnitions of these bijections. To any crossed extension of G by A

µ 0 −→ A −→ M −→ P −→ G −→ 1 we associate a map φ : C3 −→ A by inductively constructing homomorphisms f1, f2 and φ such that

/ ∂3 / ∂2 / ∂1 / / ··· C3 C F G 1

φ f2 f1 1 µ p 0 / A / M / P / G / 1 commutes. The associated cohomology class [φ] ∈ H3(G, A) is well deﬁned and it is called the Postnikov invariant of the given crossed extension (or of the crossed module).

∗ Let us detail how to get such extensions. To get f1, we need a map s : G → P such that ps = 1 and this is done by choosing elements s(g) ∈ P such that p(s(g)) = g for all g ∈ G. The map f1 is the only homomorphism of groups extending s and satisﬁes pf1 = ∂1. ∗ ∗ ∗ As before, to get f2 we need a map κ : G × G → M. Since we want µf2 = f1∂2, for each g1, g2 ∈ G , we have to choose κ(g1, g2) ∈ M such that

µκ(g1, g2) = f1∂2(g1, g2).

This is possible because pf1∂2(g1, g2) = ∂1∂2(g1, g2) = 0 and Ker p = µ(M). Then the map f2 is the only homomorphism extending κ and satisﬁes µf2 = f1∂2. −1 −1 It is interesting to develop the equation deﬁning κ. Since f1∂2(g1, g2) = f1([g1g2] [g1][g2]) = s(g1g2) s(g1)s(g2), we have that κ(g1, g2) ∈ M is an element satisfying

s(g1g2)µκ(g1, g2) = s(g1)s(g2). 335

So κ measures the diﬀerence of s from a homomorphism.

∗ ∗ ∗ As a last step let us see that there is a unique map u : G × G × G → A such that iu = f2∂3, i.e.

g3 −1 −1 iu([g1|g2|g3]) = f2([g1|g2] ) f2([g1g2|g3]) f2([g1|g2g3])f2([g2|g3])

= f2∂3([g1|g2|g3]).

As before, it exists because µf2∂3([g1|g2|g3]) = f1∂2∂3([g1|g2|g3]) = 0 and the uniqueness follows form the injectivity of i.

Remark 13.3.1 To make it easier to remember the formula for ∂3, let us check that the condition µf2∂3 = 0 comes from associativity. Let us compute both triple products. On one hand

s(g1)(s(g2)s(g3)) = s(g1)s(g2g3)µf2(g2, g3)

= s(g1g2g3)µf2(g1, g2g3)µf2(g2, g3).

On the other hand

(s(g1)s(g2))s(g3) = s(g1g2)µf2(g1, g2)s(g3)

s(g3) = s(g1g2)s(g3)µf2(g1, g2)

s(g3) = s(g1g2g3)µf2(g1g2, g3)µf2(g1, g2) .

s(g3) Thus µf2(g1, g2g3)µf2(g2, g3) = µf2(g1g2, g3)µf2(g1, g2) .

Remark 13.3.2 Notice that as seen in general, a change of the map s does not change the Postnikov invariant and also that all the cocycles in the Postnikov invariant of a crossed extension may be got through a particular choice of s just by changing f2.

Let us develop some examples. Recall that the zero cohomology class is represented by the crossed extension

1 0 1 0 / A / A / G / G / 1.

2 Example 13.3.3 Let Cn2 denote the cyclic group of order n , written multiplicatively, with generator u. Let n µn : Cn2 → Cn2 be given by µn(u) = u . This deﬁnes a crossed module, with trivial operations. 3 This crossed module represents the trivial cohomology class in H (Cn,Cn), in view of the morphism of crossed sequences / 1 / 0 / 1 / / 0 Cn Cn Cn Cn 0

1 λ λ 1 0 / C / C 2 / C 2 / C / 0 n n µn n n

n where if t is the generator of the top Cn, then λ(t) = u .

n 2 Example 13.3.4 The dihedral crossed module µ : Dn → Dn where Dn has a presentation given by hx, y; x , y , xyxyi, represents the trivial cohomology class in H3(Coker, Ker). 336

For n odd, we know that µ is an isomorphism, so we have the morphism of crossed sequences

/ 1 / 0 / 1 / / 0 Dn Dn Dn Dn 0

1 0 1 0 / 0 / / 0 / / 0 0 Dn µ Dn 0 0

∼ ∼ For n even, we have Ker µ = Coker µ = C2 and we simply construct a morphism of crossed sequences as in the following diagram / 1 / 0 / 1 / / 0 C2 C2 C2 C2 0

∼= f2 f1 ∼= / / / / / 0 C2 Dn µ Dn C2 0

n/2 where if t denotes the non trivial element of C2 then f1(t) = x, f2(t) = x .

The method used for the calculation of the cohomology class here is also of interest, since it involves a small free crossed resolution of the cyclic group of order n in order to construct an explicit 3-cocycle corresponding to the crossed module. This indicates a wider possibility of using crossed resolutions for explicit calculations. It is also related to Whitehead’s use of what he called in [175] ‘homotopy systems’, and which are simply free crossed complexes.

Let us describe such free crossed resolution of Cn.

Set F1 = C∞, with generator written w, and ∂2 : F2 → F1 the free crossed module with one generator w0 n ∼ ∼ ∼ over the map w0 7→ w . It is clear that Coker ∂2 = Cn. As G-module F2 = ZCn and Ker ∂2 = Ker ². We can extend to the left getting the crossed free resolution

M φ F∗ = ··· ZCn −→ ZCn −→ C∞.

ti Let us call w the generator of Fr as free G-module. As abelian group, Fr has free generators wi = (w0) .

n n2 Theorem 13.3.5 Let n > 2, we denote the following cyclic groups as follows Cn = ht : t i, Cn2 = hu : u i n and let ι : Cn → Cn2 denote the injection sending t to u . Let µˆ : ι∗Cn → Cn2 be the induced crossed module 3 of 1 : Cn → Cn by ι and let An denote the Cn-module which is the kernel of µˆ. Then H (Cn,An) is cyclic of order n and has as generator the class of this induced crossed module.

Proof Let us describe An as Cn-module n By Corollary 5.6.11 the abelian group ι∗Cn is the product V = (Cn) . As a Cn-module it is cyclic, with ti generator v, say. Write vi = v , i = 0, 1, . . . , n − 1. Then each vi is a generator of a Cn factor of V . −1 ti −1 The kernel An ofµ ˆ : ι∗Cn → Cn2 is a cyclic Cn-module on the generator a = v0v1 . Write ai = a = vivi+1. n As an abelian group, An has generators a0, a1, . . . , an−1 with relations ai = 1, a0a1 . . . an−1 = 1.

We use the free crossed resolution of Cn described above

n N n M n φ F∗ = ··· (C∞) → (C∞) → (C∞) → C∞

n where φ(wi) = w . 337

We deﬁne a morphism n N / n M / n φ / (C∞) (C∞) (C∞) C∞

0 f3 f2 f1 n 0 / A / (Cn) / C 2 n νn n as follows:

1. f1 maps w to u, inducing the identity on Cn

2. f2 maps the module generator w0 of F2 to v = v0.

3. f3 maps the module generator w0 of F3 to a0.

The operator morphisms fr over f1 are defined completely by these conditions. n The group of operator morphisms g :(C∞) → An over f1 may be identified with An under g 7→ g(w0). t −1 Under this identification, the boundaries ∂4, ∂3 are transformed respectively to 0 and to ai 7→ ai(ai) . So the 3-dimensional cohomology group is the group An with ai identified with ai+1, i = 0, . . . , n − 1. This cohomology group is therefore isomorphic to Cn, and a generator is the class of the above cocycle f3. 2

The following is another example of a determination of a non trivial cohomology class by a crossed module. The method of proof is similar to that of Theorem 13.3.5, and is left to the reader.

0 Example 13.3.6 Let n be even. Let Cn denote the Cn-module which is Cn as an abelian group but in which 0 0 the generator t of the group Cn acts on the generator t of Cn by sending it to its inverse. For n = 2, this gives 3 0 ∼ the trivial module. Then H (Cn,Cn) = C2 and a generator of this group is represented by the crossed module n νn : Cn × Cn → Cn2 , with generators t0, t1, u say, and where νnt0 = νnt1 = u . Here u ∈ Cn2 operates by switching t0, t1. However, it is not clear if this crossed module can be an induced crossed module for n > 2.

Remark 13.3.7 The reason for the success of the previous determinations is that we have a convenient small free crossed resolution of the cyclic group Cn. Let us extract some more consequences.

Now, let us study the induced crossed module through the inclusion of a normal subgroup when the crossed module µ is just the inclusion of another normal subgroup. Now we have a new candidate for the induced crossed module. It is ζ : M × (M ab ⊗ I(Q/P )) → Q where for m, n ∈ M, x ∈ I(Q/P ), and I(Q/P ) denotes the augmentation ideal of the quotient group Q/P . The map ζ is deﬁned by ζ(m, [n] ⊗ x) = m ∈ Q and the action of Q is given by

(m, [n] ⊗ x)q = (mq, [mq] ⊗ (q − 1) + [nq] ⊗ xq) where q denotes the image of q in Q/P .

Remark 13.3.8 It might be imagined from this that the Postnikov invariant of this crossed module is trivial, since one could argue that the projection

ab ab pr2 : P × P ⊗ I(Q/P ) → P ⊗ I(Q/P ) 338

ab should give a morphism from ι∗P to the crossed module 0 : P ⊗ I(Q/P ) → Q/P, which represents 0 in 3 ab the cohomology group H (Q/P, P ⊗ I(Q/P )). However, the projection pr2 is a P -morphism, but is not in general a Q-morphism, as the above results show. In fact, in the next Theorem we give a precise description of the Postnikov invariant of ι∗P when Q/P is cyclic of order n. This generalises the result for the case P = Cn,Q = Cn2 in 13.3.5.

Theorem 13.3.9 Let P be a normal subgroup of Q such that P/Q is isomorphic to Cn, the cyclic group of order n. Let t be an element of Q which maps to the generator t of Cn under the quotient map. Then the ﬁrst Postnikov invariant k3 of the induced crossed module of the inclusion BP → BQ lies in a third cohomology group 3 ab H (Cn,P ⊗ I(Cn)). This group is isomorphic to ab P ⊗ Cn, and under this isomorphism the element k3 is taken to the element

[tn] ⊗ t.

Proof We have to determine the cohomology class represented by the crossed module

ab ξ : P × P ⊗ I(Cn) → Q.

ab Let A = P ⊗ I(Cn). As in 13.3.5 for the case Q = Cn2 ,P = Cn, we consider the diagram

N / M / φ / ZCn ZCn ZCn C∞

0 f3 f2 f1 0 / A / P × A / Q. i νn

Here the top row is the beginning of a free crossed resolution of Cn. The free Cn-modules Z[Cn] have generators n y4, y3, y2 respectively, C∞ has generator y1 and φ(y2) = y1 ,M(y3) = y2.(t − 1) (here C∞ operates on each 2 n−1 ZCn via the morphism to Cn); N(y4) = y3.(1 + t + t + ··· + t ). Further, we deﬁne f1(y1) = t, f2(y2) = n n (t , 0), f3(y3) = [t ] ⊗ (t − 1), and i(a) = (1, a), a ∈ A. Thus the diagram gives a morphism of crossed complexes, and the cohomology class of the cocycle f3 is the Postnikov invariant of the crossed module. 3 As in 13.3.5 since Z[Cn] is a free Cn-module on one generator, the cohomology group H (Cn,A) is isomorphic to the homology group of the sequence M N AAo o A where N is multiplication by 1 + t + t2 + ··· + tn−1 and M is multiplication by t − 1. It follows that N = 0, and it is easy to check that I(Cn)/I(Cn)(t − 1) is a cyclic group of order n generated by t − 1. The cocycle f3 n determines the element f3(y3) = [t ] ⊗ (t − 1) of A, and the result follows. 2

13.4 Dimension 2 cohomology with coeﬃcients in a crossed module and extension theory

In this Section we generalise the Deﬁnition of the second cohomology group to use coeﬃcients not in a G-module, but in a crossed module. 339

Definition 13.4.1 Let G be a group and let C(G) be a free crossed resolution of G, so that we are given a morphism of groups φ : C(G)1 → G inducing an isomorphism π1(C(G)) → G. Let M be the crossed module M = µ : M → P . As we have hinted above, we can define the second cohomology of G with coefficients in the crossed module M as

[C(G), E2(M); φ] the set of homotopy classes of morphisms of crossed complexes over φ where E2(M) is

· · · → 0 → · · · 0 → 0 → M → P, i.e. trivial in dimensions > 2 and M up to dimension 2.

Our aim is to analyse this cohomology group and relate it to other useful group theoretic information. We ﬁrst outline the classical theory of extensions of M by G1 of the form:

p 1 → M −→i E −→ G → 1.

Choose a section of p, i.e. a function s : G → E such that s(1) = 1 and ps = 1G. Then s deﬁnes a bijection of sets α : E → G × M

−1 −1 −1 by e 7→ (pe, i ((sp(e)) e)). Note that p((sp(e)) e)) = 1 since ps = 1G. The problem is to deﬁne a multiplication on G × M so that α is a morphism (and so an isomorphism). The section s deﬁnes a function k1 : G → Aut(M) using the fact that i(M) is normal in E. Assume there is a multiplication of the form

k1h (g, a)(h, b) = (gh, k2(g, h)a b), where k2 : G × G → M. In order for this multiplication to be a group structure and α to be a morphism the functions k1, k2 have to satisfy two conditions which make them what is called a ‘factor set’. The most complicated condition is that requiring the multiplication to be associative. This is the classical extension theory of Schreier, of which an exposition can be found in many places, such as books on group theory. We rephrase this in a more conceptual way using the notion of free crossed resolution. Also we generalise the theory (following Dedecker [73]) by considering extensions of the type of a crossed module (see also Taylor [164]).

Deﬁnition 13.4.2 Let M denote the crossed module µ : M → P . An extension (i, p, σ) of type M of the group M by the group G is:

1. an exact sequence of groups i p 1 / M / E / G / 1

so that E operates on M by conjugation, and i : M → E is hence a crossed module; and

2. a morphism of crossed modules i 1 / M / E

σ µ M / P i.e. σi = µ and me = mσe, for all m ∈ M, e ∈ E. Thus the action of E on M is also via σ.

1The literature uses the words ‘of’ and ‘by’ in also the opposite sense to that given here! 340

We shall write i p σ 1 / M / E / G / 1, E / P.

Two such extensions of type M

i p σ 1 / M / E / G / 1, E / P.

0 i0 p σ 1 / M / E0 / G / 1, E0 / P. are said to be equivalent if there is a morphism of exact sequences

σ 1 / M / E / G / 1, E / P

φ φ

σ 1 / M / E0 / G / 1, E0 / P. such that also commutes the right hand square. Of course in this case φ is an isomorphism, by the 5-lemma, and hence equivalence of extensions is an equivalence relation. We denote by

ExtM(G, M) the set of equivalence classes of all extensions of type M of M by G.

The usual theory of extensions of a group M by a group G considers extensions of the type of the crossed module χM : M → AutM. The advantages of replacing this by a general crossed module are ﬁrst that the group AutM is not a functor of M, so that the relevant cohomology theory in terms of χM appears to have no coeﬃcient morphisms, and second, that the more general case occurs geometrically, as in [165] and in [54] We shall use in this situation a much more general result of Brown and Porter ([55]).

Theorem 13.4.3 Let N = (ν : N → Q) and M = (µ : M → P ) be crossed modules, and let φ : Q → G be an epimorphism with Ker φ = µ(M). Let [N , M]0 denote the set of homotopy classes of morphisms k = (k2, k1): N → M of crossed modules, such that k2(Ker µ) = 1. Then there is a natural injection

0 E :[N , M] → ExtM(G, M) sending the class of a morphism k to the extension

1 → M → E(k) → G → 1 where E(k) is the quotient of the semidirect product group Q n M, in which Q acts on M via P . The function E is surjective if Q is a free group.

Let us recall the notion of homotopy between morphisms of crossed modules. For two crossed modules N = (ν : N → Q) and M = (µ : M → P ) a morphism k : N → M of crossed modules is a pair k = (k2, k1), satisfying conditions A homotopy h : k ' l between two morphism of crossed modules is a l1-derivation h : P → A such that for all p1, p ∈ P, m ∈ M, (k1p) = (l1p)(αhp) and k2m = (l2m)(hµm). These conditions are consistent with k1, k2 being morphisms of groups, given that k, l are morphisms of crossed modules. 341

These homotopies define an equivalence relation on morphisms N → M of crossed modules. This definition of homotopy is more special than that given in [175, 44]. See [43] for a discussion of the use of the more general homotopies in the classification of extensions. Let us compare this with the definition of cohomology with coefficients in a crossed module. Write [F stG, M] for the set of pointed homotopy classes of morphisms F stG → M.

Theorem 13.4.4 There is a bijection

st ∼ [F G, M] = ExtM(G, M).

Proof Clearly a morphism between crossed complexes C → E2(µ) is just a morphism between the corresponding crossed module part satisfying the extra condition on Ker δ2. 2

Let us examine this in relation to the usual exposition of extensions. Here we take the crossed module M to st be the bottom part of the standard free crossed resolution F∗ (G), so that given an extension we are considering the diagram:

st δ3 / st δ2 / st φ / / F∗ (G)3 F∗ (G)2 F∗ (G)1 G 1

k2 k1 i p 1 / M / E / G / 1

ω / M µ P

st where the part M → E is regarded as the beginning of an exact (acyclic) crossed complex. Since F1 (G) is free on the elements of G and with generators [g], g ∈ G, the choice of a morphism k1 such that pk1 = φ is equivalent to the choice of a section s of p as detailed above (except for the condition s(1) = 1). Because φ(G) is free and the middle row is exact, we know that we can obtain a morphism of crossed complexes k = (k1, k2) as shown, and the morphism k2 is determined by its values on the elements [g, h], g, h ∈ G. 1 We set k = ωk1. Then we have the two factor set conditions: 1 (FS1) k δ2 = µk2;

(FS2) k2δ3 = 0.

These can be expanded out in terms of the formulae for δ2, δ3.

The extension E(k) corresponding to k has elements written [[p, a]], p ∈ C1(G), a ∈ M and is the quotient of C1(G) n M by the elements 2 −1 (δ2[g, h], (k [g, h]) ), g, h ∈ G.

There is a bijection G × A → E(k)

(g, a) 7→ [[[g], a]] 342

Then

[[[g], a]][[[h], b]] = [[[g][h], a[h]b]] [h] = [[(δ2[g, h])[gh], a b]] [gh] [h] = [[[gh](δ2[g, h]) , a , b]] (*) = [[[gh], (k2[g, h])[gh] a[h] b]].

This is near to the usual multiplication on the extension determined by a factor system. If we had chosen the −1 2 [h] convention that δ2[g, h] = [gh] [g][h], then we would have obtained the usual rule [[[gh], k [g, h] a b]] for the right hand side of (*).

13.5 Obstructions to existence and classiﬁcation of extensions

We now show there is an obstruction to realizability, the same as the classical result of Eilenberg-Mac Lane ( [129], Ch.IV, Thm.8.7). sprinkle more references The point is that crossed complexes allow for methods analogous to those of chain complexes as in standard homological algebra, but including non-abelian information of the type given by crossed modules. The obstruction result arises from an exact sequence of a ﬁbration of crossed complexes. This allows us to give a proof analogous to that given for the classical case using topological methods by Berrick in [14]. A direct proof may also be given by extending the methods of Mackenzie [136] to more general crossed modules than M → Aut(M).

Deﬁnition 13.5.1 An extension of M by G of type M = (µ : M → P ) determines a morphism

θ : G → Q = Coker µ, which is dependent only on the equivalence class of the extension, and θ is here called the abstract M-kernel of the extension. The set of extension classes with a given abstract M-kernel θ is written Ext(M,θ)(G, M).

n Notice that an abstract kernel θ : G → Q = Coker µ gives an action of G on Ker µ = A. We write Hθ (G, A) for the cohomology groups with coeﬃcients in the G-module A.

3 Theorem 13.5.2 There is an obstruction class k(M, θ) ∈ Hθ (G, A) associated to any abstract M-kernel θ : G → Q. The vanishing of this obstruction class (k(M, θ) = 0) is necessary and suﬃcient for there to exist an extension of M by G of type M with abstract M-kernel θ. Further, if the obstruction class is zero, then the equivalence classes of such extensions are bijective with 2 Hθ (G, A).

Proof Recall that M is the crossed module µ : M → P , and A = Ker µ, Q = Coker µ. Let ξM, ζM denote the crossed complexes in the following diagram of morphisms of crossed complexes 343

... / 1 / 1 / M / P M

id id i ... / 1 / A / M / P ξM

id q q ... / 1 / A / 1 / Q ζM where q is determined by the quotient morphism q : P → Q. Since q : ξM → M is an epimorphism in each dimension, it is also a ﬁbration of crossed complexes and therefore, since F stG is free, the induced morphism of morphism complexes

st st q∗ : CRS∗(F G, ξM) → CRS∗(F G, ζM) is also a ﬁbration of crossed complexes (9.4.6). Since F stG is free and ξM is acyclic, there is an identiﬁcation

st ∼ π0CRS∗(F G, ξM) = Hom(G, Q).

3 Further, each morphism θ : G → Q determines an action of G on A and so a cohomology group Hθ (G, A). st Then π0CRS∗(F G, ζM) is the union of all these cohomology groups for all such θ.

The function π0(q∗) takes a morphism θ to a cohomology class

3 k(M, θ) ∈ Hθ (G, A), called the obstruction class of (M, θ). If k : F stG → ξM is a realisation of θ, then qk represents k(M, θ). If this class is 0, then there is a 0 homotopy h : qk ' l, say, where l1 = qk1, l3 = 0. Hence k3 = h2δ. So there is a homotopy k ' k where 0 0 0 k 1 = k1, k 2 = k2 − δh2, k 3 = 0.

st Let F be the ﬁbre of q∗ over l. Then π0F may be identiﬁed with the set [F G, M] of homotopy classes of morphisms F stG → M, and so with the classes of extensions of A by G of type M. The exact sequence of the

fibration q∗ with fibre F yields, given the above identifications, the exact sequence

2 3 0 → Hθ (G, A) → ExtM(G, M) → Hom(G, Q) → Hθ (G, A)(?) where the three right hand terms have base points the class of the split extension, the morphism θ, and zero respectively. The obstruction part of the Theorem follows immediately. The exact sequence of a ﬁbration of 2 crossed complexes given in Theorem 9.4.2 yields that the group Hθ (G, A) operates on ExtM(G, M) so that the classes of extensions of type M with abstract kernel θ are given by this group. This completes the proof of the Theorem. 2

13.6 Local systems

The homotopy classification result of Theorem 9.2.14 suggests that if A is a crossed complex, and X is a space, with singular complex SX, then the set [Π|SX|,A] may be thought of as singular cohomology of X with 344 coefficients in A, and written H0(X,A) [43]. With our present machinery, it is easy to see that this cohomology is a homotopy functor of both X and of A. We show in this section that H0(X,A) is (non-naturally) a union of Abelian groups, each of which is a kind of cohomology with coefficients in a generalised local system. Let C and A be crossed complexes. In examples, C is to be thought of as ΠX for some CW -complex X. By a local system A of type A on C we mean the crossed complex A together with a morphism of groupoids

A : C1 → A1 such that A(δC2) ⊆ δA2. This last condition ensures that A induces a morphism of groupoids π1C → π1A. It is also a necessary condition for there to exist a morphism C → A extending A. The morphism A induces an operation of C1 on all the groupoids An for n ≥ 2. By a cocycle of C with coeﬃcients in A we mean a morphism f : C → A of crossed complexes such that f1 = A. By a homology of such cocycles f, g we mean a homotopy (h, g): f ' g of morphisms of crossed complexes such that h0x is a zero for all x ∈ C0, and δh1 = 0. The set of homology classes of cocycles of C with coeﬃcients in A is written [C,A]A.

Proposition 13.6.1 Let C, A be crossed complexes and let A be a local system of type A on C. Let H = π1A and let A0 be the crossed complex which is H in dimension 1 and agrees with A in higher dimensions, with H as groupoid of operators and with trivial boundary from dimension 2 to dimension 1. Let A0 be the composite

A C1 −→ A1 → H.

Then a choice of cocycle f with coeﬃcients in A determines a bijection

0 [C,A]A → [C,A ]A0 , and hence an abelian group structure on [C,A]A.

Proof We are given f extending A. Let g be another morphism C → A extending A. Then g1 = f1 and δg2 = δf2. For such a g we deﬁne rgn = gn − fn , n ≥ 2. Clearly rgn is a morphism of abelian groups for n ≥ 3. We prove that it is also a morphism for n = 2. Let c, d ∈ C2. Then

rg2(c + d) = g2(c + d) − f2(c + d)

= g2c + g2d − f2d − f2c

= g2c − (f2c) + g2d − f2d

= g2c − f2c + g2d − f2d since δg2 = δf2

= rg2c + rg2d.

[1] Clearly rgn is a C1-operator morphism where C1 acts on A via A. Also δrg2 = 0 and for n ≥ 3, 0 δrgn = rδgn−1. So we may regard rg as a morphism C → A extending A. It is clear that r defines a bijection between the morphisms g : C → A extending A and the morphisms g0 : C → A0 extending A0. Next suppose that (h, g) is a homology g ' g as defined above. Then δh1 = 0. Hence h1 defines uniquely k1 : C1 → Kerδ2. Further we have for n ≥ 2

gn = gn + hn−1δn + δn+1hn.

For n ≥ 2 let kn = hn . Then kn is a C1-operator morphism. Now for x ∈ Cn and n ≥ 3, we have hn−1δnx + δn+1hnx lies in an abelian group, while for n = 2 it lies in the centre of A2 and so commutes with f2x . It follows that (k, rg) is a homology rg ' rg. Conversely, a homology rg ' rg of A0-cocycles determines uniquely a homology g ' g of A-cocycles. It 0 follows that r deﬁnes a bijection [C,A]A → [C,A ]A0 as required. 345

0 Notice also that the set [C,A ]A0 obtains an Abelian group structure, by addition of values, and with the class of rf as zero. 2

Let C be a reduced coﬁbrant crossed complex, and let A be a reduced crossed complex. We are interested in analysing the ﬁbres of the function

η :[C,A]∗ → Hom(π1C, π1A).

α −1 We write [C,A]∗ for η (α). This set may be empty. We will elsewhere analyse the ﬁrst obstruction to an element α lying in the image of η . Here our aim is to show that if f : C → A is a morphism realising −1 α : π1C → π1A then f determines an Abelian group structure on η (α). We recall from [45] the relations between crossed complexes and chain complexes with operators. There is a category Chn of chain complexes with groupoids as operators and two functors

∆ : Crs À Chn :Θ such that ∆ is left adjoint to Θ . Hence if D is a chain complex with a trivial group of operators, then

r (NΘD)r = Crs(Π∆ , ΘD) = Chn(∆Π∆r,D).

r r In this last formula, ∆Π∆ consists of the chain complex C∗∆˜ of cellular chains of the universal covers r r r of ∆ based at the vertices of ∆ , with the action of the groupoid π1∆ . Since D has trivial group acting, it follows that

r r Chn(∆Π∆ ,D) = Chn(C∗∆ ,D),

r r where C∗∆ is the usual chain complex of cellular chains of ∆ . This shows that ND coincides with the simplicial Abelian group of the Dold-Kan Theorem [75]. [1] [1] Let H = π1A. In the last example we deﬁned A . We consider the pair (A ,H) to be a chain complex with H as group of operators.

Proposition 13.6.2 Let C,A be reduced crossed complexes such that C is coﬁbrant. Let f : C → A be a morphism inducing α : π1C → π1A on fundamental groups. Then f determines a bijection

α ∼ [1] α [C,A]∗ = [∆C, (A ,H)] , where the latter term is the set of pointed homotopy classes of morphisms which are morphisms of chain complexes with operators and which induce α on operator groups.

Proof Let p : A → A(1) denote the ﬁbration of the last example, so that A[1] is the kernel of p. Let

G = π1C. Recall that X(H, 1) denotes the crossed complex which is H in dimension 1 and is zero elsewhere. The (1) (1) projection A → X(H, 1) is a trivial fibration. Since C is cofibrant, the induced morphism CRS∗(C,A ) → (1) CRS∗(C,X(H, 1)) is also a trivial fibration. It follows from Proposition 9.3.8 that CRS∗(C,A ) has component set Hom(G, H) and has trivial fundamental and homology groups. 346

Suppose that f : C → A induces α : G → H on fundamental groups. Let F (f) be the ﬁbre of CRS∗(C,A) → (1) CRS∗(C,A ) over pf. Then the exact sequence of this ﬁbration yields an exact sequence

η 1 → π0F (f) → [C,A]∗ −→ Hom(G, H)

such that the ﬁrst map is an inclusion with image η−1(α). 0 Let A = f1. Then π0F (f) = [C,A]A. So by Proposition 13.6.1, π0F (f) is bijective with [C,A ]A0 . But in terms of the functors relating crossed complexes and chain complexes with operators given in [45] recalled above, we have A0 = Θ(A[1],H). The proposition follows directly from the adjointness of ∆ and Θ. 2

Corollary 13.6.3 If α : π1C → π1A is realisable by a morphism f : C → A, then a choice of such morphism α determines an Abelian group structure on [C,A]∗ .

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