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Acyclic Models

May 1997

Department of Mathematics ancl S tatistics McGill University, Montréal Canada

A thesis submitted to the Faculty of Graduate S tudies and Research in partial fuEllment of the requirements for the degree of Master of Science National Library Bibliothique nationale I*m of Canada du Canada Acquisitions and Acquisitions et Bibliographie Seivices services bibliographiques 395 Wdtimgton Street 395, nie Wellington ûttawa ON K1A ON4 ûttawaON K1AON4 Canada i;ariada

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The author retains ownership of the L'auteur conserve la propriété du copyright in this thesis. Neither the droit d'auteur qui protège cette thèse. thesis nor substantial extracts fiom it Ni la thèse ni des extraits substantiels may be printed or othefwise de celle-ci ne doivent être imprimés reproduced without the author's ou autrement reproduits sans son permission. autorisation, Dedicated to my husband Robi and daughter Jita, with love and affection. Acknowledgement s

I wish to express my deepest gratitude to my superviser Prof. M. Barr for his guidance, invaluable suggestions and encouragement throughout the preparation of this thesis. He also greatly helped in making the diagram. 1 wish to express my sincere gratitude to Prof. S. Majumder. who introcluced me to homology group and to Prof. A. H. Beg, Prof. S. Bhattacharjee and Prof. A. S. A. Noor, mho encouraged me to corne here. 1 would like to thank my student friends Lassina Dembele and Gülhan Alpargu for their help and numerous advices on cornputer materials. 1 specidly acknowledge their friendly attitude and the nice and pleasent atmosphere they made during the years we spent togather in this department. 1am also grateful to Lassinn, who reads my thesis very carefuly and made some grammatical corrections and translated the abstract into French. Very special thanks go to Hasiba Islam for her warmest friendship Ùuring my studying and to my parents and my brothers for their mental support and for t&ng care of my daughter during my absence. I thank rny husband for his understanding and also 'apologize' to him for dl the hard time he endured when 1 was away. My studies at McGill University are supported by Canadian Commonwealth Scholarship and Felloship Program. Abstract

In this thesis we describe a new version of acyclic models, which was first given by Barr, that gives the Theorem of Barr and Beck, and of André as special cases. We begin the thesis with Beck's definition of module and we describe how he used this definition, in conjunction with the theory of triples, to define homology t heories. The t heory described here is based on the notion of acyclic classe^. An acyclic class is a class of objects in a category of chah complexes and corresponds to a class of arrows, whose mapping cones they are. We also give an answer to the question. where do acyclic classes corne fkom. We conclude the thesis by showing that Cartan- Eilenberg cohomology of groups, of associative algebras. and of Lie dgebras are the sarne as cotnple ones, using proofs based on those presented by Barr. Résumé

Dans ce mémoire, nous décrivons une nouvelle version de modèles acycliques ini- tialement introduite par Barr. Cela conduit aux Théorèmes de Barr et Beck, et de André comme cas particuliers. La thèse commence par la définition des modules. par Beck, qui en conjonction avec la théorie des triples conduit aux théories de l'homologie. La théorie décrite ici est basée sur la notion de classes acycliques. Une classe acyclique est une classe d'objets dans une catégorie de chaines complexes correspondant à une classe de flêches dont elles sont les 'mapping cones'. Nous répondons également à la question à savoir d'où viennent les classes acycliques. Nous concluons le mémoire en montrant que la cohomologie de Cartan-Eilenberg pour les groupes. les algèbres associatives. et les algèbres de Lie sont les mêmes que celles de leurs cotriples. La preuve est basée sur des résultats de Bm. Contents

Acknowledgements iii

Abstract

Résumé

Introduction

1 Triple Cohomology 1.1 Simplicial objects

2 Acyclic Models 2.1 Acyclic classes 2.2 Properties of acyclic classes 2.3 Examples of acyclic classes

3 Cohomology of Groups and Algebras

4 Calculuses of fractions 4.1 Some more theorems on double complexes

5 Application to homology on manifolds

6 Cartan-Eilenberg cohomology and cotriples

6.1 Regular epimorphisms and regular categories 6.2 The Cart an-Eilenberg setting 6.3 Themaintheorem 6.4 Applications of the main theorem

Bibliograp hy Introduction

The main purpose of this thesis is to fill in details in papers [3] and [4],we also add some new material. Chapter 1 presents triples and cotriples, and their relation with a pair of adjoint furictors [19]. It also describes Beck module and the way a cotriple G gives rise to a cohomology theory in a tripleable category dT [SI. In Chapter 2, we define e-presentable and G-contractible fùnctors, where G is a endofunctor and E is a counit. We also define Acyclic classes and Mapping cone sequence, and discuss some of their properties; some examples of Acyclic classes are given. Almost all results are fkom [41. The question of where do acyclic classes corne fkom is discussed, and the way they match with functors as weU. The Barr-Beck Theorern and its proof by Barr (41 using acyclic classes is also presented. In Chapter 3, we discuss the naturality of the isomorphism between the cotriple cohomology of groups and Eilenberg-Mac Lane cohomology group [13],and between the Hochschild cohomology group [18] and cotnple cohomology of associative algebras. These are results from [5]. Chapter 4 discusses the properties of C, the class of arrows whose mapping cone is a member of an acyclic class. Most of the results are from Barr [4]. Here we prove C-"& E C-'(%/.-), where - is defined by (f,o) .- (f',~') (each pair has same codomain) if a f a'f and aa a'of E C and Y'%?is a category of fractions [Ir]. We also present some properties of double complexes 131. In Chapter 5, we state André's Theorem and then find the homology of manifolds in detail. We follow the proofs given by Barr in [4]. In Chapter 6. we present Cartan-Eilenberg cohomology of groups. associative algebras and Lie algebras [8] are the same as (with sorne shifted dimensions) the cotriple cohomology of groups, associative algebras and Lie dgebras respectively. The discussion follows [3]. Chapter 1 Triple Cohomology

Let J3/ be a category. T = (T, q, p) is a triple in d if T:s;/ -t .d is a functor.

7: ld -t T and p: TT -+ T are natural transformations such that p Q qi? = p 0 TV=

1~:T -, T,p 0 pT = p O Tp: TTT + S. The natural transformation rl is called the unit and p is called the multiplication of T.

1.0.1 Relation between adjoint functors and triples Let F be a left adjoint of U. wbere F: d + 9, then there is a natural isomorphism Homd(.l, [TB) 9 Homa(FA,B)for A E Ob(&), B E Ob@'). Putting B = FA, we get a natural transformation q: ld -t UF and for -4 = UB we have s: FU -, la: and they satisfy EFoF~= IF, UEO~U= lu. P. Huber [19] observed that T=UF:d-td T= -A' and G=FU:&?+&?

t:G+ lS 6=FlrU:G-GG T is a triple in d and G is a cotriple in 9.

1.0.2 Definition Let V be a category and S E Ob(%). The cotego~y(%.-.Y) qwho~eobjects are aiiowJ C X and anows are commutative triangles -y + C

-, C' -t X; C,C' E Ob(%) is called a dice category over -y.

Here is Beck's definition of a module. His definition is appropriate for the kind of module that is a coefficient module for cohomology. For groups, commutative 3 algebras and Lie dgebras, these are left modules; for associative algebras, the appropriate notion is that of two-sided module.

1.0.3 Definition Let a! be a category and let A E Ob(&). The category of Mod(A) is the category of abelian group objects of (d,-4)

Assuming that a category d has finite products, by the abelian group objects of a! we mean that for each A E Ob(&) there are riaps rn: A x -4 -t A, i: .1 -4 and u: 1 -t A called multiplication, inverse and zero maps respectively. They are assumed to make the following dia.grarns are commutative.

-4 x -4 *4 x -4 .4 x -4 -lxa -2x1

A~1~~4x.A-1x4uxl

Here A: A + A x A denotes the arrow whose components are the ideutity on each 11 coordinate and O: A + A denotes the composite -4-, 1 -4 A. So the objects of Ab(&) are (A, m, i, u) which satisfy the above conditions. An arrow f: (A,n, u) -t (B, m, u) is f: 4 + B, such that the following diagrams conunute.

Now, we show how a category of Mod(R) is the category of abelian group objects of (9,R), where 9 is a category of commutative rings and R is a commutative ring with unity. For this, let M be a two sided R-module. S = R x M is an abelian group under the coordinatewise addition. The multiplication on S is defined by

This makes S into a ring. Then S -r R, projection on the first coordinate is in Ob(9,R ). In (9,R) the product is the fiber product and the terminal object is R. So the product of S with itself is

Then the abelian group object of (9:R) is determinecl by the arrows: the niulti- plication takes the pair ((r, s), (r, 5')) to the element (r, s + s'); the inverse map is given by (r, s) H (r, -3) and the zero map takes r to (r, O). Thus an R-module S is an object of Ab(l,R). The case of associative dgebras is similar. Here we will show that Beck modules over an associative algebra are two-sided (although, for the previous case Ive get two- sided modules; the fact that the larger ring is commutative forces the operations on the two sides to coincide). For this, let A be an associative K-algebra. Ii a commutative ring and B A an abelian group object in (d,A), where d is a category of associative K-algebras. So we need to show that the A--algebra B is. as a K-module, B S A x fil, in which the multiplication is given by

(a,m)(af, mi) = (aa', am' + ma') for a two sided A-module M.

By the definition of abelim group object, t here is a zero map tr: .A -+ B such that 4 -, B -+ A is the identity. Thus, as K-module at least, B Z -4. x M. where il/l = ker(B + A). We mal- si-ppose that B = A x Ad, so the map B

+ A is the projection on the hst coordinate. Since the projection is an algebra homomorphism, the product in B is given by the formula

(a, rn)(at,m') = (au', s(a, m, a', ml)) where s is a function of that variables. The distributivity of law then gives iis

~(a,rn, a', m') = s(a, O, a', O) + s(a,O,O,mf) + s(0, m, al.O) + 40-m. O. m')

Since u is also an algebra homomorphism, it follows from u(auf)= u(a)u(a')that

(aa', O) = (a,O)(at, O), so s(a,O, a', O) = 0. Putting s(a, O, O, ml) = am'. ~(0,rn, a', O) = ma' and s(0, m, O, m') = mm', the above formula becomes

(a, n)(af,m') = (ua', am' + ma' + mm') and using the associativity of multiplication we see that M is a two sided A-module. -4gain by the property of abelian group objects, chere is ao algebra hiiioluor- phism we denote a: B XA B B. Since the product is the fiber product. the element of B x A B cm be represented either as ((a, rn)(a. m')) or as 3-tuples (a, m, ml). Define a(a, rn, m') = (a,m + m'). The fact that rr preserves multiplication, implies that a((l,m,O)(l,O,mt))= a(l,m,O)a(l,O,rn'); wLch gives that (1,rn + m') = (1,rn + ni' + mm1), hence mm' = O. This verifies the multiplication in B which we needed.

Be& modules in other categories are just one sided. In case of group if K is a group and Y - n is a n-module, that is Y -+ ?r is an abelian group object of (g,R), where (B is the cotegory of groups, then there is a zero section n -t Y such that r -t Y -r ir is the identity. Thus Y E M x rr: where M = ker(Y --+ R) is a left K-module. rr acts on M by xrn = u(x)mu(x)-',where u is the zero map. 1.1 Simplicial objects

Let -/Ibe the category of the classes b] = (0, 1, . . . , p} where p is a non negative integer and the arrows between [pl -t [q] are order preserving functions.

A sirnplicial object in a category %' is a contravariant functor S: A? -+ 55. By a simplicial set we mean a sirnplicial object in the category of sets. A simplicial R-module is a simplicial object in the category of R-modules. Denote S[O] by So, S[1] by S1 and so on. Thus a simplicial set is a sequence So?SI, Sz, . . . with Sn - Sn-*and Sn-, satisfiing the identities:

Each simplicial module S determines a chain complex C = C(S)with C, = Sn and the boundary operator d: C, -t Cn-iis the alternating sum of the face operators; d =dO -dl+**- +(-l)"dn. Sadd: Cn -t CnV2is

i< j- 1 isk Let T be a triple in d.A T-algebra is a pair (X, e), where 9 E Ob(&) and J: TX -, X is an arrow in d such that

F is called the T-st~wtweof the algebra. A map f: (XI5) -+ (2- 0) is an arrow of T-algebra if f: X -t Y is an arrow in d and 8 0 Tf = f 06 The category of T-algebras will be denoted by dT and dTis said to be tripleable. For example. if d is the category of abelian groups and T is the triple () @ R, R a ring. Shen a

T-structure (: A Qi R -t R satisfy the unitary and associative properties.

Naw let dT be tripleable. There is then an adjoint pair of functors F -( Ir. where F:d * dT is the free T-dgebra functor defined by F'i = (Tri,PX) ancl if f: X -, Y is an arrow in d,then F(f): (TX, PX) -t (Tl;pY) is an arrow in d,because pY 0TTj = f .PX. The forgetful functor Li: dT -, d is given by U(X,{) = X. So by the above relation T = UF , q: Id - T is the unit and p = UeF is the multiplication of T, G = FU is a cotriple in dT,the counit E: G -, 1,~ is defùied by c(X, () = (: (TX,pX) (.Y. f) and the comultiplication

6 = FqU: G -+ GG is defined by &-Y, 5) = Fqu'(X, () = FqX = FTX = (TTX,pTX). The cotriple G = (G,c, 6) giws a cohomology theory in dT.The theory tvill have coefficients in an (X, c)-module, for a 1-algehra (X, 5). To simplify notation we cal1 it an .Y-module, for a T-algebra X. -4 cotnple G in dTnaturally operates on (dT,X) as well. The resulting cotnple (G, X) has the folloniing properties :

P €TV P (G,X)(W -t -X) = GW -- CV -+ X P (c, X)(W- X) = thernapGW + W such that the triangle

P commutes and (6 X)(W -4 X) = the map GW + G2Wwhich makes Let Gn be the nth iterate of G, then we get an augmented simplicial object

in the category (dT,X), where Gn+'W is the n-dimensional component.

and the usual simplicial identities hold. Each GntlW is an algebra over .Y. and each face operator Ei as well as degeneracy operators bi are wows over X.

Now applying Hornx(-, Y) to (Gnfi)n>-l, - we get an augrnented simplicial object in the category of abelian groups:

Hornx(W, Y)-t Hornx(GW, Y)=t ~orn~(G~W.Y) --t ~orn,(C;~~.Y)- * The associated cochain complex of abelian groups is:

with coboundary operators d, = C(-l)'Hom(ciW, Y). The cohomology of this coiiiphx is denoted by HiW, Y)xwhere 15' + X is s T-algcbra over .Y and Y - X is an X-module. Chapter 2 Acyclic Models

Suppose 3Y is a category, G: 3- + 3Y is a functor and E: G -t Id is a natural transformation. Let d be any abelian category and F:3 -+ d a functor; then there is a chain complex functor FG' that goes Îrom 55" to the category of chah complexes over d,whose n-th term is FGnfl and whose n-th boundary operator is Cy=-,(-1)' FGieGn-'. This chain complex is augmented over F where the augmentation is Fe: FG -t F, for the naturality of E gives the following commutative

FG-F where the composition F@G - FGe) = O holds. Wow if Ii = {Ii, 1 n 2 O) is another chah complex functor from 2-to d together with a boundary operator d, then KG' is a double complex functor. We Say that IC is e-presentable if for a11 n 2 0, the augmented chain complex &Go -, IC,, -, O is contractible and that IC is weakly e-presentable if for each n 2 0, the augmented chain complex IC,tG'

-, hfn-, O is acyclic. If L -, Li -, is a chah complex functor. we Say that L is G-contractible if the chain cornplex functor LG -t L-*G - O is contractible and Gacyclic if LG -t LIG-, O is acyclic.

Let U:%' + a be a forgetW functor. An exact sequence O -t L + C -t K O of objects and arrows of '& will be cded U-split if O UL - UC + CTK + O is split in 28; we denote the category of chain complexes by % and the category of graded objects by 8. If K = {& 1 n 2 O) is a chain complex we let SI< be the chain complex defined by (SK),= with boundary operator 4.where d is the boundary operator of K. SK is called the suspension of K.

2.1 Acyclic classes

A class r of objects of '& will be cded an acyclic class provided : AC-1. The O complex is in r. AC-2. The complex C is in I' if and on1y if SC is. AC-3. If the complex K E I' and is homotopic to a complex L?then L E r. AC-4. Every complex in I' is acyclic.

AC-5. If %' is a double complex and all of its rows are in r. then the total complex * of %' belongs to I'.

2.1.1 Mapping cones Suppose that f: I< -, L is a chah map in CE. A mapping cone of f gives an example of an exact sequence in the following way: construct a complex C = CI by letting C, = Ln 3 rin-[. Then C is a chain complex witli the boundary operator (O jd)and the requenre 0 - C SI< + 0 is exact.. C is called the mapping cone of f.

We note that the exact sequence O -t L + C -t Sh-+ O is lT-split. This turns out to characte~zemapping cone sequences. &O, we know that exact sequence induces a long exact sequence of homology with the connecting homomorphism En: Hn(SK)+ Hn-I(L). But H,(SK)= Hndi(h')?thus En is just Hn-i(f ).

2.1.2 Proposition AU-split ezact sequence

is i~omorphicto the mapping cone of a unique rnap S-'K -t L. Proof. Let O + L + C + O be a CI-split exact sequence. Then, up to isomorphism, C,, = Ln $ & and the inclusion and projection maps are

( O 1 ), respect ively. The boundary operator on C have a matrix say

Now from

and

we conclude that d' = dL, g = O and d" = dri, where dL is the boundary operator on L?dK is the boundary operator on K, g: L -t SIC and f: S-III: -+ L. Hence the boundary operator on C is

implies that j 0 dK + dt 0 f = O, so that f: 9% -t L is a chah map. where the boundary operator on S-'I' is -dK. Thus C is the mapping cone of f. O

2.1.3 Proposition Let C be the mapping cone of f: K -, L. If I< has tîiuial homology, then the inclusion i~ a homology equivalence. If Ii is contractible then L C has a left inverse that is a homotopy inuer~e.

Proof. If has trivial homology, then fiom the exact homology triangle. we have

L + C is a homology equivalence.

Now suppose K is contractible. Let s: & + be the contracting homotopy, which means that sd + ds = 1. in all dimensions n 2 O. The inclusion i: L -t C on C the boundary operator is (: jd)*sO

Thus j is a chah map and the composite j i = ( 1 js ) = 1. Now let s: C,

-' Cn+li n 2 O, have matrix (O Q). Tbeo are have

w hile

'I'hus ds + sd = 1 - ij; which means that ij is homotopic to the identity. Sirnilarly, if L has trivial homology, then the map C:+ S-l Ii is a homology equivalence and if L is contractible then this map is a homotopy equivalence. O

2.1.4 Theorem A map of conzplezea is a homology equivalence if and only if its mapping cone is acyclic and it i~ a homotopy equivalence if and only if its mapping cone cont~uctibte.

Proof. First let j: K -, L be a map such that H(f) is an isomorphism and let C be the mapping cone of f. Then the exact homology sequence gives us Hn(C) = 0, for ail n 2 O. Conversely, if C is acyclic then again from that exact homology sequence we have H(f) is an isomorphism.

Next suppose that f: Ii -+ L is a homotopy equivalence. Shen there is a map g: L -, K and maps s: K -r IC and t: L -r L such that td+ dt = 1 - fg and sd + ds = 1 - g f. Let d be an abelian category. If F: 2' -t d is any functor, then for any object Z of d,there is a functor we denote Horn(Z. F):2.

-P d defined by Horn(Z, F)(X) = Hom(Z,FX). Similady, if F = ({F,).d) is any chain complex functor, we get a chah complex functor Hom(Z.F) whose nth term is Horn(Z, F,) and boundary operator is Horn(& d). We can carry out this construction successively with Z = li, Z = L and Z = C. Horn@, f)is the map Hom(Z,K) -, Hom(Z, L). Thus

is the mapping cone of sequence of Horn(& f) and it is the homotopy equivalence using Hom( 2, y), Hom(Z, s) and Hom( 2,t ) and a homotopy equitalence is certainly a homology isomorphism. Hence fiom the exactness of the homology triangle of (A)

Ive have the complex Hom(Z, C) is exact. Let Zn = ker(d: C. + C,- 1 ) and denote the inclusion map Zn -, Cn by in. Since di, = O, in is a cycle in the cornplex Horn(&, C) and since that complex is exact, in is also a boundary, so that there is a map 2,: Zn + C,+*such that dz, = in. Now the image of d: C. -, C.-, is included in so that we can form the composite and using this we get cZ~,-~d= d, so d(l - ;,-id) = O. We can then compose 1 - with r, and get dz& - ~,-~d)= 1 - zn-,d. Let s, = z,(l - then we get

which means that s, is a contracting homotopy in C. Conversely, suppose that C is contractible. Let the contracting homotopy u have matrix (i :) . ~hcnthe matrix du + idis /dt+ fg+td dr- fs+tf -rd\

If this is the identity matrix, we conclude that dt + f g + td = 1, -dg + gd = O and ds + g f + sd = 1. Thus g is a chah map and a homotopy inverse of f. 13

j 2.1.5 Proposition If O -+ L - C 5 K + O is a U-split ezaet sequence of chain complezes, then I' id homotopic to the mapping cone CI and L is h,omotopic to the mapping cone Cg.

Proof. Since the two parts are dual, we only need to show that Iï is homotopic to Cf.The sequence O + UL -* WC -+ UI' -t O is split esnct. so let u: LX

-i UL and v: UIi + UC be the maps such that trolTf = 1. Ugou = 1 and

U f 0 u + v 0 Ug = 1. Thus UV = O. Since the mapping cone of f and that of LT f /V\ are equivalent, we prove that IC is homotopic to CUI. Now we have ( -UdU :Ii- -r Cu/ is a chain map, for

The product (CTg O) = 1 and the other composite is Now we have

Hence # is homotopic to Cf.

2.1.6 Arrows determined by an acyclic class Let r be an acyclic class and C denote the clasa of all arrows f whose mapping cone belongs to r. So if f is in C then kom AC-4 CI is acyclic. Let ri = the class of chain cornples mith nul1 homology and ro = the class of contractible chain cornplex, then roC I' C rI.Let Cland Co be the corresponding class of rl and ïorespectively. Thus Co Ç S C Ci. Hence by Theorem 2.1.4 the class C lies between the class of homotopy equivalences and that of homology equivalences.

2.2 Properties of acyclic classes

2.2.1 Proposition If O + L + C + K -, O id a U-~plitexact seqoence of chain complexe^ and if any tuio belong to I', then so dues the third. Proof. First suppose that L and K belong to I'; hence by AC-2, S-'K belongs to r, and since the sequence O -, L + C - I< -+ O is a U-split exact, C is the mapping cone of f: 27% _t L. We cmwrite this as a double complex as follows:

The boundary operator on C is (O jd), the square of this operator ir zero rives the squares cornmute. If we repl=ce -d by d the squares anticornrnute. So this can be considered as a double complex. Each row of this complex is in r, hence from BC-5 the total complex belongs to r. But here the total complex is defined by C, = Ln$ Ii, which is the mapping cone of f, thus C belongs to r. Next suppose that L adC belongs to r. We have just seen that the mapping

cone of L -t C belongs to I'. And since the given exact sequence is Cr-split by Proposition 3.1.5, li is homotopic to the mapping cone of L -+ C. Thus by AC-3.

I< belongs to P. Dually, if C and K are in î, so is L. O

2.2.2 Proposition C is closed under composition.

Proof. Suppose f: K -r L and g: L -t M are in C. Then CIand Cgare in I'. The n-th term of S''Cg is Mn+'$Ln and the n-th term of Cl is L., lï,,di.So we can define h: S%', -r CIby . The n-th term of Chis L,$Kn-I 3~%f~$L,-i.

Thus there is a U-split exkt seclience O -t Cf + Ch -, Cg + O and hence by Proposition 2.3.1 Chbelongs to I'. The boundary operator of Ch have matrix Also the n-th term of the mapping cone of g f is 4, IL-1 and the boundary

-+ operator is ( ( 5). N6w let -id: Ln Ln be n rnnp, then ),, = Lni in-, ? where the boundary operator of Cid is - hGd,s=(O ( d). -1 O contracting hornotopy. Thus Cid belongs to r. The constniction of Ch gives that the sequence

is U-split exact, where

Matrix multiplication shows that these are chain maps. Thus by Proposition 3.2.1 we have Cgfis in I' and hence gf is in S. O

2.2.3 Theorem Suppose C = C.. = {Cm, 1 n 2 O, n 2 O) iu a double cornplez augmenied over the single complez CI.and that for each n 3 0, the complex

' -' Cmn+ Cm-ln-' + Con-' C-ln + O belongs tu l?. Then the induced map f: C -+ Ci. is in C.

Proof. We can mite f: C -, Ci. as a double complex as follows: Al1 rows are in r, hence by AC-5 the total complex is in î. But here the total complex is : Co = CIO,Ci = Cil@ Cao, C2 = 9 Col 3 CIO,..., C, = C'-ln $ Ci+j=,-rCij7 which is just the mapping corie of f:C + C-!.. Hence f E C. O

2.3 Examples of acyclic classes

2.3.1 Acyclic complexes Let î consists of the acyclic complexes, in that case C consists of homology isomorphisms. AC-1, AC-2, AC-3 and AC4 are obvious. So we only need to show that I' satisfies AC-5. For this, let C be a double complex such that each of whose rows belong to r , let &(C) denote the pth row and TJC) the tnincation above the pth row of C. That is Tp(C)is Hence there is an exact sequence of double complexes

This sequence splits as a sequence of bigraded objectç. It follows that the associated single complex is split exact. Since To(C)= &(C), To(C)E r; by induction each Tp(C)E î. We know the n-th homology of C depends only on the fragment Cn+i + C, + Cn-iand by the definition of total complex Cn depends only on T,(C). Thus the above fragment of C is constant &ter Tn+i(C).So the inclusion Tn+[(C) + C induces an isomorphism on n-th homology, that is H,(C) 2 Hn(Fn+i(C')). Since Tn+I(C)is acyclic, its n-th homology is trivial and therefore H,(C) = O and since this is true for al1 n, it follows that C is acyclic.

2.3.2 Contractible complexes Let l? consists of contractible complexes. AC-1 and 2 are obvious. Since contractible complexes are exact, AC-4 holds. To see AC-3, suppose Ii is homotopic to L and K E r. then there are chain rnaps f: I<

-r L and g: L + K and maps s:Kn -, Kn+,and t:Ln -t Ln+,.for n 2 O such that ds + sd = 1 and dt + td = 1 - fg. t + fsg is a contracting homotopy in L. for

Thus L is contractible ând it is in î. To prove -4C-5, let Cm, be the double complex, rn 2 O and n 2 -1, but that Cm, = O for n = -1. This avoids there are any special case for lower dimension.

Suppose d: Cm, -t Cm,-i is one boundary operator and the other is d:C,. + Cm-i, with da = -3d. Further suppose that each row is in r. Then for each m and n, there is a contracting homotopy s: Cm, + Cm,+i such that ds + sd = 1. The n-th term of the associated single complex is C,, = Ci+j=,Cij and is O when n = -1. The boundary operator D:C, -, C,-1 has the matrix

dd = -ad gives D2= O. The map S:C, + C,+i is defined by the matrix

We need to show that SD + DS = 1. For this, we block D into an upper triangular rnatrix and a single colümn and block S into an upper triangular matris and a single row of zeros. The multiplication of the upper triangular matrix is upper triangular and the product of the single column of D with the sin& row of zeros of S is zero. Thus SD + DS is upper triangul~and has sd + ds = 1 in each diagonal entry, the last diagonal entry is ds but in this case sd = O, so ds = 1. So we only have to show that the above diagonal entries are zero. We claim that for i 2 0, ds(ds)' = (as)' + (sa)'ds. In fact, for i = 1,

Assuming that this is true for i - 1, NOW suppose we choose i < 3, the i, j-th entry of SD is

and the i, jth entry of RS is

Hence the i, j-th entry of SD+ DS is (-l)j-'-*(~~)j-~ds+(-l)~-~(s8)j-~ds= 0. Thus SD + DS = 1.

2.3.3 Quasi-contractible complexes Let C& be a f'ctor category of chain complexes. A chain complex C is quasi-contractible if for each object -Y of X. the complex CX is contractible. A map f:IC -+ L is a quasi-homotopy equidence if at each object X, fXKX -+ LX is homotopy equivalence. Thus if II: is a quasi-contractible complex, so is L. Also if f: K -, L is a quasi-homotopy equivalence, the mapping cone CX of fX is contractible and if CX is contractible, f is a quasi-homotopy equivalence, that is J is a quasi- homot opy equivalence if and only if its rnapping cone is quasi-contractible. So we cmcarry over each of the previous results on contractible complexes to these quasi-contractible ones, except that in each case the conclusion is object by object. Thus the quasi-contractible complexes constitute an acyclic class.

2.3.4 Acyclic classes come from If someone asks. where do acyclic classes come from, here is an answer, at least in known examples. Let 9 be a fùced class of additive functors d Ab. Then a chah complex C belongs to r if and only if #(C)is acyclic for dl + E 0, n 2 0) be a double complex such that

for alI n. Thus H,&(C.,)) = O for all rn and n. Let C be the total complex of the double complex Ce.. So C,,= Ci+j=,CG and hence +(C,) = Ci+j=,+(Cij) since $ is additive. Thus homology of the left hmd side is equal to the hornology of nght hand side and this is Ci+j=,H(4(Cij)) which is zero. It follows that H,,(#(C))= 0. Hence d(C) is acyclic and C E r. Now we will see what kinds of functors give the different lcind of acyclic class.

If 9 is the class of al1 exact functors, then 4(C) is acyclic for any exact C and for dl 4 E O. Thus exact functors give that r is the class of acyclic complexes. If @ is the class of al1 Hom functors, then for any Z E d,Horn(& C) is acyclic if and only if C is contractible. Thus Hom functors give î is a class of contractible complexes.

Finally, if @ is the class of all functors of the form Hom(2, - ) 0 evn (Hom functor evaluated at X), then for any complex functor C: 3- - d,we have Hom( 2. CX ) is acyclic if and only if Cx is contractible. Thus I' is a class of quasi-contractible complexes. The Barr, Beck theorem is:

2.3.5 Theorem Let I< -r Ii-i -t O and L -t L-, -t O be augmented chah

cornplez finctors sach that IC is c-presentable and L -t LWI+ O id G-contractible.

Then any natural transformation f-,: ILi + L-i extenh tu a natural chain trans- fonnatzon f:K + L and any two extension3 off_[ are natarnlly homotopic.

They prov~dit by indiiction. Later Barr proved it by usine acyclic classes. Let us suppose that r is an acyclic class on %? and C is the associated class of maps. We denote by C-'V the category of fractions gotten by inverting all the arrows in

S. This category is characterized by the fact that there is a functor T: Ce -+ Y'V

such that o E E implies that T(o)is an isomorphism and if S:(e + G@ is any functor such that S(a)is an isomorphism for d cr E 2, then there is unique functor

S': C-'Y -, 9 such that S'T = S. By AC-4, each cornplex C E I' is acyclic. Also we know each C is the mapping cone of o:O -t C. Norv since homology H, :'&

+ 2)is a functor; by Theorem 2.1.4, H&) is an isomorphism for all a E E. So by

the construction of C-"Y, there exists a unique functor H': C-'V -+ 9 such that H = H'T. Now suppose that G: .% + X is an endofunctor and e: G + Id is a natural transformation. If -t hf-l is an augrnented chah complex functor, then we denote by KGe the double complex that has IcmGn" in bidegree m,n. This is augmented in both directions, one is KG' -t and the other is KG' -, Ii-lGe. K is called epresentable with ~espectto ï if for each n 3 0, the augrnented chain complex &Go i Ii, -r O belongs to r. I< is G-acyclic with respect to r if the augmented complex KG -* ICIG + O belongs to r.

2.3.6 Theorem Svppo~ea: K + hLl and 8: L + LI are augmented chain complex finctors. Suppose G is an endofinctor on 2-and É: G -, Id a natural tian~formationfor whzch h" zs c-presentable and L -t L-i is a G-acyclzc. both u>ith revpect to r. Then given any natu~aft~ansfo~mation f-1: 1'-1-, Lei thsre is. in C-'59, a unique amow f:Ii -+ L that extends f-,.

Proof. Since IC is epresentable, the augmented cornplex ICmG0 -, II, -, O belongs to F for ail m 2 0, which means that each row of the double complex IieGe with augmentation term belongs to I'. Hence by AC-5, the total augmented complex

KG' -t li -, O of that double complex is in T'. But if we consider K as a complex concentrated at O, then the rnapping cone of the chain map hetwepn t,wo complexes ... -+KGn+' +... +I{G2 +I{G +O afid +O-+ +O-+ I< + Ois the complex KG' -, K -t O. Thus the arrow KE:KG' + Ii is in C. Again LG

-, L-iG -t O belongs to î. So rnaking G the same as h- we have LG' --+ L-1Ge

+ O belongs to I' and that is the mapping cone of PG': LG' + L.-1Ge, where L-IG' is considered as a complex concentratecl in degree -1. Thus BG' E 2. We can summarize the situation in the diagram

f-i G'

L,lG' LG' 4 -PG' LE the maps Ka and ,BG' are in C. So in C-' we have the inverses of Iie and @Ga. Now define f:h" -, L by

We claim that this map extends f-1 in the sense that f-1 ocr = 130 f and f is' unique with this property. The naturality of a, @ adE gives the following commutative diagram:

For uniqueness, let g: IC -t L be another map in the fractions category for which f-, ocu = ,D og. This gives fdlG*oaGo= ,dGm0gGe. which then implies that

(BGo)-' 0 f-lGa~aG*= yG* and then

from which we get

Thus f is unique with that property. CI

2.3.7 Corollary Suppose that K and L are each E-presentable and G-acyclic on modeb wdh ~espectto r. Then any natuîal WomorphWm f-i: I<-i -+ L-1 extends to a unique isomorphisrn f: h' -, L in C-'%'. Moreouer if g: I< L is a naturaz t~amfomtationfor which P 0 g = f4 .a, then g = f in Y'%'

Proof. Since K is é-presentable and L is G acyclic, by the previous theorem there exists a unique map f:K -+ L which extends f-'. Moreover, jl1is an isomorphism implies there exists a rnap h-i: L-I -t ILl such that f-l 0 h-l = 1 = h-, of-,. Again since L is E-presentable and IC is G-acyclic, by the previous theorem, there is a unique map h: L -r I< extends h- 1. Then h 0 f extends h- 1 0 f- 1 = id. as does the identity. So that by the uniqueness of the preceding, we see that in S-'%. h 0 f = id. Sirnilarly f 0 h = id in C-IV. Thus f is an isomorphism. O

To recover the form of Barr, Beck Theorem for acyclic models. we require:

2.3.8 Theorem Suppo~eG: 3- + X is a Qnctor and E: G -t Id is a natu~al tran;rfomzation. Then for any ftlnctor F: 2'- d,FG* -t F -r O zs contractible if and only if FE spk?s

Proof. First suppose that FGe + F -, O is contractible. Then there is a contracting homotopy s: FG" -+ FGn+', for al1 n 2 O. So for n = O, we have s: F

+ FG such that FE= id. Conversely. suppose that FEsplits. Let O: F -t FG such that Fe00 = id. Let Y = $Gn:FGn + FGnC1,then for di = FG'ÉG~-'.we have 8s= FEG"0 9Gn = id and for i > 0,

by naturality of 0. Hence, with d = ~:=,(-l)idi,

2.3.9 Corollary Let K -t ILl + O and L 4 Ll -+ O be augmented chah cornplex funetor3 ~uchthat Gehn 4 & is splat epirnorphi~mfor al1 n 2 O and

L + L-l -+ O is G-contractible. Then any natural tranrrfomation h--l+ L-l eztends to a natu~ulchain transfomation f:h- -+ L and any two extensions off are naturally homotopic. Chapter 3 Cohomology of Groups and Algebras

Let d be the category of sets and '3' the category of groups. Then there is a trip12

T = (T, q,p) arising from the adjoint pair of functors F -4 Ir. where F: d + Y is the fkee group functor and U:Y -t d is the forgetful functor. So if -Y is a set then TX is the underlying set of the fiee group on X. Now let <: TX + .Y be an T-structure of the algebra dT.Denote C((zi)(z2))= .rlxz and E((r)-')= z-~. Since < makes the following diagram commutative

it follows that ( is also a T-algebra homomorphism fiom ( TX,pAy ) to ( ri, <). This

where n is finite and €1, Q ,. . . , E, takes only values f1, then obviously is a T structure. Thus, if X is a set there is a one-one correspondence between group laws on X and T-stmctiue (. Hence the category of T-algebras dTis isomorphic to the category of groups Y.

If n is a group, let Y -t a be a *-module. By above Y M x T, where the left n-module Ad = kr(Y -, T)and the multiplication in Y is in tems of the left a-operators on M. Now let f: W -+ T be an object of the slice category (Y, n). A map g: W -r Y = M x a over n must have second coordinate f and first coordinate a map we cdd: IV -+ M. Thus

t~(ww')= gw 0 gw' = (dw,fw)(dwf, fw') = (dw + f w O dw', f tu of ri?')

which means that d(ww1)= dw + fw 0 dw' and this is the definition of a derivation of

W + M over R. Thus Hom,(W, Y) % Der(FVl M) and the isomorphism is natural.

3.0.10 Theorem There is a natural isomoiphism

Der(r/V, M), n = 0, Hn(W, Y)-, Hn+l(W, LM). n > 0, where Hn+'(W, iki) id the Eilenberg-Mac Lune cohomology gr0.u~.

Proof. Let G: Y -+ Y be the cotriple defined by GW. the free goup generated by the underlying set of W. The natural group epimorphism E: GW -r W is its counit. Thus, if tV is a group over rr the commutativity of

gives GVV is a group over T. Define a cochain complex K:(g, a)' -t Ab by

and Y + n is a a-module. The cohomology of this complex is Hn(. Y),, n 2 0. When n = 0, the cohomology is Ho%(W, Y) and it is Iid'Fv.

Let L be another cochain complex functor fiom ('Y. n)' -r d where LnW is the abe1ia.n group of functions fiom the caxtesian power Wnf' + Ml n > O, M = ker(Y + n) as a left R-module and L-' W = Der(tK, M). The coboundary d: LnW + Ln+'W is defined by

+(-I)"+~f (~0,.. . , W,) where n 2 0 and W acts on M via W -+ ir, and d: L-l -, LoW is the inclusion. The cohomology of this cochain complex is Eilenberg and Mac Lane cohomology group. We akeady have K-I - L-LeThus if K and L is G-acyclic and e-presentable, where G is a functor with counit acting on (Y, T), then by the Theorem 3.3.5 there is a natural cochain equivalence K "- L and this proves the theorem. G-acyclicity of K: The cotriple G = (G, c, q) induces a simplicial object

If we apply the functor Hom(-,Y), the simplicial object becomes

Putting W = GW,the original simpiicial object becornes

Since eGVV 06W = id the above simplicial object has a contraction

where h, = 6G"W; these operatoa satisfy the equation eohn = Gn+W, eihn = h,-i~i-t, for O 5 i 5 n, n > O. Thus the simplicial object

_I) K-~GW KOGW a PGW-+ . - * is contractible. Hence there is a contracting homotopy s: KnG -, Kn-' in the corresponding cochain complex KG. G-acyclicity of L: Let f E LnGW. A contracting homotopy s: LnW -+ Ln-' is defined by induction on the length of the word go E GW. We wnte the word in GW in letters, Say jw) where w E W. Then If go = (rw)g where g E GW,let

If go = (~~'9)where y E GW?let

If go = 1, let

fs(Lg~,--,g~-d= f(lAg1, w.--~n-ij

This homotopy s is natural with respect to TV -r W' in (Y,rr). epresentability of K: We -have the contractible simplicial object **+G~w=tG2W +GW + Applying the functor ICn on the above line we get - --+ I-~G*3 1iv3--+ . ..

_3 is constructible. Thus the corresponding augmented cochain complex

is constructible. Hence there is a arrow 0": KnG -t such that 8" Q K'E = 1

epresentability of L: The map 0": LnW + Ln is defined by

The proof is thus complete. Let d be a category of K-modules, and T = Ii 0 ( ) then a T-structure on a module 4 is (: Ii 8 A -+ A, satisfying

Thus dTis a category of associative li-dgebras.

3.0.11 Theorem There is a natural bomorphkm

where î + A is an algeb~aoveî A, Y. -, 11 as a A-module, and Hn+'(ï,hi) id the Bochschdd cohomology group.

Here Y -t A is a A-module is equivalent to saying Y -r A is a split K-algebra extension by a kernel M with iM2 = 0, and M is a two- sided A-module. For if

Y + h is a A-module, then Y = A x M = i\ @ M. so O -+ ALI -t Y -t :i -t O is split. Since (A, rn) o A; (O, m) are the elements of the kernel. that is we can identify the elements of M with (O, n).Thus the product of two elements in A4 is [O, ml)(O, m2)= (0, O + O) and this implies M2 = O. The product (A, 0)(0. rn) = (0, A4

(Al, O)((&, O)((),m)) = (Al, 0)(0, X2m) = (O? Al(X27-4)

By associativity (XJ2)m = Xi(X2m); this implies that hl is a left A-module. On the other hand, the product (O, m)(A, O) = (O, mX), so

And again by associativity m(X1X2) = (rnAi)A2. So M is a right il-module. Conversely, if Y -+ A is a split K-algebra extension by a kernel M with M2= O and Ad is a two sided A-module. Then from the splitting, Y = A @ :VI = i\ x M and the multiplication in Y is Chapter 4 Calculuses of fractions

We dready clefined category of fraction and denoted it by Ç-'%?. Here is the contraction of C-l

for which,

(i) cod(ol) = C;

(ii) cod( f,) = Cf;

(iii) dom(fi) = dom(q) for i = 1,. . . , n;

(ivj cod( fi)= CO~(O~+~)for i = 1,. . . ,il - 1

In picture an arrow is :

Composition is juxtaposition so that the identity arrow is the empty string. The equivalence relation - is the srnallest one closed under juxtaposition such that f 0 ü1 r-Log whenever T 0 f -- g 0 cr and such that CT 0 ad' is the empty string. The functor T:V + C-'5.9 is the identity on object and T(f) = f .id-' = f.

The arrow T(a) is invertible for all o E C and if S:% -r 9' inverts every elernent

34 of C, then there exists a unique functor SI:C-'%' + 9 defined by

clearly this functor extends S. We fùc an acyclic class I' of contractible complexes and let C be the corresponding class of arrows.

f O 4.0.12 Theorem Suppose L --+ N + kf are mapa of chain complexe^ wzth - - 7 a E C. Then there is a chain complez K and chain maps L t IC that T E C and the square * !VI

zs homotopy commutative.

Proof. Define Ii, = Ln @ 11/1, CJ Nn+1.The boundary operator of I< is given by

. Letr=(l O O):K-+Landy=(O 1 0):IC

-t M. Clearly these are chain maps. The short exact sequence

is U-split, so by the Proposition 2.1.5, S%', is homotopic to C,. Since C, E î, AC- 2 and AC-3 together imply that C, E r. Thus r E C. Now let h = ( O O 1 ) : K which shows that the square is homotopy commutative. O f 4.0.13 Theorern Let L 5 hi =t N be rnap of chain complezev with o E S 9 mch that fo e go. Then there exists a chain complex K and a chain map r such that rf = rg with T E C.

Proof. Since fa z go, there exists a map s: L -, N such that fo - go = ds + sd. We put h', = N, $ il.fn-l @ Ln-*;then K is a complex with boundary operator

d f-g -s dt+td-(; ;)(%)+(%). Thus r f is homotopic to rg. We also have to show that r E S. But

is a split exact sequence, hence by above r E S. O

From Proposition 22.2, Theorem 4.0.12 and4.0.13 we conclude that S has a homotopy calculus of left as well as right fractions.

4.0.14 Two pairs (f,a) and (f', of),each pair has same codomain, are said to be equivalent if there exist a homotopy commutative diagram

tliot is uf r (LI fl, ao n: a'd E S. By the definition of homotopy calculus of fractions this forms an equivalence relation: the refiexivity and symmetry are obvious. To prove transitivity let (f,o) - (fi,oi) and (f', d)- ( f",oit). Here af cz alf, au = da1 E C and bfl bUf", bat E b'd' E S. NOW we have bu' a'd B' - D -B. Hence by the property of homotopy calculus of fractions, C there exist B' 5'E - B such that c E S and D-B a'd

0' ca' is homotopy commutative. Thus we get D -t A' --> E such t.hat cn'a' cz c'ba'. c'b So again by the property of 2, there exists r: E -t B" such that rcal ci rc'b. Thus rcaf z rca'f'~rc'bf'z rcbI I f II andrcaae rca'a' E rc'ba'~rcba 1 1 '1 E S. Dehe then a new category (&/-, whose objects are objects of 'l: and arrows are the equivalence classes of (f,O), we denote it by üLf. The composition of maps: let adLf and T-'g be two arrows in the fom

then the composite is defined by r-lg 0 ülf= r-l~'-~f'f, where a' is an elexnent of C and f' a morphism which make the diagram

homotopy commutative. The definition of homotopy calculuses of fractions shows that the above formula makes sense and Y/-is a category. 4.0.15 Proposition Homotopic maps becorne equal in C-'(&. 1 1 Proof. We apply the previous theorem to L --+ L - L,then we get the homotopy commutative square Ii- -P L

4 h is a chain map such that p. k = 4 0 k = id. In YIV,p and d are invertible, whence k = p-' = 6-' so that p = d. If p. q: C -, L is a chain, . map and s: C -t L

chain map for which pou = p and Q Q u = q, so that in C-%, we have p = q. O

4.0.16 Proposition Let Co be the class of homotopy r:quivalences. Then C-'V = C-L(Co-l~)S C-'((if/-)? where - Z'J the homotopy relation.

Proof. Let V -r Co-"& and V + %/- be the cornparison functors and suppose that a f 5 bg, a

Thus by universal mapping property there exists a unique functor %'/- -t Xi1% such that the diagram %' commutes. Again if cr: C -+ D is a homotopy equivalence and r is its homotopy

inverse, then in '&/yT is the inverse of o. Thus again by the universal mapping property there exists a unique functor CôlV-+ a/- such that the diagram ci

commutes. Hence c;' Z Y/-and hence Y'%?S C-'(%'/-).

4.0.17 Corollary Eves, rnap in C-lg ha, the fo~mf ,where f E V and o€T:

Proof. Since every map in C-l'B has the form

f* 00;' 0 j,,-p;!, O . . of, oa;l and cod(fi) = cod(ai+i), by the definition of homotopy calculus of right fractions, there are maps ri E C and gi E C such that CT,;'~ fi = 9i~;+:. Again cod(gi) = cod(~i+?),so there exist pi E C, hi E C SUC~that TG\~~= hip$2. Continuing the process and substituting the value, the form of the map we get is

Since X is closed under composition, we are done. 13

Dually, by using the definition of homotopy calculuses of left fractions, we get :

4.0.18 Coroilary Every rnap in C-'C& has the form ülo f where f E %? and

O E C.

Thus we see that these facts hold despite the fact that there is no calculus of (left or right ) fractions. For example, in the proof of proposition 4.0.15, the rnap k: L + I< is a homotopy equivalence, because p 0 k is homotopic to id and k 0 p is also homotopic to id. Thus k E C and for this k we have p 0 k = qb 0 k, but only the O map coequalizes them and that is a homotopy equivalence if and only if C is contractible. 4.1 Some more theorems on double complexes

4.1.1 Proposition Let

be a sequence of differential module^ (re~p.chah complez). Suppose that C is the

colimit of the sequence. If each C, -+ Cn+l is a hornology equivalence. then 30 2s

each Cn + C.

Proof. The colimit C is a differential module. For, let gi: Ci -t C be a map. then gidi is also a map from Ci to C. Hence by the universal mapping property there exists a unique map d: C * C such that dgi = gidi. Thus 8gi = dgidi = gidf = 0. We also have colimit of H(Cn,dn) Y H(C,d). Since H(Cn)2 H(Cn+i) for dl n. X(C)i H(C.). O

4.1.2 Theorem Let

be a sequence of differentzal e es p. differeniial g~aded) module^. Suppose that C' ZJ the colirnit of the sequence. If each C, + C,+l ha3 a left inverse that is also a homotopy inverse, then the Jarne is tme for each C, -, C.

Proof. Note that, if C, -t Cn+i and Cacl-t Cn+2have homotopy inverse then the composite Cn -, Cnczhas a homotopy inverse. So suppose that f,": Cm + C, denotes the composite arrow for n 2 m. Let gz:C, -t Cm,for n 1 rn be the composite of the left inverses which are also homotopy inverses. Then glof," = 1 and for each n, there is an h,: C, + Cnsuch that do h, + h, ~d = 1 - f:-:-'ogn-,.

We claim that there is a sequence of maps 12,: C, -+ C, such that d 0 Ln + k, O d =

1 - ff 0 gg and % 0 f,"-' = fz-' 0 Indeed, suppose we have defined km for m < n and let and k: C -+ C by k 0 fn = f" 0 k,. These are compatible families for? g, O fm O km = f; km = f;, if m 5 n and g, 0 fm O km = g: O km = gr, if rn > n; and on the hand, f" 0 kn 0 fz = f" 0 fr0 km = fm 0 km, for n < n. To complete the proof we

- need to show d 0 k + k 0 d = 1 - f0 0 g". Cornposing the right side with fn, we have dohofn+todof"=d*fnok,+ko fnod= f"od~k,+r~k~d

The hypothesis for this theorem rnay be too strong, but the hypothesis that

each Cn -t Cn+thave a homotopy inverse, is not sufficient. We give an example to show this. Note that any morphism between contractible complexes is a homotopy equivalence; the O map in the opposite direction is a homotopy inverse. We claim that if a map from a contractible complex to another complex is invertible. then the second complex is also contractible. Indeed, suppose K is a contractible complex

and f:K + L has a homotopy inverse, Say g. Then we have ds+sd = Ili, tdfclt = IL - fg and ud + du = lx - g f. So we get gf = gds f + gsdf = dgs f + 9sfcZ and

this implies 421 + gs f) + (71 + gsf!"! IL. Thiis L is contractible and it is sufficient to exhibit a sequence of contractible objects whose colimit is not contractible. We

let C, -r C,+I be the map from the top row to the bottom of

O Zn+l --s Zn+ 2 (1 2 ... y+'jZ-O The unlabeled horizontal maps are simply the inclusion of the kernels and map between them is the induced map kom one kedto the other. The colimit of this sequence is the complex The group 2-'2 is the group of all rationals of the form h/2". The rvay to this is to let ~12":Z -, Q denote the map that takes k to k/Zn. Then the cocone

commutes and is, in fact. the colimit. Here each C,,is contractible because Z is projective, but the colimit sequence is not (2-lZ can't be projective, for exarnple k./2",1/2" E 2-l2 and they are linearly independent. )

We can also use this example to show that the limit (as opposed to the colimit ) of a sequence of acyclic complexes is not acyclic. In fact if we apply the functor Hom(-, 2)then each cornplex Hom(C,, 2)is contractible but the limit Hom(C.2) is not even acyclic. Consider a double complex C..: Let Tm = Tm(Ce.)be the double complex tnincated above the rnth row and R, be the mth row of the complex with the negative of the boundary operator and with the grading reduced by 1, so that the degree of the elements of C,,, have degree n + m - 1 as elements of R,. If d denotes the horizontal and d denotes the vertical boundary operators in the double complex. then the identity d 0 d + 8 0 d = O irnplies that do t3 = a O(-d), so that 0: R, -t Tm-l is a chain map. Since C,, is the elenlent of R, as of degree rn + n - 1, Tm is the rnapping cone of the chah rnap 8 and the rnapping cone sequence is

We then conclude,

4.1.3 Theorem Let Ce. be a double complex as above. Suppo~ethat eveq row R,, rn 2 O is acgclic (respectively contractible). Then the inclzl~ion map R-l + Ce. i~ a homology (~espectivelyhomotopy) epuiualence.

Proof. In the above double complex, we have R-1 = T-i. So for m = O the mapping cone sequence becomes O -t R-i -t To + Ro - O. Since & is acyclic, R-1 -+ To is a homology equivalence. Again Ri acyclic gives us To is homology equivalence to Ti. Thus Rdi -+ TIis a homology equivalence. Inductively this is true for R-1 + T.Now by the definition of T,, Hk(Tn)"- Hk(Tn+i),n 2 k + 3. Hence by the Proposition 4.1.1 T, + Ce,is a k-homology isomorphism. So is R-1 -+ Ce.. If R, is contractible, then Hom(C.., 2)+ Hom( R-1, 2)is a homology isomorphism.

Thus the inclusion Ki -t Ce. is a homotopy isomorphism. O

4.1.4 Corollary Let Ce. be a double complez as above. Suppose that every rodw ezcept the bottom and every colvrnn ezcept the ight are acyclzc (resp. contractible). Then the bottom row and the rightmo~tcolurnn are homologow (resp. homotopic). Chapter 5

Application t O homology on manifolds

The statment of André's theorem:

5.0.5 Theorem Let IC -, ALl + O and L -, L-i -r O be augmented chain complez fùncto~s~uch that both K and L are uteakly c-pre~entableand both Ii -t

ILl + O and L -+ L-l -+ O are G-acyclic. If ILI "- L+ then for each object .Y of X and any n 2 0, H,(I

Let M be a manifold of class CP,p 2 2. For p 5 p, let C':(Ad) be the group of singular n simplexes of class Cq in 1çT. The homology of the singular chain cornplex Cq(M)does not depend on p. We will prove this by using double complex and for this we need the following theorem of algebraic topology. We are not giving the proof of the theorem.

5.0.6 Theorem A C? manifold LW,p 3 2, ha3 u P-J~LP~ÇUWT.

A cover 9V is cded simple if any finite intersection of elernents of Q is either contractible or empty. The cover W is cded psimple if the finite intersections are contractible by a map of class CP. A map t: itl x [O, 11 -t M is called a contraction on the space hl, if t is constant on the bottom face LW x {O) and the identity on the top face iM x 11). Now let 9V be a psimple cover of a manifold M. Define (Ml = M; &(LU)= U,,* u, disjoint union of member of W.Thus &(Al) -+ M is a projection. Define ICl(lLf)= &(Al) x~ &(hl),pullback over M. Shen And &(M) = {(tO,. . . ,xn) E Ki '(hl)1 d(lo) = - = tj(x,)} where h,"+'(M)

denotes the n + 1st product of &(M) and r# is a map from KO(-M) M. Thus

A simplex a: A, -+ M is called 9-smd if its image lies entirely in some tri,

iri E 9,that is o factors through some member of 9.The set of all V-smd q times differentiable singular n-chains generate a subgroup of Cn(ibl),we denotc this by Cn(-VI,%). Let rro(M)= set of connected component of 113,

For fixed n the sequence

is exact because it is the singular chah complex of a space which is a disjoint union of contractible components. Also for fix m the complex

is exact (in this case it is contractible), but the contraction is not natural in M. For each Q-smd o: A, -, M choose, quite arbitrary, a u(o) E 4? such that o factors as A, -+ u(a)-, M. Let (uo,.. . , un): uo n n un -P itr be the inclusion. Then let a: A, + uo n n u, be an m simplex in Horn,(A,, Iïn(,(M)).Let u(a) denote u((uO,.. . ,un))' O. Let t(a)denote the induced simplex A, -t uo n n unn u(o). Then t is the contraction at the simplicial level. Thus the homologies corne fiom and

Z7ro(Ii,(n/1)) + z7r(,(h;-i(n/r))+ Z7ro(&(lId)) -r O But by André's theorem the homology of the fist sequence is isomorphic to that of the second sequence. The first one depend on q; the second one does not. Finally. in the first row if we replace (M,U) by -41 the homology remain unchanged (up to isomorphism) because S(n/l,U) -, S(M)is a homotopy equivalence [9,Proposition 111.7.31. Thus the homology of Cq(11.I)is isomorphic to that of CP(M).But the isomorphism is not known to be induced by the inclusion of CP C Cq and it is not knnwn to be natural. However if we take p = oo and q = O, then by Stokes theorem we have the singular cohomology is naturally isomorphic to de Rham cohomology on CCQmanifolds and the isomorphism is induced by the restriction of singular cochains to the C" cochains. We define a specid category .% to deal with this situation. An object is a 4- tuple (X,I, U,u). Here X is a topological space; I is a set; U is a fiinction from I to B(X), the open set lattice of X; and u is a function that chooses. for each small singular simplex o:A -t X, an element u(a)E I such that the image of cr is included in U(u(a)).Suppose that {U(i):i E I} is a simple open cover of X. .4n arrow ( f,F): (X, 1,U, u ) (Y,,J, V, u) consists of a continuous map f: -Y I' and a function F: 1 -t J subject to two conditions. First, for eadi i E 1. there mtist be a unique arrow f (i): U(i)-t V(F(i))such that

commutes, where (i) : U (i) + X for the inclusion a,rrow. The second condition that for each singular simplex O, F(u(cr)) = v(f 0 O). If we take (g, G):(Y, J, V,II) + (2,K, W,w) to be another arrow, then we can see easily that the composition (g,G)(f,F) = (gf,GF):(.Y, 1,U,u) -, (2,I<, W,W) is again an arrow. Thus this definition makes X into a category. Now we define a functor H: E -, 3 by

This requires some explanation. Let us temporarily denote by .f the disjoint surn

h CicrU(i) For i E 1, let (i,i): U(i) + -Y denote the inclusion. Let e: ..Y 'C

denote the unique map such that e ~(i,i) = (i). ?or i. it E I, let (if.2): U(if) n U(i) (i,i) - - 2 denote the composite U(if)n U(i) -r U(i)- .Y, the first arrow being inclusion. Thus (a", i) is a inclusion. Shen U n U denotes the map that takes (il, i) E I x 1 to U(it) n U(i). Before defining û, we introduce a bit more notation. Suppose the singular simplex o factors through U(i). Let us denote by o(i):A -, U(i)the unique singular simplex such that a = (i) oa(i). In that case, we denote the singular simples

(i' i) 0 o(i ) in .Y by [a;il. ive daim that every singular sirnplex in ..has the form /. [o; i] for a unique O and i. For if r: A -t X is a singular simplex, since the image of n T lies in a connected component of IC and component are disjoint, r factors through a unique comected component U(i) and then o = e 0 r is a singular simplex of ri whose image is included in U(i), thus e .[a;i] = e o(i,i) 0 o(i) = (i) 0 a(i) = o. so that T = [O; il. Npw we cm define G[a;i] = (u(o),i). This makes sense because if o factors through U(i) and U(u(o)), it factors through U(i)n U(u(o)). Now we have to define H on arrows. If (f: F) is a map as above, then we define

A A h H(f, F) = (7,F x F), where f: X -, Y is the unique map such that the square

commutes for each i E I. To çee that (f,F x F) is an arrow in 9-we need

h to show that (F x F)(û[o;il) = C( f o[o; il). We have a = (i)0 a(i),thus f 0 a = f ~(i)0 o(i) = (F(i))0 f (i)0 o(i) by the commutativity of the first diagram. Also we have (f 0 o)(F(i)):A + V(F(i))is a singular simplex, sn that (f 0 o)(F(i))=

J(i)0 O(;). Then,

and on the other hand,

Û(&[o; il) = ~(To(i,i)0 o(i))= Û((F(i),F(i)) 0 f (i)0 o(i))

= (v(f 0 O), F(i)) = (F(u(o)),F(i))

Define a rnap c = (e, E):H -, Id as follows. We already defined e on -? with e 0 (i, i) = (i) and E: I x 1 -t I is first projection. This definition satisfies the requirements for being a map. For, let e((ir,i)):U(ir) n U(i) - U(i) be the inclusion, i, ir E I. First, we have to show that e O(?,i) = (i) 0 e( (il. i)). tVe have

e O(?,i) = e ~(i,i) 0 e((if,i)) = (i) 0 e((il,i))

Since the composite of inclusions is an incliision. e O(?. i) is the inclusion of U(il)f~

U(i) into X, which is also (il) 0 e((ir,i))Second, we show that for all singular simplexes [a;i] of 2, E(Û[a;il) = u(e .[O; il). But we have

5.0.7 Theorem Ler: K be the chain complex finctor that has in degree n the free abelian group generated by the ~mallsingulaî n-simplexes, azlgrnented mer the free abeliun group on the connected components. Then ho is E-contractible and Ii

+ -t O U II-contractzb~evath re~pectto the clas of contractzons.

Proof. Let be the chah cornplex functor on X that has above property. Since every simplex in X E .T factors through some open subsets of X. the functor K depends only on the space and cover; not on the index set or the choice function, that is Kn(Xl1, U, U) = Kn(X,U). The inclusion &(X, U)+ ri,(-Y)is a homotopy equivalence, [9].Thus they have same homology (up to isomorphism).

Now since the covers are simple, the complex IiH -, ILIH -+ O is contractible.

We clairn that for each n 2 O, there is a natural transformation en:& -, li,H such that IC.EO& = id. Define $,(O) = [O; u(a)], where 0:A, -+ X is a small simplex. It will be natural if the diagram

is commutative. For a E &(X, 1,Ul u), we have

A So, to prove that the diagram is commutative, we show that f .[O; u(o)]

= [foa;F(u(o))]. But the latter has two properties; firstly, its composite Nith

h e is f 0 o and secondly, its component is F(i). For the first, we get e 0 f .[a;i] = f 0 e .[O; i] = f 00 while it fdlows hmthe definition of 7 that the component of f .[O;i] is F(i). Thus by the Theoreni 2.3.8 K is E-contractible. IJ

Now let p > O be an integer and let 2-p denote the subcategory of 3"cconsisting of those (X, IlU, u) for which X is a manifold of class CP and the open cover is simple with contracting homotopies of class CP. The maps are the pairs (f,F) for which f has class CP. Every differential manifold has that kind of cover 121, Section 46 of Chapter VIII]. So that the results we get are true for all class CP manifolds for all p > O. Suppose that q 5 p and define the chin cornplex functor IP as the free abelian group generated by the smdsimplexes of class C'L .As usual, I

we get Kq is e-contractible and Kq + I(ol -, O is H-contractible. In particular for q = p, A? is E- contractible and Kp -t Kf, - O is H-contractible. Since p times differentiable functions are also q times differentiable. we have an inclusion IP + 1P and since the cornponent does not depend on charts, lï[ = 1-.SO id: Kr

-r KP is an isornorphism, and by André's theorem Hn(K(X)) 2 H,(IP(X)). By the Barr, Beck theorem any two extensions of id are homotopic. Since inclusion is also an extension of id, P(X)-t IP(X)is a homotopy equivalence.

5.0.8 Theorem Let q a p be non-negative integers. Then on the catego~yof manifolds of C~UJJ CP. the inclusion of the IP - IP' i~ a quasi-hornotopy eq~~iua- Chapter 6 Cart amEilenberg cohomology and cotriples

6.1 Regular epimorphisms and regular cat egories

In this section we find some facts about regular epimorphisms and regular categories. For details see Chapter 1 of [6]. An arrow in a category is cded a regular epimorphism if it is a coequalizer of two arrows into its domain and instead of "epimorphism" we mite %pi'?. If the mow ha. a kernel pair (the pullback of the arrow with itself), then it is a. regular epi if and only if it is the coequalizer of that kernel pair. A category is called a regular category if in any pullback diagram

C-D f f regular epi implies g is too. Regular epi have many properties, one of them is they are strict epi, which means they factor through no proper subobject of their codumain.

6.1.1 Proposition Let a' be a ~egularcategory. Then the forgetfd fimctot

IA:Mod(A) -, d/Apreserues regular epU. Proof. Let f: M 4 M' be a regular epi in the category Mod(A). In a category of

modules strict epis are regular epis, if we suppose f:M 4 Mt is a strict epi then it will be sufficient to show that it is regular in d and hence in d1.4.

Red that an object of Mod(A) is an object B -, A togather with mows

m. i, u, the most important being m:B XA B + B which defines the addition (multiplication). The argument we give actually works in the gcnerality of the

models of a finitary equational theory. So suppose f:Mt -» M is a strict epi in Mod(A). If the map IAf is not a strict epi, then it ccn be factored as B' = IAMt-» B" -B = I,lll.l in &/A. Since d is regular, so is &/A. The map BI -H B" is a extrema1 epi in d/Awhich implies that this is a regiiliu epi in dl.4

Hence BI XA BI -' B" x A BI1 is also regdar epi and we have the commutative diagram

The regular factorkation system in a regular category has one property that if the

is commutative, where e is epi and m is monomorphism, then there exists a unique arrow along diagonal fiom the target of e to the source of rn called diagonal fill-in which makes the diagram commutative. Since we have a commutative diagram, we get an arrow m": B" xa Bu -, B" and for this both of the mows Br -t B" -, B preserve the new operation. A similar argument works for any other nnitary operation. As for the equations that have to be satisfied, this follows kom the usual argument that shows that subcategories defined by equations are closed under the formation of subobjects. For example, we show that rn" is associative. This requires

to show that the two arrows B" x~ B" XAB" BU(firstapply m" on the left side

of the product and keep the rest fix; to get other arrow apply ml1 on the right side and keep fix the left B") are the same. But we have the diagram

which cornruutes with either of the two left band arrows. Since the bottom arrow is rnonomorphism, it is left cancellable. Thus m" is associative. u

We have seen in Chapter 3, how cotriple cohomology corresponds with Eilenberg- Mac Lane cohomology. Here we will try to set up a relation between Cartan- Eilenberg cohomology (homology) and cotriple cohomology (homology).

6.2 The Cartan-Eilenberg setting

Cartan and Eilenberg give in [8] more or less uniform definition of cohomology theories in various algebraic categories that cm be described as follows. In each of the Mnous categories, they associate to each object A the enveloping associative algebra Ae together with a left Ae-module we cdZ(.4.). They construct a %tan- dard projective resolution" C.(A) of Z(A). Shen for any A-module M, applying the functors M @ - and Hom(-, 1I.I) they defmed hornology and cohomology as TO~~(M,Z( A)) and Extl. (Z(A),hl), respectively. This standard resolution allows us to compare the CartanEilenberg theory with the cotriple theory. When applying the first functor, M acts as a right Ac-module and for second, 1VI acts as a left Ae- module. However, Ac is always isomorphic to its opposite ring and the notions of A-module (in the sense of Beck), left Ae-module and right Ac-module coincide. For example, in the case of associative rings, an A-module in the sense of Beck is a two-sided -4 in the usual sense and if hI is such a rriodule, it is a left A' = A @ AOP-module according to (a @ b)m = amb and a right .-le-module by dehing m(a @ b) = bma. In their procedure they used two elements without giving any description. One is the definition of module (and therefore of the enveloping algebra) and the other is the definition of Z(A). Beck gave a systematic description of the definition of a module. He showed that if an A-module is an abelian group object in d/A. then for any A-module 21.1 and any arrow 8 -, A, the Hom set d/A(B,I,.&I)is ui abelian group and it is the groiip of derivations of B to M in dl the cases-groups,

Lie, associative and commutative algebras, where B acta on Ad via B -t -4. We assume the inclusion IA has a left adjoint ~iff~.Then for nny A-module hl, for ~iff"(~)is the cokernel of A Qi A @ -4 3 A - .4 c>7 -48 -4. The image of this map isab@c@e-a@bc@e+a@[email protected] f:A@A@A +M. Definedr:d -t M by d,(a) = f(l@a@l),then dl(cb) = f(l~ab@l)= f(a@b@l)+ f(I@a@b)? because f dsheson a@6@1-1@ab@l+l@a@b. So the latter isaf(l~b@l)+f(l~a~l)b= adl(b) + df(a)b. Conversely, suppose d: A -, M is a derivation. Define fd: A@ A 0-4

-t 1\.1 by fd(a@I b @ c) = ad(b)c. Then

Thu c@e-a@bc@e+a@b@ceis the kernelof fd. Also, Beck indirectly solved what is Z(A). The key is actually in the standard complex. Let be the projective resolution of Z(A), where C'.(.A) is the (n+ -)-folcl tensor product of A. Since IIiffA(~)is the cokernel of A @ A @ A @ A + A 8 A 3 il, there is an exact sequence

which is a projective resolution of ~iff~(~).Thus the module Z(.4) is rather arbitrary, but in every case, the kernel of Co(A) + Z(A) is the module ~iff"(rl). Now we define the shifted Cartan-Eilenberg hornology and cohomology to be HFES(~,M) = Tor.(n/l, D~E*(A))and HEEs(d,M) = ~xt'(~iff"(d),M) respectively. Here we discuss the connection between the shifted and original theories.

6.2.1 Relation between the shifted and original CE (co)hornolog~r For any object A of one of the Cartan-Eilenberg categories adany A-module M. If,CES(~,Ad)2 ~,"5(.4,M),for n > O, H~~~(.I,M)2 :bI .I+ ~iff'(.-l) and HfE(A, M)is a subgroup of M@.~=D~E"(A). Similarly, HFES(-4,Ad) - (A,M), for n > O, HEEs(A.M) E Der(A, 1LI) and H&(A, M) is a quotient group of Der(A, M). And we have nothing in the shifted Cartan-Eilenberg homology (coho- ruology) to connect with H,CE (HEE). What we will be showing is that, in the appropriate setting, the cotriple homology ( cohomology) groups are equivalent to the shifted Cartan-Eilenberg groups.

6.2.2 The standard setting Here we will describe what we mean by a standard Cartan- Eilenberg or CE setting. We begin with a regular category d. For each object d of d,we denote by Mod(A) the category Ab(d/A) of abelian group ohjects of d/A.We assume that the inclusion 4:Mod(.4) + &/A has a left adjoint we denote ~iff*?Let f: B - A be an arrow of d,the composite with f determines a functor fi s/ -+ 9: defined by F(0 -t B) = D -r B A and for if D -t B and D + C are two arrows making the diagram

B-A commutative, then by the definition of pullback there exists a unique arrow D

+ B xa C which makes the triangle cornmute. Thus B x.4 - is a right adjoint of A we will denote it by f;. The right adjoint induces a functor from Mod(il) + Mod(B); we will also denote it by f;. We will assume that it has a left adjoint and we will denote it by f#. The diagram

d/B4 IB Mod(B) is made by the left adjoint to the respective arrow of the diagram

of the right adjoint. Clearly the diagram of the right adjoint commutes. Thus by Yoneda lemma, the diagram of the left adjoint commutes. In CE category every il-module in the sence of Be& is a left -4'-module. Hence Mod(A) is equivalent to a category of A'-module Now let fe: Be - Ae be an arrow of a category of associative enveloping algebras. Then the functor Mod( Ae) -t Mod(BC)has a left adjoint A' @B. (-). (If f: R + S is a ring homomorphism then the functor from category of S-module to category of R-module has a left adjoint RDs.) Thus f# is actually the functor Ae @B. (-) and A' becomes a right Be-module via f and the tensor product is an A'-module.

6.3 The main theorem

For the purposes of this theorem, we define an object .A of d to be U-projective if UA is projective in 2" with respect to the clam of regular epis. That is. if -Y

-t UAis any regular epi, then it splits.

6.3.1 Theorern Suppose that, in the context of a CE setiing, zvhen -4 iu U- projective,

I (i) GA is U-p~ojective;

(ii) Ct(.4) is a projective resohtion of ~iffl'(rl);

(iii) For each n 2 0, there *s a ftmcto~Et: 9-/UA Mod(.A) mch that the

cornmut es.

Then the complexes Ct(A)and ~iff"(~'+%)are ehain eqz~iualent.

The last condition of the theorem means that the modules in the projective resolution depend only on the underlying object of -4. For notational convenience, we fiu A and write C, and Diff for C: and ~iff",respectively in the proof of the t heorem. Proof. We prove this by using the Corollaq 4.1.4 complexes. To apply this corollary, we must show that in the double complex

all rom except the bottorn and al1 columns except the right hand one are contractible. By the property (iii) the column

Here we need two lemmas, the kst one stating that columns satise the above property while rows do by the second one.

6.3.2 Lemrna Let the fîmctor U:d cr' X have left adjoint F and let G be the ~e~ultantcot7iple on d.Then for any object A of d,the ~zmplicialobject Proof. Let s = r7UGmA:UGmA -t UGm+'A. Then

while, for O < i < m,

and the last term equals, by naturality of q,

This shows that s is a contracting hornotopy in the sirnplicial object. The corresponding complex of the sikqdicial object is also contractible. Now if we apply the additive functor En to this contractible cornplex, we still get n contractible complex, which shows that the columns of the double complex, escept for the rightrnost one, are contractible. O 6.3.3 Lemma Let P be a ~egalurp~ojective object of X. Then, for any P - Li-4. Diff (FP)is projective.

Proof. By the way, an object P is regular projective if, whenever S -t P is regular epi, it splits. Let P be a projective in S, then P - X is a projective in .T/X Denote by the class of regular epis in X. We claim that when L: 3- +

9 has a right adjoint R:Y -t Z,then L takes a projective object in 3' to a projective object in 9 provided R presenres the epimorphic class that defines the projectiveness. hdeed, let P be 8' projective and A + B E gy. then R(rl)

-, R(B)is in gX.Hence for any arrow P R(B)there exists an arrow P + R(rl) such that the triangle P -R(A) cornmutes. If we apply the functor L to that triangle, we will get that the diagram

which cornmutes, because L is a functor and the r has natural property. Thus L(P) is gY projective.

Now take R to be UIA:Mod(A)-, XIU.4, we have assumed U preserves regular epis and hence UIA. By the Proposition 6.1.1, IA preserves regular epis. Thus U IA does.

So if P is projective in X,since Diff 0 F is a left adjoint of UIa4?DiffFP is projective in Mod(A). Thus by properties (i) and (ii) of the theorem we have DiffG.1, DiffG2A, . . . , DiffGm+'.4 are projective; and by (iii) we conclude that every row of the double complex except the bottom one is contractible. This estab- lishes the t heorem.

6.4 Applications of the main theorem

6.4.1 Groups Let Gp be the category of groups and U:Gp -+ Set be the underlying functor. In the category of sets every epimorphisms are splits. This implies that Un is projective for any group R and hence G(H) is U-projective.

Now for a fked group T, the functor k:SetlUx + Mod(a) defined by ci(g:S -+ Un) is the free r-module generated by the (n + 1)st cartesian power of S. Let g = Z7 f for a group homomorphism f:H -t a. To define the boundary operat.or a on C~(Uf: UH -, Un), it is sufncient to define it on the generator (xO,q, . . . 5,) of C,"(U~:UH + Ur). So d((so,q, . . . ,r,)) is Here we see that c: deppends only on the arrows in Set, but the boundary operator depends on the group structure in H. Thus this defines the functor C: on Gp/a.

If we put f = id: x -r n, then the definition of B shows that C',"(id: UT -r UT) is the standard Cartan-Eilenberg resolution. We may denote Cf(id: Un -, Un)as C.(T). Cartan-Eilenberg showed in [8] that CJn)is a projective resolution of Z(n) which in this case is the group of integers with trivial action by R. Hence C.(lr) is a projective resolution of DifF(r). Thus the conditions of Theorem 6.3.1 are satisfied and we conclude that the group cohomology is the cotriple cohomology.

6.4.2 Associative algebras In this case, we assume that d is a category of ho- algebras, K a commutative ring with unity and X is the category of K-modules. Let U:d -, 3? be the underlying functor and A be a K-algebra. Assume U(A)= M be a projective K-module; then G(A)= FU(.4) = FM, the free algebra generated by a K-module M; is the tensor algebra

where M(") denotes the n-th tensor power of M. The projectivity of M implies that FM is IC-projective. We have already sccn that n-hcn -4 is a K-algebra, the category Mod(A) is the category of two sided A-modules. Now if we assume M is a two sided A-module, then for any m E ll.i and any a E Al am and ma belongs to M. This gives us 1I.1 is a left Ae-module;because for any a @ b E A'. (a@ b)m = amb is in M. Fix a K-algebra il and dehe the functor Et:S/UA + Mod(A) by If g = U f for some algebra homomorphism f: B -, A, then the boundary formula is

which depends on the algebra structure in -4.So applying a similar argument as in the case of groups, we see that the algebra cohomology is the cotriple one.

6.4.3 Lie Algebras To apply Theorem 6.3.1 to the category of Lie algebras. first we need to prove that when M is projective K-module, then FM is also Ii- projective. In [8, Chap. XIII, Ex. 81 they stated that when Ad is a K-free module. then the free Lie algebra generated by Ad is I<-free. But we can not conclude from there that if M is Ii-projective then the free Lie algebra generated by .VI is even ri'-projective. We begin by seeing what needs to be done to go fiom free modules to projectives.

In (SI this is used in tvo places. The &st is the Poincaré-Witt Theorem, which pays that the enveloping associative algebra generated by a Ii-free Lie dgebra is &fkee. The enveloping K- algebra gC has the property that each (left) g-representation cm be regarded as a (left) gc-moduleand vice-versa. In other words ge cornes from the functor g H gel adjoint of the underlying functor gC w g which replaces the multiplication in an associative algebra by the Lie bracket [x,y] = xy - yx. But if g is IC-projective, then any epimorphism to g in the category of Lie algebras splits. Let the IC-module go denote the kernel of that epimorphism, then the K-module g @ go is kee. Now suppose our go is a central ideal, then g @go is a Lie algebra, for the central property of go gives: [(x,01, (Y,O)] = ([x,YI, 0)

[(G01, (0,YO)] = (070)

[(O, xo), (Y, 011 = (070)

[(O, xo), (O?YOJO)] = (0, 0)

So the Lie bracket of g defines the Lie bracket for g $ go. This irnplies that g is. as a Lie algebra, a retract of g $ go. Since a.li functors preserves retracts. ge is a retract of (g $ go)'. SO if the latter is K-free, then the ge is Ii-projective. The second place where freeness is used is in the proposition that if f) is a Lie subalgebra of the Lie algebra g, and if g, I) and g/t) are Ii-free, then g' is a free 9'-module. We would like to prove this with "free" replaced everywhere by "projective".

6.4.4 Proposition Let O * 9 -r g -+ g/(t -, O 6e an exact sequence of I<-p~ojectiveh'-Lie algeb~as. Then ge is projective as an ije-module.

Proof. The proposition is valid when 9 and g/b are both Ii-free [S, Proposition XIII.4.11. To making notations simple we will denote g/b by 1. Since i) is Ii- projective, tliere is a Ir'-modulc bu such that f) % is IC-&ce. If $ve b0 the structure of a central ideal, t hen $ ho is a K-free K-Lie algebra. Similady, we can choose fo so that f @ fo is &free. Now we have a commutative diagram

In the bottom sequence, the two ends are %fkee kom which it follows that the middle is as weU. Applying the enveloping algebra functor to the left hand square we get the above diagram, in which go = bo $ fo

By 18, chap. XIILl.P], for any two Lie algebras gl and gz, there is an isomorphism

(81 $ gz)' S $7 8 8; This can be proved directly, as there. We can also prove it by showing that they represents the same functor. Here it is: Denote the right adjoint of g H ge by L. Let the functor Alg -, Set be given by A H Hom(g,Ll.4)). The universal mapping property of ge implies that the functor is represented by 8'. Now, given any two Lie algebras 01 and gz, the associative algebra A maps to the subset of Hom(gl, L(A))x Hom(gz,L( A)) consisting of al1 pairs (ft, fi) such that

[fizl, f~x2]= O, hence fi and fi commute. This functor is represented by g; @ 8;.

On the other hand if we denote L(A) = g, then the functor gi $ gz ++ g is consisting of two pair of functors fi:gl ++ g and fi: g2 H g which commute. The functor

(f1,fi):gl @g2 -, g is given by (filf2)(xi,x2) = flXl+ f2x2. We show that this preserves Lie bracket if and only if fiand fi commute pointwise. The definition

In order to preserve Lie bracket we have [fZx2,fi fi] = 0, [f 1 XI,fZ y~]= 0, this implies that fi and fi commute pointwise and conversely. Moreover, is IC-projective since ho is. If & is K-free, then [Ig Z K, and 9' @ 9; 1 t)' @ h-xII' @ K Z t)'. Thus (4 $ Bo) is a free be-module. Also if 9: is K-projective, then it is a retract of a free K-module, Say Plthat is t): -, P -t 115 is identity. This gives us 4' (3$a

-r Ile8 P -+ b' @ t): is identity and P is K-free gives us t)' 8 P is Oe-free. Hence (I) @ ilo)' is projective as an ff-module. Also we have g" is (I] $ So)'-projective,

These two together imply that g' is I)e-projective. O

With these two results, the entire chapter XII1 of [8] becomes valid with fiee replaced by projective.

6.4.5 Proposition Let M be a projeclive K-modde. Then the free Lie algebra FM is also Ii-piojective.

Proof. The base of the proof is the hint to Exercise 8 on page 286 of [8].We consider first the case of a free K-module. Thcn the following diagram Ka- Lie(h') - .Lie(Z)

Set of categories and adjoints cornmute. It is clear frorn this diagram that if we show that the fiee 2-Lie algebra generated by a free 2-module (that is. abelian groiip) is a free abelian group, then by applying the functor IC @z-, it will follow that the free &Lie algebra generated by a free K-module is K-fkee. So let M be a fiee abelian group and let F(M)be the free Lie algebra generated

Then by the commutation of adjoints in the diagram we have Hom(GM,A) P Hom(M,VA) P Hom(ll1,ULA) E Hom(FM,LA) Z Horn(F(M)=,A). Hence by Yoneda Lemma FA1 Z F(M)', so F(llil)=is Z-free.

The inner adjunction is a rnap i: F(M)-+ L(F(M)'),where L is the functor from associative algebras to Lie dgebras. If we can show that i is monic, then F(M) will be a subgoup of a free abelian group and will therefore be free abelian. Let Ml be a finitely generated subgroup of M. If Ml, 1V12 are two finitely generated subgroup of M then so is 14+ M2,and M is the filtered union of these subgroups. We can also Say 1VI is the colimit of the Mi.Since all these functors commute with filtered colimits, it is sufficient to prove that i is rnonic when M is finitely generatecl. Also, F(M) is the free nonassociative algebra generated by M modulo the identities of a Lie algebra. The fiee nonassociative algebra is a graded algebra whose nth gradation- is the surn of as many copies of LW"+')as there are associations of

(n + 1) elements, which happens to be - , thir is finite, in any case. The n + 1 ( ) identities are the two- sided ideals generated by the homogeneous elements x :s x and x @ (y 0 2) + r @ (x@ y) + y Q (r 8 x). Thus if LW is finitely generated, F(M) is a graded algebra, so is the nth homogeneous component. Let F,(M) denote the sum of all the homogeneous components of F(1Z.l) up to the nth. Let N be the kernel of i and N, = N n F,(M). Shen l\iR is finitely generated. If LV # O. then for some n, lVn f O since N is the union of them. Thus .Vn is a non-zero finitely generated abelian group and it is a standard result that there is some prime p for which 2, @ Nn # O. Here we have an exact sequence O -t N,, # O FF,(M)

-, L(F,(M)') -r O. Tensoring that sequence with Zp, we get an exact sequence

But the module L(Fn(&l)e)is fice which implies that Tor(Z,, L(F,(M)' )) is zero.

So that 2, @ Fn(M) -t 2, Q L(Fn(iW)') is not injective. But the Poincaré-Witt theorem gives the explicit form of the fiee basis of 2, @ L(F,(M)') since 2, is a field and 2, @ - conunutes with ()' and L. This contradicts the fact the above rnap is not injective. Thus the adjunction map is injective. O Now for a K-module M, let An(M)denote the nth exterior power of M. Shen g = U f for a Lie algebra hornomorphism f: II -, g, the boundary is described on generators as follows, where, as usual, the denotes the omission of an argument.

Thus by Theorem 6.3.1 we conclude that the Lie algebra cohomology is the cotnple cohomology. Bibliography

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