On the Geometry of 2-Categories and Their Classifying Spaces ∗

On the geometry of 2-categories and their classifying spaces ∗

M. BULLEJOS ([email protected]) and A.M. CEGARRA ([email protected]) Departamento de Algebra,´ Universidad de Granada

Abstract. In this paper we prove that realizations of geometric nerves are classifying spaces for 2-categories. This result is particularized to strict monoidal categories and it is also used to obtain a generalization of Quillen’s Theorem A.

Keywords: 2-category, monoidal category, simplicial space, nerve, classifying space, homotopy equivalence.

Math. Subjects Classiﬁcation (2000): 18D05, 18D10, 55U40, 55P15, 55U10

Introduction and Summary

The construction of nerves and classifying spaces seeks to associate geometrical objects to categorical structures, in such a way that those objects should keep all the categorical structure information. This process of taking classifying spaces leads to the homotopy theory of those categorical structures, whose interest is well recognized; for example, let us recall that Quillen [13] defines a higher algebraic K-theory by taking homotopy groups of the classifying spaces of certain categories and also recall that classifying spaces of symmetric monoidal categories pro- vide the most noteworthy examples of spaces with the extra structure required to define a Ω-spectrum [18, 22]. It was Grothendieck [10] who first associated a simplicial set NC to a small category C, calling this simplicial set the nerve of C. The p- simplices of NC are diagrams in C of the form x0 → x1 → . . . xp−1 → xp. The classifying space of the category is the geometric realization [19] of its nerve, BC = |NC|. Later, Segal [21] extended the realization process to simplicial (topological) spaces. He observed that if C is a topological category then NC is, in a natural way, a simplicial space and he defines the classifying space BC of a topological category C as the realization of the simplicial space NC. This general construction given by Segal provides, for instance, a definition of classifying spaces of 2-categories. A 2-category C is a category endowed with categorical hom-sets C(x, y), for any pair of objects, in such a way that the compositions C(x, y) ×

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C(y, z) → C(x, z) are functors. By replacing the hom-categories C(x, y) by their classifying spaces BC(x, y), we obtain a topological category BC (with discrete space of objects). Then, the classifying space BC of the 2-category C is deﬁned as that of the topological category BC. However, there is another convincing way of associating a space to a 2-category C. This way goes through what Duskin called [5] the geometric nerve ∆C of the 2-category and it was developed, among others, by Street [23] and Duskin himself. This geometric nerve ∆C is a simplicial set, which encodes the entire 2-categorical structure of C, whose simplices have the following pleasing geometrical description. The vertices of ∆C are the objects x0 of C, the 1-simplices are the x0,1 arrows x0 / x1 of C and the 2-simplices are triangles:

= x1 C x {{ CC x 0,1 {{ CC 1,2 {{ ⇑ x CC {{ 0,1,2 C! x0 / x2 x0,2

with x0,1,2 : x0,2 ⇒ x1,2 x0,1 a deformation (2-cell) in C. For n ≥ 3, an n-simplex of ∆C is a diagram in C with the shape of the 2-skeleton of an oriented standard n-simplex,

x o x = F OnV- l6 < nH O−1 {{ - lll yy {{ ll--l yy {{ lll - y { ll l -yy {ll yy- x0 y - 1EE yy -- 11 E yy - 1 yEyE - 1y EE - yy1 E -- 11 EE - x1 RR 1 EE - CC RR1R EE - CC 1RRR E -- CC 11 RRR EE - CC1 RRR EE - ! RR(" x2 / x3

whose vertices are then objects x0, . . . , xn, whose edges are arrows xi,j xi / xj, for 0 ≤ i < j ≤ n, and whose faces are triangles:

x > j B x }} BB x i,j }} BB j,k }} ⇑ x BB }} i,j,k B xi / xk xi,k

geometry.tex; 19/02/2002; 9:46; p.2 The geometry of2-categories 3 with xi,j,k : xi,k ⇒ xj,k xi,j a deformation, for 0 ≤ i < j < k ≤ n. These data are required to satisfy that each tetrahedron

> xO ` Ca x || CC x x : x ⇒ x x i,` || CC k,` i,j,k i,k j,k i,j || CC || C xi,j,` : xi,` ⇒ xj,` xi,j x / xi A i,k >xk AA || xi,k,` : xi,` ⇒ xk,` xi,k AA xj,` || AA || x : x ⇒ x x xi,j A || xj,k j,k,` j,` k,` j,k xj for 0 ≤ i < j < k < ` ≤ n, is commutative in the sense that the following square of deformations

xi,j,` xi,` +3 xj,` xi,j

xi,k,` xj,k,` xi,j xk,` xi,j,k xk,` xi,k +3 xk,l xj,k xi,j commutes. This simplicial set ∆C becomes coskeletal in dimensions greater than 3 [23, Theorem 23]. Since we have two diﬀerent spaces associated to a 2-category C (namely BC and |∆C|) and both have shown their utility, it seems natural to compare them. To our knowledge, the only comparison that has been given is in the case in which all arrows and deformations of C are invertible, that is, when C is a 2-groupoid. For this case, Moerdijk and Svensson [20] prove that the two classifying spaces associated as above to a 2-groupoid C are homotopically equivalent. In their proof, the restriction on C is essential, so that it can not be translated to general 2-categories. The main purpose of this article is to prove the following: THEOREM 1. For any 2-category C there is a natural homotopy equivalence BC ∼ / |∆C|.

The above Theorem 1 can be applied to strict monoidal categories when these are considered as 2-categories with only one object. As an application we also prove Theorem 2 below, which is a generalization of Theorem A of Quillen [13] to 2-categories. In our theorem we replace the notion of homotopy ﬁber category of a functor by Gray’s notion of homotopy ﬁber 2-category y0//F of a 2-functor F : C → C0 at an object y0 ∈ C0 [9] (see Section 1 for details).

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THEOREM 2. Let F : C → C0 be a 2-functor. If the classifying space B(y0//F ) is contractible for every object y0 ∈ C0, then the induced map BF :BC → BC0 is a homotopy equivalence. This paper is organized into three sections. The first one is dedicated to stating some concepts, results and terminology we are going to use by reviewing definitions and basic facts concerning 2-categories, lax- functors, nerves, realizations and classifying spaces. The material in this first section is pretty standard, except perhaps Proposition 1.1, so that an expert reader could skip this section. Section 2 is the main one and it includes the proof of Theorem 1. In Section 3 we prove Theorem 2.

1. Preliminaries and Notations

For the general background on 2-categories used in this paper we refer to [2, 15], and on simplicial sets to [16]. Throughout this paper all categories are assumed to be small.

2-Categories. A 2-category C is just a category enriched in the category of small categories. Then, C is a category in which the hom-set between any two objects x, y ∈ C is the set of objects of a category C(x, y), whose arrows are called deformations and are denoted f : u ⇒ v. Moreover, the composition is a bifunctor C(x, y) × C(y, z) → C(x, z) which is associative and has identities Idx ∈ C(x, x). This bifunctor produces compositions of arrows and deformations respectively, both denoted here by juxtaposition. On the other hand, composition in each category C(x, y) is denoted by “◦”. Besides, we will usually identify an arrow w : y → z with the identity deformation on w in the category C(y, z), hence we write wf : wu ⇒ wv for the composition of f with the deformation identity on w. The underlying category of a 2-category C is the category obtained by leaving out the deformations of C. A 2-functor F : C → C0 between 2-categories is an enriched functor and so it takes objects, arrows and deformation in C to objects, arrows and deformations in C0 respectively, in such a way that all the 2-category structure is preserved. Any 2-functor induces, by restriction, an ordinary functor between the corresponding underlying categories and another one C(x, y) → C0(F (x),F (y)), for any pair of objects x, y ∈ C. Given a 2-functor F : C → C0 and an object y0 ∈ C0 by the homotopy ﬁber 2-category y0//F we mean Gray’s lax comma category py0q//F where py0q : 1 → C0 is the “name of an object” 2-functor [9]. Its objects

geometry.tex; 19/02/2002; 9:46; p.4 The geometry of2-categories 5 are pairs (u0, x) where x ∈ C is an object and u0 : y0 → F (x) is an arrow in C0. Its morphisms are also pairs

0 0 0 (f , u):(u0, x0) → (u1, x1) 0 0 0 where u : x0 → x1 is an arrow in C and f : u1 ⇒ F (u)u0 is a deformation in C0. Finally, its deformations

0 0 f :(f0, u0) ⇒ (f1, u1) 0 0 0 are deformations f : u0 ⇒ u1 in C such that f1 = F (f)u0 ◦ f0. Lax-functors. Given a category I and a 2-category C, a (normalized or strictly unitary) lax-functor x : I → C consists of three maps which assign to each object i ∈ I an object xi ∈ C, to each arrow τ : i → j in I an arrow xτ : xi → xj in C and to each pair of composable arrows τ σ i−→j−→k in I a deformation xσ,τ : xστ ⇒ xσ xτ in C, respectively. These data are required to satisfy the following normalization and coherence conditions:

• for any object i ∈ I, xIdi = Idxi ,

• for any arrow τ : i → j in I, xτ,Idi = xτ = xIdj ,τ , γ • for any triple of composable arrows i−→j−→τ k−→σ ` in I,

xσ,τ xγ ◦ xστ,γ = xσxτ,γ ◦ xσ,τγ . (1)

By a deformation f : x ⇒ y, between lax-functors x, y : I → C, we mean a (normalized) lax-natural transformation. Then, f consists of a pair of maps which assign to each object i ∈ I an arrow fi : xi → yi in C and to each arrow τ : i → j in I a deformation fτ : fjxτ ⇒ yτ fi in C, respectively. Such that:

• for any object i ∈ I, fIdi = fi, • for any pair of composable arrows i−→τ j−→σ k,

yσfτ ◦ fσxτ ◦ fkxσ,τ = yσ,τ fi ◦ fστ . (2)

When two lax-functors x and y coincide over a set of (base-) objects Θ ⊆ I, a deformation f : x ⇒ y is qualiﬁed as “relative” to Θ whenever fi = Idxi , for all objects i ∈ Θ. For a ﬁxed (possibly empty) set of objects Θ, lax-functors from I to C and deformations relative to Θ between them form a category that we write here as `Func(I, C)relΘ.

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Note that lax-functors x : I → C such that all deformations xσ,τ are identities are just ordinary functors to the underlying category of C, to which we refer simply as functors from I to C. We write

Func(I, C)relΘ for the full subcategory of `Func(I, C)relΘ whose objects are the functors from I to C. Observe that the coherence condition (2) for a deformation f between functors x, y : I → C reduces to the equality

fστ = yσfτ ◦ fσxτ . (3) Let us now replace category I by a (directed and reflexive multi- ) graph G. By a morphism x : G → C we mean a (directed and reflexive multi-) graph morphism from G to the underlying graph of the underlying category of C. Then x consists of a pair of maps which assign an object xi to each vertex i ∈ G and an arrow xτ : xi → xj to each edge τ : i → j in G, respectively, such that the arrows associated to degenerated edges are identities, that is, xIdi = Idxi . Given a set Θ of vertices in G and two morphisms x, y : G → C, by a deformation relative to Θ, f : x ⇒ y, we mean a pair of maps that assign an arrow fi : xi → yi in C to each vertex i ∈ G and a deformation fτ : fjxτ ⇒ yτ fi to each edge τ : i → j, respectively, such that fi = Idxi for any vertex i ∈ Θ and fIdj = Idfj for any vertex j ∈ G. Morphisms form G to C and deformations relative to Θ form a category that we write as Gph(G, C)relΘ. Let us now suppose that the category I = I(G) is the free category generated by the graph G. Then, any morphism x : G → C has a unique extension to a functor xe : I → C [15, Theorem 1]; moreover, any deformation relative to Θ, f : x ⇒ y, between morphisms has a unique extension to a deformation relative to Θ, fe : xe ⇒ ye, between σ1 σ2 σn the corresponding functors. On a string i0−→i1−→i2 . . . in−1−→in of edges in G, the deformation fe is inductively defined, by using (3), by the formula e e fσn···σ1 = yσn fσn−1···σ1 ◦ fσn xeσn−1···σ1 . Therefore, by restriction, we have an isomorphism of categories ∼ Func(I(G), C)relΘ = Gph(G, C)relΘ. (4) Bellow we shall also comment on lax-functors from free categories.

PROPOSITION 1.1. Let I be the free category generated by a graph G and let Θ be a subset of objects of I. Then, for any 2-category C, the subcategory Func(I, C)relΘ is reﬂexive in `Func(I, C)relΘ (i.e., the inclusion functor has a left adjoint).

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Proof. The reﬂector functor `Func(I, C)relΘ → Func(I, C)relΘ takes a lax functor x : I → C to the unique functor xe that extends the morphism restriction of x to the graph. Similarly, a deformation f : x ⇒ y is taken to the unique deformation fe : xe ⇒ ye that extends the deformation restriction of f to the graph (according to (4)). The counit of the adjunction is the identity and the unit takes any lax functor x to the deformation ε : x ⇒ xe which is inductively deﬁned, using coherence (2), by the equalities:

• εi = Idxi , for any object i ∈ I,

• ετ = Idxτ , for all τ ∈ G and τ σ • εστ = εσxτ ◦ xσ,τ , for any composable pair of arrows i−→j−→k with τ ∈ G and σ ∈ I.

Simplicial spaces and their realizations. The simplicial category ∠∆ has as objects the ordered sets [n] = {0, 1, . . . , n}, n ≥ 0, and as arrows the (weakly) monotone maps α :[n] → [m]. This category is generated by the directed graph with edges all maps

σ δ [n + 1] i / [n][o i n − 1] , 0 ≤ i ≤ n,

where ½ ½ j j ≤ i j j < i σ (j) = and δ (j) = i j − 1 j > i i j + 1 j ≥ i .

A simplicial object in a category E is just a functor S :∠ ∆ op → E. Such a functor amounts to a collection {Sn = S[n], n ≥ 0} of objects in E together with morphisms di = S(δi): Sn → Sn−1 and si = S(σi): Sn → Sn+1, 0 ≤ i ≤ n, called face and degeneracy operators respectively, which satisfy the simplicial identities described, for example, in [16, Definition 1.1]. A simplicial morphism f : S → S0 is just a natural transformation from S to S0, it then consists of a 0 family {fn = f[n] : Sn → Sn+1, n ≥ 0} of arrows in E which commute with the face and degeneracy operators. If f, g : S → S0 are simplicial morphisms, then a simplicial homotopy H : f ⇒ g is a system 0 {Hm : Sn → Sn+1, 0 ≤ m ≤ n} of arrows in E that satisfies the set of homotopy identities described, for example, in [16, Definition 5.1]. A simplicial space is a simplicial object in the category Top, of topological spaces, and a simplicial set is a simplicial object in the category Set, of sets. By regarding a set as a discrete space, any simplicial set is viewed as a simplicial space. Given a simplicial space S, Segal [21] defines its realization |S| as follows: for each n ≥ 0, let

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∆n denote the standard n-dimensional affine simplex and for each n m map α :[n] → [m] in∠ ∆ let α∗ : ∆ → ∆ be the induced affine 0 0 0 P map (i.e., α∗(t0, . . . , tn) = (t0, . . . , tm) with ti = α(i)=j tj). Then ` n the space |S| is defined from the topological sum n≥0 Sn × ∆ by n m identifying (S(α)(x), t) ∈ Sn × ∆ with (x, α∗(t)) ∈ Sm × ∆ , for all n x ∈ Sm, t ∈ ∆ and α :[n] → [m] in∠ ∆. This construction is functorial and any simplicial homotopy H : f ⇒ g, between simplicial space morphisms f, g : S → S0, determines a homotopy He : |f| ⇒ |g|, [17, Corollary 11.10]. A simplicial map that induces a homotopy equivalence on realizations is called a weak homotopy equivalence. A bisimplicial set is a functor S :∠ ∆ op ×∠∆ op → Set. This amounts to a family of sets {Sp,q = S([p], [q]); 0 ≤ p, q} together with horizontal and vertical face and degeneracy operators

h h sv dv si di j j Sp+1,q o Sp,q / Sp−1,q and Sp,q+1 o Sp,q / Sp,q−1, with 0 ≤ i ≤ p and 0 ≤ j ≤ q respectively, such that, for all p and q, both Sp,∗ and S∗,q are simplicial sets and the horizontal operators commute with the vertical ones. We now note that, on the one hand, any bisimplicial set S provides two simplicial objects in the category of simplicial sets: the horizontal h v one S :[p] 7→ Sp,∗ and the vertical one S :[q] 7→ S∗,q. Then, by taking h realization, S gives rise to two simplicial spaces |S | :[p] 7→ |Sp,∗| and v |S | :[q] 7→ |S∗,q|, respectively. On the other hand, by composing with the diagonal functor diag :∠ ∆ op →∠∆ op ×∠∆ op, the bisimplicial set S also provides another simplicial set diagS :[n] 7→ Sn,n, whose face and degeneracy operators are given in terms of those of S by the formulas h v h v di = di di and si = si si , respectively. It is known (e.g. [13, Lemma in page 86]) that there are natural homeomorphisms

||Sh|| =∼ |diagS| =∼ ||Sv||. (5)

We also point out (see [3, 1.2,4.3] for example) that if f : S → S0 is a 0 bisimplicial morphism such that the induced maps |fp,∗| : |Sp,∗| → |Sp,∗| 0 (resp. |f∗,q| : |S∗,q| → |S∗,q|) are homotopy equivalences for all p (resp. q), then so is the map |diagf| : |diagS| → |diagS0|.

Nerves of categories and their classifying spaces. Any ordered set [n] can be considered as a category with the elements of [n] as objects and only one arrow i → j when i ≤ j. After this identiﬁcation, the category∠ ∆ can be considered as a full subcategory of the category Cat of small categories. The nerve NC, of a category C, is the simplicial set deﬁned as the restriction of the Yoneda

geometry.tex; 19/02/2002; 9:46; p.8 The geometry of2-categories 9 embedding Func(−, C): Catop → Set to the category∠ ∆ op; thus NnC = Func([n], C), for each n ≥ 0. Since the category [n] is the free category generated by the graph 0 → 1 → · · · n − 1 → n, to give a functor x :[n] → C, say x(i → j) = xi,j xi / xj, is equivalent to giving a string of composable arrows in C

x0,1 x1,2 xn−1,n x0 / x1 / x2 → · · · xn−1 / xn. (6) Therefore we can make the identiﬁcation a NnC = C(x0, x1) × ... C(xn−1, xn) x0,...,xn∈Ob(C) and thus NC can be described as the simplicial set whose set of n- simplices are strings of n composable arrows in C and N0C = Ob(C). The i-face (resp. degeneracy) operator of an n-simplex such as (6) in NC is obtained by deleting the object xi, putting xj at place j − 1 for all j > i and using composition, if needed, to complete the new (n − 1)-simplex (resp. replacing the object xi by the arrow identity

Idxi : xi → xi). By applying realization to NC, we obtain the classifying space BC of the category C, that is BC = |NC|. This construction is functorial so that we have a functor B from Cat to the category of CW-complexes and cellular maps. We recall that functors related by natural transformations go to homotopic simplicial maps [11, VI, Proposition 2.2.2] and then to homotopic cellular maps. As a consequence, if a functor F : C → C0 has a left or right adjoint then the induced map BF :BC → BC0 is a homotopy equivalence.

Nerves of 2-categories and their classifying spaces. For any 2-category C, its nerve NC is the simplicial category (i.e., the simplicial object in Cat) obtained by restricting the functor Func(−, C)relOb(−) : Catop → Cat to the category∠ ∆ op; thus

NnC = Func([n], C)relOb[n], for each n ≥ 0. Observe that the simplicial set of objects of NC coincides with the nerve of the underlying category of C. Moreover, since [n] is a free category, by the isomorphism (4), then giving an arrow f : x → y in NnC (i.e., a lax natural transformation relative to the objects of [n]) is equivalent to giving the string of deformations in C,

x0,1 x1,2 xn−1,n & & ( x0 ⇓ f01 8 x1 ⇓ f12 8 x2 xn−1 ⇓ fn−1n 6 xn .

0 0 0 x0,1 x1,2 xn−1,n

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Therefore we can make the identification (cf. [12]) a NnC = C(x0, x1) × C(x1, x2) × · · · × C(xn−1, xn) x0,...,xn∈C and N0C = Ob(C), as a discrete category. After the above identification, the face and degeneracy functors are defined in the standard way. The classifying space BC of the 2-category C is, by definition, the realization of the simplicial space obtained by composing NC :∠ ∆ op → Cat with the classifying space functor B : Cat → Top, that is,

BC = |BNC|.

Note that, for each n ≥ 0, BNnC = |NNnC|. Then, we have a natural homeomorphism BC =∼ |diagNNC|, (7) where NNC : ([p], [q]) 7→ NpNqC is the bisimplicial set double nerve of the 2-category C. This construction of classifying space deﬁnes a functor from the category of 2-categories to the category of CW-complexes and cellular maps.

2. Realizations of geometric nerves are classifying spaces

It is not diﬃcult to see that the data for an n-simplex x of the geometric nerve ∆C of a 2-category C, as described in the introduction, is the same as the data for a lax functor x :[n] → C. That is, x consists of a family

x = {xi, xi,j, xi,j,k}0≤i≤j≤k≤n with xi objects, xi,j : xi → xj arrows and xi,j,k : xi,k ⇒ xj,k xi,j deformations in C, respectively, such that

• xi,i = Idxi , 0 ≤ i ≤ n,

• xi,j,j = Idxi,j = xi,i,j, 0 ≤ i ≤ j ≤ n, and

• xk,`xi,j,k ◦ xi,k,` = xj,k,`xi,j ◦ xi,j,`, 0 ≤ i ≤ j ≤ k ≤ ` ≤ n. Thus, the simplicial set ∆C can be described as the functor that takes each ordered set [n] to the set of lax functors from [n] to C. The face ∗ operators dm = δm : ∆nC → ∆n−1C act by deleting all the elements indexed by m and renumbering every index ` > m as ` − 1, that is, for any x ∈ ∆nC,(dmx)i = xδm(i),(dmx)i,j = xδm(i),δm(j) and (dmx)i,j,k = ∗ xδm(i),δm(j),δm(k). Analogously, the degeneracy operators are sm = σm : ∆nC → ∆n+1C.

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As we said in the introduction, we devote this section to proving Theorem 1. To this end, we are going to build a diagram of simplicial sets and natural maps,

Z diagNNC / TNC (8)

i T(i) 0 η Z / ∆ diagN∆Ch T CFb FF FFψ φ FF F ∆C in which both the square and the triangle placed at the bottom of the diagram are commutative (i.e., Z0i = T(i)Z and ψ = Z0φ) and the other triangle is commutative only up to homotopy (so that we will ﬁnd a simplicial homotopy H : T(i) ⇒ ψη). Then we will observe that the maps i, Z, Z0 and φ all induce homotopy equivalences on realizations and we conclude that so does every other map in the diagram. In particular, we will have that the composition ηZ induces a natural homotopy equivalence BC−→|∼ ∆C| and Theorem 1 will be proved. The remainder of Section 1 is devoted to carrying out the above objectives, and so we naturally divide it into ﬁve subsections.

2.1. Using the Artin-Mazur T-construction: the simplicial set TNC and the maps Z and η

There is another way of associating a simplicial set TNC to a 2-category C, whose simplices also have a pleasing geometric description. This simplicial nerve is deﬁned by applying Artin-Mazur “Total simplicial set” construction [1, Section III] to the bisimplicial set NNC double nerve of C that we have already introduced. Recall that, for a bisimplicial set S, the Artin-Mazur total simplicial set T(S) is deﬁned as ( ) Yn v h Tn(S) = (x0, x1, . . . , xn) ∈ Si,n−i; d0xi = di+1xi+1, 0 ≤ i < n , i=0 n ≥ 0, with face and degeneracy operators given by

Si Di Tn+1(S)To n(S) / Tn−1(S) ,

v v v h h h Di(x0, . . . , xn) = (di x0, di−1x1, . . . , d1xi−1, di xi+1, di xi+2, . . . , di xn),

v v v h h h Si(x0, . . . , xn) = (si x0, si−1x1, . . . , s0xi, si xi, si xi+1, . . . , si xn).

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Also recall that Zisman proves that there is a natural comparison map Z = ZS : diagS → T(S), which takes a n-simplex x ∈ diagS to ³ ´ h n h n−1 v h n−i v i v n Z(x) = (d1 ) x, (d2 ) d0x, . . . , (di+1) (d0) x, . . . , (d0) x .

This simplicial map induces a homotopy equivalence on realizations |Z| : |diag S|−→|∼ T (S)| (cf. [8]). The simplicial set TNC is deﬁned as TNC = T(NNC). We then have the canonical map

N Z = ZNNC : diagNNC −→ T C, (9) which induces a homotopy equivalence

|Z| :BC ∼ / |TNC|, therefore the simplicial set TNC also realizes the classifying space of the 2-category C. The simplices of TNC have the following geometric description: the 0- and 1-simplices of TNC are the objects and arrows of C respectively, its 2-simplices are diagrams in C of the form

1 x2 x0 & x 1 / x 1 x 0 1 ⇑ x¯2 8 2 (10)

0 x2

1 0 1 withx ¯2 : x2 ⇒ x2 a deformation in C, its 3-simplices are diagrams in C of the form

2 x3 x1 2 2 0 ⇑ x¯3 x1 & x / x x¯1 x x1 / x 0 1 ⇑ 2 8 2 3 B 3 ⇑ x¯1 0 3 x2 0 x3 and so on. More precisely, an n-simplex x of TNC is a family

i i x = {xi, xj, x¯j}, (11)

i consisting of objects xi ∈ C, for 0 ≤ i ≤ n, arrows xj : xj−1 → xj, for i i−1 i 0 ≤ i < j ≤ n, and deformationsx ¯j : xj ⇒ xj, for 0 < i < j ≤ n.

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N N The face operators Dm :Tn C → Tn−1C, 0 ≤ m ≤ n, are given by the formulas   xδm(i) j 6= m i δm(j) (Dmx)i = xδ (i), (Dmx)j = m  i i xm+1xm j = m, and  δm(i)  x¯ i 6= m 6= j  δm(j) (D x)i = x¯i x¯i j = m m j  m+1 m   m+1 m x¯j+1 ◦ x¯j+1 i = m, N N and the degeneracy operators Sm :TmC → Tm+1C, 0 ≤ m ≤ n, are given by   xσm(i) j 6= m + 1 (S x) = x , (S x)i = σm(j) m i σm(i) m j  Idxm j = m + 1, and   x¯σm(i) i 6= m + 1 6= j  σm(j) i (Smx) = Id j = m + 1 j  Idxm  Id m i = m + 1. xm+1 We are now ready to deﬁne the simplicial map η :TNC → ∆C. (12) This map η is the identity at levels zero and one, at level two it takes a 2-simplex as (10) to the 2-simplex of ∆C represented by the triangle

= x1 C x0 {{ CC x1 1 {{ CC 2 {{ ⇑x¯1x0 CC {{ 2 1 C! x0 / x2 0 0 x2x1 and, in general, η takes an n-simplex x ∈ TNC as (11) to the geometric n-simplex ηx given by

(ηx)i = xi, i i (ηx)i,j = xj . . . xi+1 j i+1 j i+1 i i (ηx)i,j,k = (¯xk ◦ ... ◦ x¯k ) ... (¯xj+1 ◦ ... ◦ x¯j+1)xj . . . xi+1

xj > CC j j (ηx) =xi ...xi }} C (ηx) =x ...x i,j j i+1 } CC j,k k j+1 }} CC }} ⇑(ηx)i,j,k CC }} C! xi / xk . i i i i xk...xj+1xj ...xi+1

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2.2. The simplicial category ∆C and the map i : NNC → N∆C

The geometric nerve ∆C of a category C is the simplicial set of objects of a simplicial category ∆C, which is deﬁned by

op ∆C = `Func(−, C)relOb[−] :∠ ∆ −→ Cat.

Thus, each category ∆nC has ∆nC as set of objects and its arrows are deformations relative to the set of objects of [n]. Therefore, in order to 0 0 have an arrow f : x → x in ∆nC we need xi = xi, for all 0 ≤ i ≤ n, 0 and in this case f consists of a family f = {fi,j : xi,j ⇒ xi,j}0≤i≤j≤n of deformations in C, which satisfy the equations

• fi,i = Idxi , 0 ≤ i ≤ n and 0 • fj,kfi,j ◦ xi,j,k = xi,j,k ◦ fi,k , 0 ≤ i ≤ j ≤ k ≤ n. ∗ The face and degeneracy functors are dm = δm : ∆nC → ∆n−1C and ∗ sm = σm : ∆nC → ∆n+1C respectively, so that they act on objects as 0 in the geometric nerve of C, and for arrows f : x → x in ∆nC,

(dmf)i,j = fδm(i),δm(j) and (smf)i,j = fσm(i),σm(j). This simplicial category ∆C deﬁnes, by taking the nerve of each category ∆nC, a bisimplicial set

N∆C : ([p], [q]) 7→ Np∆qC.

More explicitly, an element χ ∈ Np∆qC can be described as a string

f 1 f p χ = ( x0 / x1 → · · · → xp−1 / xp) , of p composable arrows in ∆qC. The vertical face and degeneracy operators

v v sm dm Np∆q+1C o Np∆qC / Np∆q−1C , 0 ≤ m ≤ q are induced by those of ∆C, that is

d f 1 d f p v 0 m / 1 p−1 m / p dm(χ) = ( dmx dmx → · · · → dmx dmx ) and s f 1 s f p v 0 m / 1 p−1 m / p sm(χ) = ( smx smx → · · · → smx smx ) . The horizontal face and degeneracy operators

h h sm dm Np+1∆qC o Np∆qC / Np−1∆qC , 0 ≤ m ≤ q

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h m are those of the nerve N∆qC, that is, dm acts by deleting x in the string χ, by putting xj at place j − 1 for all j > m and by using composition, if needed, to form the new string dmχ. In the same way, h m sm duplicates x at place m + 1 and introduces Idxm on the string. We note now that the simplicial category NC is a simplicial subcategory of ∆C via the inclusion functors

iq :NqC = Func([q], C)relOb[q] ,→ `Func([q], C)relOb[q] = ∆qC, q ≥ 0. Then we have a bisimplicial inclusion i : NNC ,→ N∆C, (13) whose restriction to the diagonal gives a simplicial inclusion, which we also denote as i : diagNNC → diagN∆C. (14) Next, Theorem 2.1 proves that this inclusion is a weak homotopy equivalence.

THEOREM 2.1. For any 2-category C, the map

|i| :BC = |diagNNC| ∼ / |diagN∆C|, induced by the inclusion (14) is a homotopy equivalence. Proof. Since the category [q] is free on a graph, by Proposition 1.1, each inclusion functor iq has a right adjoint. Therefore, the induced map on classifying spaces ∼ / Biq : BNqC = |NNqC| B∆qC = |N∆qC| is a homotopy equivalence, for all q ≥ 0, whence Theorem 2.1 follows.

2.3. Using the Artin-Mazur T construction again: the simplicial set T∆C and the maps Z0 and T(i)

We now consider the simplicial category ∆C, which was introduced in 2.2, and then the total simplicial set T∆C = TN∆C, associated to the bisimplicial set N∆C. At dimensions ≤ 1, this simplicial set TNC coincides with ∆C (and also with TNC). Thus, the 0- and 1-simplices of TNC are the objects and arrows of C, respectively. For n ≥ 2, an n-simplex ζ of TNC is a list

n−1 1 n n f n−1 2 f 1 ζ = (x , d0x / x , ··· , d0x / x ), (15)

geometry.tex; 19/02/2002; 9:46; p.15 16 BULLEJOS and CEGARRA in which each

m m m m m m m m m x = (xi , xi,j : xi → xj , xi,j,k : xi,k ⇒ xj,kxi,j) ∈ ∆mC is a geometric m-simplex of C, 1 ≤ m ≤ n, and each

m m m+1 m m+1 m f = (fi,j : xi+1,j+1 ⇒ xi,j): d0x → x

m m+1 is a morphism in the category ∆mC (so xi = xi+1 ), for all 1 ≤ m < n. The face and degeneracy operators

N o Si N Di / N Tn+1C Tn C Tn−1C , 0 ≤ m ≤ n, on an n-simplex ζ ∈ T∆C as (15) are given by the formulas:   (xn−1,f n−2,...,f 1) m=0 n n−1 n−m+1 n−m−1 n−m n−m−2 1 Dmζ= (dmx ,dm−1f ,...,d1f ,f d0f ,f ,...,f ) 0

0 ∆ Z = ZN∆C : diagN∆C → T C, (16) which is a weak homotopy equivalence. The simplicial map

T(i):TNC → T∆C, (17) is that induced by the bisimplicial inclusion (13) i : NNC → N∆C on total complexes. It can be described as follows: Given an n-simplex x = i i N {xi, xj, x¯j} ∈ T C, let us write ςnx ∈ ∆C for the geometric n-simplex given by the family

0 0 (ςnx)i = xi, (ςnx)i,j = xj ··· xi+1, and (ςnx)i,j,k = Id(ςnx)i,k , and let ςen−1x : d0ςnx → ςn−1d0x be the arrow in ∆n−1C given by the deformations

1 1 0 0 1 1 (ςen−1x)i,j =x ¯j+1 ··· x¯i+2 : xj+1 ··· xi+2 ⇒ xj+1 ··· xi+2.

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N Then, T(i) maps a simplex x ∈ Tn C to n−2 ∆ T(i)x = (ςnx, ςen−1x, ςen−2d0x, ··· , ςe1d0 x) ∈ Tn C. At this point we have completely described the commutative square

Z diagNNC / TNC

i T(i) Z0 diagN∆C / T∆C which forms part of the diagram (8) (the commutativity follows from the naturality of Zisman’s maps). We also know that both Z and Z0 are weak homotopy equivalences and, by Theorem 2.1, so is i. Therefore we conclude the following:

THEOREM 2.2. For any 2-category, the map

|T(i)| : |TNC| ∼ / |T∆C|, induced by (17) on realizations, is a homotopy equivalence.

2.4. The maps φ and ψ

As we showed in Subsection 2.2, the geometric nerve ∆C of a 2-category C is the simplicial set of objects of the simplicial category ∆C. Hence, if we regard ∆C as a discrete simplicial category (i.e., with only identities), we have an inclusion simplicial functor ∆C ,→ ∆C that induces a bisimplicial map φ : ∆C ,→ N∆C, (18) where ∆C is considered as a bisimplicial set that is constant in the horizontal direction (i.e., (∆C)p,q = Np∆qC = ∆qC). Thus, for any x ∈ ∆qC,

h p Id / Id / φp,q(x) = (s0 ) (x) = (x x → · · · x x) ∈ Np∆qC. We shall also write φ : ∆C ,→ diagN∆C, (19) for the induced simplicial map on diagonals.

LEMMA 2.3. For any 2-category C and any p ≥ 1, the simplicial set Np−1∆C is a simplicial deformation retract of Np∆C, via the simplicial h injection s0 :Np−1∆C ,→ Np∆C.

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h h Proof. Since d0 s0 = Id, to prove this lemma we only have to show h h the existence of a simplicial homotopy H : id ⇒ s0 d0 . In order to deﬁne H, we ﬁrst associate a geometric (q + 1)-simplex µmf ∈ ∆C to any arrow f : x → y in ∆qC and any 0 ≤ m ≤ q. This simplex µmf is given by ½ yi,j j ≤ m (µmf)i = xσm(i) = yσm(i) , (µmf)i,j = xσm(i),j−1 m < j and   yi,j,k k ≤ m (µ f) = x f ◦ x j ≤ m < k , m i,j,k  j,k−1 i,j i,j,k−1 xσm(i),j−1,k−1 m < j

Observe that µ0f = s0y and dq+1µqf = y, for all f : x → y, and that for f = Idx, any identity in ∆qC, we have µ(Idx) = smx, for all 0 ≤ m ≤ q. Next, we consider the arrows

hmf : smx → µmf , in ∆q+1C, deﬁned by ½ Idyi,j j ≤ m (hmf)i,j = , fσm(i),j−1 m < j

v v (note that h0 = s0 :N1∆qC → N1∆q+1C, and that dq+1hqf = Idy for all f : x → y). And ﬁnally we deﬁne the homotopy H by the maps

Hm :Np∆qC → Np∆q+1C, 0 ≤ m ≤ q,

f 0 f p which send a q-simplex χ = ( x0 / x1 / ··· / xp−1 / xp) in N∆C to the (q + 1)-simplex

h f 1 s f 2 s f p 1 m / 1 m / / p−1 m / p Hmχ = ( µmf smx ··· smx smx ) .

We leave it to the reader to check that H is indeed a simplicial homotopy as required.

By an iterative application of above Lemma 2.3, the simplicial map

φp,∗ : ∆C = N0∆C → Np∆C induces, for all p ≥ 0, a homotopy equivalence on geometric realizations, and therefore so does the simplicial map φ : ∆C → diagN∆C. We have then proved Theorem 2.4 stated below.

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THEOREM 2.4. For any 2-category C, the map

|φ| : |∆C| ∼ / |diagN∆C| induced by the above simplicial map (19) is a homotopy equivalence.

To fully complete the data in diagram (8) it only remains to intro- duce the simplicial map

ψ : ∆C → T∆C, (20)

φ 0 which is deﬁned to be the composite map ∆C / diagN∆C Z / T∆C, of the above φ (19) with Zisman map Z0 (16). Explicitly, ψ maps a simplex x ∈ ∆nC to

Id / n−1 Id / n−1 ψ(x) = (x, d0x d0x, ··· , d0 x d0 x). Since φ and Z0 are both weak homotopy equivalences, we have

THEOREM 2.5. For any 2-category C, the map

|ψ| : |∆C| ∼ / |T∆C| induced by the above simplicial map (20) is a homotopy equivalence.

2.5. The map |η| is a homotopy equivalence

We are now going to prove that the triangle

TNC DD DD DDη T(i) DD DD DD ψ ! T∆C o ∆C is commutative up to a simplicial homotopy. Then, since T(i) and ψ are both weak homotopy equivalences, by Theorems 2.2 and 2.5, we will have proved that |η| is also a homotopy equivalence.

PROPOSITION 2.6. There is a simplicial homotopy H : T(i) ⇒ ψη.

Proof. In order to deﬁne H, we ﬁrst build a geometric (n+1)-simplex N hmx ∈ ∆C, associated to any n-simplex x ∈ T C and any 0 ≤ m ≤ n.

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This simplex hmx is given by

(hmx)i = xσm(i),  i i  (ηx)i,j = xj ··· xi+1 j ≤ m (hmx)i,j =  (ς x) = x0 ··· x0 m < j n σm(i),j−1 j−1 σm(i)+1 and

(hmx)i,j,k =  j i+1 j i+1 i i  (ηx)i,j,k = (¯xk ◦ · · · ◦ x¯k ) ··· (¯xj+1 ◦ · · · ◦ x¯j+1)xj ··· xi+1 k ≤ m  0 0 i 1 i 1 xk ··· xj+1(¯xj ◦ · · · ◦ x¯j ) ··· (¯xi+1 ◦ · · · ◦ x¯i+1) j ≤ m < k   Idx0 ···x0 m < j . k−1 σm(i)+1

Observe that h0x = s0ςnx, h1x = s1ςnx and dn+1hnx = ηx. Next, we consider the arrows e hmx : d0hmx → hm−1d0x in ∆nC, for m ≥ 1, deﬁned by the family of deformations   Id j < m e (ηd0x)i,j (hmx)i,j =  x¯1 ··· x¯1 m ≤ j. j σm(i+1)+1 And, ﬁnally, the homotopy H is given by the maps

N ∆ Hm :Tn C → Tn+1C, 0 ≤ m ≤ n, i i N which send a n-simplex x = {xi, xj, x¯j}, in T C to the (n + 1)-simplex

Hmx =  n−2  (h0x,Idςnx,eςn−1x,eςn−2d0x,···,eς1d0 x) m=0 m−1 m n−2 (hmx,ehmx,ehm−1d0x,···,eh1d x,Idς dmx,eςn−m−1d x,···,eς1d x) 0

(note that H0 = s0T(i) and H1 = s1T(i)). To prove in full that H is a simplicial homotopy between T(i) and ψη, some veriﬁcations are needed, which are straightforward (though tedious) and we will therefore leave them to the reader.

As an immediate consequence of the comments at the beginning of this Section 2 and the above Proposition 2.6, we obtain Theorem 2.7 below, from which the proof of Theorem 1 in the introduction is completed.

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THEOREM 2.7. For any 2-category C, the map

|η| : |TNC| ∼ / |∆C| induced by the simplicial map (12) is a homotopy equivalence.

Remark. A strict monoidal category C = (C, ⊗,I) [15] can be viewed as a 2-category with only one object, say I, the objects x of C as arrows x : I → I and the arrows of C as deformations. It is the horizontal composition of arrows and deformations given by the strict tensor ⊗ : C×C → C and the vertical composition of deformations given by the composition of arrows in C. Then, the classifying space BC of the monoidal category is just the classifying space of the 2-category it deﬁnes. Then, by Theorem 1, the geometric nerve ∆C realizes the classifying space of the monoidal category C, that is, BC ' |∆C|. The geometric nerve ∆C is a 3-coskeletal reduced simplicial set whose simplices have the following simpliﬁed interpretation: the 1- simplices are the objects of C, the 2-simplices are morphisms of C of the form x0,1,2 x0,2 / x1,2 ⊗ x0,1 and the 3-simplices are commutative squares in C of the form

x0,1,3 x0,3 / x1,3 ⊗ x0,1

x0,2,3 x1,2,3 ⊗ Id Id ⊗ x0,1,2 x2,3 ⊗ x0,2 / x2,3 ⊗ x1,2 ⊗ x0,1 , the face operator di is obtained by deleting the data in which the index i appears. When the strict monoidal category C is a categorical group, that is, when all objects and arrows are invertible, it is largely known that BC ' |∆C|. Indeed, in that case C is just an internal category in the category of groups and the nerve NC is a simplicial group. Thus, we can apply Kan’s classifying complex functor W [16] to NC and the resulting simplicial set is isomorphic to the geometric nerve of C, that is, W NC =∼ ∆C. We also recall that strict categorical groups are equivalent to crossed modules (see [15] for example) and that classifying spaces of crossed modules are defined as the classifying spaces of the categorical groups they define ([4, 14]). We finally note that geometric nerves of non-necessarily strict categorical groups have been treated in [6, 7] and, more generally, geometric nerves of arbitrary monoidal categories (even of bicategories) have been treated in [5]

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3. A suﬃcient condition for a 2-functor to induce a homotopy equivalence

In this section we prove Theorem 2, stated in the Introduction. Let us recall that Quillen’s Theorem A [13] asserts that a functor F : C → C0 induces a homotopy equivalence BF :BC → BC0 whenever the classifying space of every homotopy fiber category y0/F is contractible, for all objects y0 ∈ C0. The homotopy fiber category y0/F has as objects the pairs (u0, x), with x ∈ C an object and u0 : y0 → F (x) an arrow in C0. 0 0 0 The arrows u :(u0 , x0 ) → (u1 , x1 ) in y /F are arrows u : x0 → x1 in C 0 0 such that u1 = F (u)u0 . In section 1 we have recalled the notion of homotopy fiber 2-category y0//F , for any 2-functor F : C → C0 between 2-categories and any object 0 0 y ∈ C . In particular, by taking F = IdC, we have the lax comma 2- categories y//C = y//IdC, y ∈ ObC. The objects of the 2-category y//C are pairs (v, x) with v : y → x an arrow, its arrows are also pairs (f, u):(v, x) → (v0, x0) with u : x → x0 an arrow and f : v0 ⇒ uv a deformation, and its deformations g :(f, u) ⇒ (f 0, u0) are deformations g : u ⇒ u0 such that f 0 = gv ◦ f. As for the case of categories, we have: PROPOSITION 3.1. Given a 2-category C,for any object y ∈ C, the classifying space B(y//C) is contractible. Proof. By Theorem 1 we have B(y//C) ' |∆(y//C)|. Let Dec(∆C) be the simplicial set obtained from the geometric nerve of C by “forgetting” the first face and degeneracy operators at each level and “renumbering” the levels [11, Chap. VI,1]. This is an augmented simplicial set over ∆0C = ObC, with augmentation given by the face operator d1 : ∆1C → ∆0C, on which the first degeneracy operators s0 define a simplicial contraction. Furthermore, the augmentation d1 : ∆1C → ∆0C gives rise to a decomposition of the simplicial set Dec(∆C) as a −1 Dec(∆C) = d1 (y), y∈Ob(C) in such a way that the above simplicial contraction of Dec(∆C) restricts −1 to the simplicial subsets fibers d1 (y), for any object y ∈ C. Therefore, −1 all the simplicial sets d1 (y) are homotopically trivial. However, it is not −1 difficult to identify each fiber d1 (y) with the geometric nerve ∆(y//C) of the lax comma 2-category y//C and so |∆(y//C)| is contractible.

We now are ready to prove our Theorem 2, an extension of Quillen’s Theorem A for 2-categories.

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3.1. Proof of Theorem 2

Let F : C → C0 be a 2-functor such that, for any object y0 ∈ C0, the space B(y0//F ) is contractible. We have to prove that the induced map BF :BC → BC0 is a homotopy equivalence. Hereafter, we shall also write F : ∆C → ∆C0 for the simplicial map induced by the 2-functor F : C → C0 between the corresponding geometric nerves. Let us consider the bisimplicial set S = S(F ), which is given by: n o 0 0 q+1 0 Sp,q = (x, y ) ∈ ∆pC × ∆p+q+1C ; F (x) = d0 (y ) , p, q ≥ 0 , with horizontal and vertical face and degeneracy operators

h h sv dv si di j j Sp+1,q o Sp,q / Sp−1,q and Sp,q+1 o Sp,q / Sp,q−1, given by

h 0 0 v 0 0 di (x, y ) = (dix, dq+i+1y ), dj (x, y ) = (x, djy ), h 0 0 v 0 0 si (x, y ) = (six, sq+i+1y ), sj (x, y ) = (x, sjy ). We ﬁrst observe that, for any p, the canonical projection

0 ϕp : Sp,0 → ∆pC , (x, y ) 7→ x defines an augmentation on the simplicial set Sp,∗ over the set ∆pC and that these augmentations ϕp, since they commute with the horizontal face and degeneracy operators, define a bisimplicial map ϕ : S → ∆C , (x, y0) 7→ x , where ∆C is considered as a constant bisimplicial set in the vertical direction. Furthermore, every augmented simplicial set Sp,∗ → ∆pC has a simplicial contraction, which is given by the maps εq+1 : Sp,q → Sp,q+1, q ≥ −1, defined by

ε0(x) = (x, s0F (x)), x ∈ ∆pC 0 0 0 εq+1(x, y ) = (x, sq+1y ), (x, y ) ∈ Sp,q , q ≥ 0. ∼ It follows that each induced map |ϕp| : |Sp,∗| → ∆pC is a homotopy equivalence and therefore the induced map |diagϕ| : |diagS| →∼ |∆C| is also a homotopy equivalence. Secondly, we note that, for any q, the map

0 0 0 0 ϕq : S0,q → ∆qC , (x, y ) 7→ dq+1y ,

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0 gives an augmentation on S∗,q over the set ∆qC and that these maps deﬁne a bisimplicial map

0 0 0 0 ϕ : S → ∆C , (x, y ) 7→ dq+1y , where ∆C0 is considered as a bisimplicial set constant in the horizontal direction. Let us now assume that we have proved that the induced map

|diagϕ0| : |diagS| → |∆C0| (21) is a homotopy equivalence. This will be proved later, after Lemma 3.2 below. Then, ﬁnally, we consider the following commutative diagram

|diagϕ0| |diagϕ| |∆C0| o |diagS(F )| / |∆C| ' BC

|diagγ| |F | BF 0 0 o |diagϕ | |diagϕ| / 0 0 |∆C | |diagS(IdC0 )| |∆C | ' BC

0 where γ : S(F ) → S(IdC0 ) is the bisimplicial map given by γp,q(x, y ) = (F (x), y0). In this diagram, all the horizontal maps are homotopy equivalences (note that, by Proposition 3.1 above, the homotopy ﬁber 2- 0 0 0 categories y //IdC0 = y //C are contractible). Therefore, ﬁrst |diagγ| and then |F | and BF are homotopy equivalences, and the proof will be complete. It remains to prove that the induced map |diagϕ0| in (21) is a homotopy equivalence. To do so, we return to the proof of Theorem 2. 0 0 For any q ≥ 0, the augmentation ϕq : S0,q → ∆q(C ) produces a decomposition of the simplicial set S∗,q as a 0 S∗,q = S∗,q(y ), 0 0 y ∈∆qC

0 0 −1 0 0 0 0 where S∗,q(y ) = ϕq (y ) is the ﬁber of ϕq at y . That is, S∗,q(y ) is the simplicial subset of S∗,q with n o 0 0 0 q+1 0 p+1 0 0 Sp,q(y ) = (x, z ) ∈ ∆pC × ∆p+q+1C ; d0 z = F (x), dq+1z = y .

0 0 Observe that, for any q ≥ 1 and any y ∈ ∆qC , the simplicial map v dq−1 : S∗,q → S∗,q−1 restricts to a simplicial map

v 0 0 dq−1 : S∗,q(y ) → S∗,q−1(dq−1y ) and that we have:

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LEMMA 3.2. The induced map

v 0 0 |dq−1| : |S∗,q(y )| −→ |S∗,q−1(dq−1y )| 0 0 is a homotopy equivalence, for all q ≥ 1 and all y ∈ ∆qC .

0 Proof. In fact, we are going to prove that S∗,q−1(dq−1y ) is a simpli- 0 v cial deformation retract of S∗,q(y ), with dq−1 as retraction and section given by the simplicial map

0 0 0 00 Γ: S∗,q−1(dq−1y ) −→ S∗,q(y ); (x, z ) 7→ (x, z ) ,

0 0 00 0 where, if (x, z ) ∈ Sp,q−1(dq−1y ), the geometric simplex z ∈ ∆p+q+1C is deﬁned as follows:  z0 q − 1 ∈/ {i, j} (  σ (i),σ (j) z0 i 6= q − 1  q−1 q−1 z00 = σq−1(i) z00 = y0 i ≤ q − 1 = j i y0 i = q − 1 , i,j  i,q−1 q−1  0 0 zq−1,j−1 yq−1,q i = q − 1 < j , and  0  z q − 1 ∈/ {i, j, k}  σq−1(i),σq−1(j),σq−1(k)  y0 k = q − 1 z00 = i,j,q−1 i,j,k  (z0 y0 ) ◦ z0 j = q − 1  q−1,k−1 i,q−1,q i,q−1,k−1   z0 y0 i = q − 1 . q−1,j−1,k−1 q−1,q It is straightforward to check that:

q+1 00 p+1 00 0 00 0 • d0 z = F (x) and dq+1z = y , so that (x, z ) ∈ Sp,q(y ), 00 0 v • dq−1z = z , whence dq−1Γ = Id, and • Γ is a simplicial map.

v Finally, a simplicial homotopy H : Id ⇒ Γdq−1 is given by the maps

0 0 0 0 Hm : Sp,q(y ) −→ Sp+1,q(y ), (x, z ) 7→ (smx, hmz ), 0 ≤ m ≤ p ,

0 0 0 where the geometric simplex hmz ∈ ∆p+q+1C is deﬁned as sq+m+1z = ∗ 0 σq+m+1z , except for the following elements:

0 0 0 (hmz )q−1,j = zq,j zq−1,q q < j ≤ q + m + 1, (h z0) = (z0 z0 ) ◦ z0 q < k ≤ q + m + 1, m i,q−1,k q−1,q,k i,q−1 i,q−1,k (h z0) = z0 z0 q ≤ j ≤ k ≤ q + m + 1, m q−1,j,k q,j,k q−1,q (h z0) = (z0 z0 ) ◦ z0 q < j ≤ q + m + 1 < k . m q−1,j,k q−1,j,k−1 q−1,q q−1,q,k−1

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An iterative use of the above Lemma 3.2 shows that the composite v q 0 q 0 simplicial map (d1) : S∗,q(y ) −→ S∗,0(d1y ) is a weak homotopy 0 0 equivalence, for all q ≥ 1 and all y ∈ ∆qC . But, for any object 0 0 0 0 ∼ 0 y ∈ ObC = ∆0C , we have that S∗,0(y ) = ∆(y //F ), and then

0 ∼ 0 0 |S∗,0(y )| = |∆(y //F )| ' B(y //F ),

0 where the latter is contractible by hypothesis. Hence, every |S∗,q(y )| is contractible. 0 0 It follows that each induced map |ϕq| : |S∗,q| → |∆qC | is a homotopy equivalence for all q ≥ 0, and consequently so is the induced map |diagϕ0| : |diagS| → |∆C0| (21). This completes the proof of Theorem 2.

We end this section by particularizing Theorem 2 to the case of a strict monoidal functor F : C → C0, between strict monoidal categories. To do so, recall that any monoidal category can be seen as a 2-category with only one object (see Remark at the end of Section 2). Given such a strict monoidal functor F : C → C0, we shall define its “homotopy fiber 2-category”, written by KF , as the homotopy fiber 2-category of F (considered as a 2-functor) at the unique object of 0 0 C . Then, KF is the 2-category whose objects are the objects y of C0, its arrows are pairs (u0, x): y0 → z0 with x an object in C and u0 : z0 → y0 ⊗ F (x) an arrow in C0, and its deformations

0 (u0,x0) % 0 y ⇓ u 9 z0

0 (u1,x1) are those arrows u : x0 → x1 in C such that the following triangle commutes z0 J 0 uu JJ 0 u0 uu JJu1 uu JJ uu JJ uz u J% 0 Id⊗F (u) / 0 y ⊗ F (x0) y ⊗ F (x1) .

Then, as an immediate consequence of Theorem 2 and the Remark in Section 2, we have the following

COROLLARY 3.3. Let F : C → C0 be a strict monoidal functor between strict monoidal categories such that the classifying space BKF is contractible. Then, the induced map BF :BC → BC0 is a homotopy equivalence.

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Address for Oﬀprints: Departamento de Algebra´ Facultad de Ciencias, Universidad de Granada 18071 Granada Spain

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