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 classify- ing 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 pro- cess 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 → . . . x_p−1→ x_p. 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) ×

∗ Supported by DGI:BFM2001-2886

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 arrows x₀ x_0,1

//x₁ of C and the 2-simplices are triangles:

⇑ x_0,1,2 x₁

x_1,2 C!!C CC CC CC

x₀ x_0,1{{{{{{{^=={

x_0,2 ^//x₂

with x_0,1,2 : x_0,2 ⇒ x_1,2x_0,1 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_noo x_n−1

x₀ {=={

{{ {{ {{

l 66l ll ll ll ll ll ll l

²²

1»»1 11 11 11 11 11 11

""E EE EE EE EE EE EE EE EE EE

x₁ C!!C CC CC CC

R ((R RR RR RR RR RR RR RR

°FF°

°°

°°

°°

°°

°°

°°

<<y yy yy yy yy yy yy yy yy yy

x₂ ^//

OO HH´

´´´´

´´´´

´´´´

´´´´

´´´´

´ x₃

OOVV-- ----

---- ----

---- ----

whose vertices are then objects x₀, . . . , x_n, whose edges are arrows x_i x_i,j

//x_j, for 0 ≤ i < j ≤ n, and whose faces are triangles:

⇑ x_i,j,k x_j

x_j,k BÃÃB BB BB BB

x_i x_{i,j }}}^>>}

}} }} }

x_i,k ^//x_k

with x_i,j,k : x_i,k ⇒ x_j,kx_i,j a deformation, for 0 ≤ i < j < k ≤ n. These data are required to satisfy that each tetrahedron

x_`

x_i x|_i,`||||||^>>|

x_i,k //

x_i,j^AAAA_ÃÃA AA

A x_k

x_k,`

aaCCCCCCCC

x_j x_j,k

>>|

||

||

||

| x_j,`

OO x_i,j,k : x_i,k ⇒ x_j,kx_i,j x_{i,j,`</sub>: x<sub>i,`} ⇒ x_{j,`</sub>x<sub>i,j</sub> x<sub>i,k,`} : x_{i,`</sub>⇒ x<sub>k,`}x_i,k x_{j,k,`</sub> : x<sub>j,`}⇒ x_k,`x_j,k

for 0 ≤ i < j < k < ` ≤ n, is commutative in the sense that the following square of deformations

x_i,`

x_i,k,`

®¶

x_i,j,`

+3x_{j,`</sub>x<sub>i,j</sub> x<sub>j,k,`}x_i,j

®¶x_{k,`</sub>x<sub>i,k</sub> x<sub>k,`}x_i,j,k

+3x_k,lx_j,kx_i,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 gener- alization 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).

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 cat- egory 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 Id_x ∈ 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 C⁰respectively, in such a way that all the 2-category structure is preserved. Any 2-functor induces, by restriction, an or- dinary 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

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

(f⁰, u) : (u⁰₀, x₀) → (u⁰₁, x₁)

where u : x₀ → x₁ is an arrow in C and f⁰ : u⁰₁ ⇒ F (u)u⁰₀ is a deformation in C⁰. Finally, its deformations

f : (f₀⁰, u₀) ⇒ (f₁⁰, u₁)

are deformations f : u₀ ⇒ u₁ in C such that f₁⁰ = F (f )u⁰₀◦ f₀⁰.

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 x_i ∈ C, to each arrow τ : i → j in I an arrow x_τ : x_i → x_j in C and to each pair of composable arrows i−→j^τ −→k in I a deformation x^σ _σ,τ : x_στ ⇒ x_σx_τ in C, respectively.

These data are required to satisfy the following normalization and coherence conditions:

• for any object i ∈ I, x_Idi = Id_xi,

• for any arrow τ : i → j in I, x_τ,Idi = x_τ = x_Idj,τ,

• for any triple of composable arrows i−→j^γ −→k^τ −→` in I,^σ

x_σ,τx_γ◦ x_στ,γ = x_σx_τ,γ ◦ x_{σ,τ γ}. (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 f_i : x_i → y_i in C and to each arrow τ : i → j in I a deformation f_τ : f_jx_τ ⇒ y_τf_i in C, respectively. Such that:

• for any object i ∈ I, f_Idi = f_i,

• for any pair of composable arrows i−→j^τ −→k,^σ

y_σf_τ◦ f_σx_τ◦ f_kx_σ,τ = y_σ,τf_i◦ f_στ. (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 f_i = Id_xi, 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.

Note that lax-functors x : I → C such that all deformations x_σ,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 func- tors from I to C. Observe that the coherence condition (2) for a defor- mation f between functors x, y : I → C reduces to the equality

f_στ = y_σf_τ ◦ f_σx_τ. (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 x_i to each vertex i ∈ G and an arrow x_τ : x_i → x_j to each edge τ : i → j in G, respectively, such that the arrows associated to degenerated edges are identities, that is, x_Idi = Id_xi. 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 f_i : x_i → y_iin C to each vertex i ∈ G and a deformation f_τ : f_jx_τ ⇒ y_τf_i to each edge τ : i → j, respectively, such that f_i = Id_xi for any vertex i ∈ Θ and f_Idj = Id_fj 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 ex : 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, f :^e x ⇒e y, betweene the corresponding functors. On a string i₀−→i^σ1 ₁−→i^σ2 ₂. . . i_n−1−→i^σn _n of edges in G, the deformation f is inductively defined, by using (3), by^e the formula

fe_σn···σ1 = y_σnf^e_{σn−1···σ1} ◦ f_σnxe_{σn−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).

Proof. The reflector functor `Func(I, C)_relΘ → Func(I, C)_relΘ takes a lax functor x : I → C to the unique functor ex that extends the morphism restriction of x to the graph. Similarly, a deformation f : x ⇒ y is taken to the unique deformation f :^e x ⇒e y that extends thee 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 ⇒x which is inductively defined, using coherencee (2), by the equalities:

• ε_i= Id_xi, for any object i ∈ I,

• ε_τ = Id_xτ, for all τ ∈ G and

• ε_στ = ε_σx_τ ◦ x_σ,τ, for any composable pair of arrows i−→j^τ −→k^σ 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

[n + 1] ^{σi //}[n]^{oo δi} [n − 1], 0 ≤ i ≤ n, where

σ_i(j) =

½j j ≤ i

j − 1 j > i and δ_i(j) =

½j j < 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 {S_n = S[n], n ≥ 0} of ob- jects in E together with morphisms d_i = S(δ_i) : S_n → S_n−1 and s_i = S(σ_i) : S_n → S_n+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 family {f_n = f_[n] : S_n → S_n+1⁰ , n ≥ 0} of arrows in E which commute with the face and degeneracy operators. If f, g : S → S⁰ are simpli- cial morphisms, then a simplicial homotopy H : f ⇒ g is a system {H_m : S_n→ S⁰_n+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 sim- plicial 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

∆ⁿ denote the standard n-dimensional affine simplex and for each map α : [n] → [m] in ∆∠ let α_∗ : ∆ⁿ → ∆^m be the induced affine map (i.e., α_∗(t₀, . . . , t_n) = (t⁰₀, . . . , t⁰_m) with t⁰_i = ^P_α(i)=jt_j). Then the space |S| is defined from the topological sum ^`_n≥0S_n× ∆ⁿ by identifying (S(α)(x), t) ∈ S_n× ∆ⁿ with (x, α_∗(t)) ∈ S_m× ∆^m, for all x ∈ S_m, t ∈ ∆ⁿand 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 H : |f | ⇒ |g|, [17,^e 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 {S_p,q= S([p], [q]); 0 ≤ p, q} together with horizontal and vertical face and degeneracy operators

S_p+1,q ^s

hi

oo S_p,q ^d

hi //S_p−1,q and S_p,q+1 ^s

v

oo j S_p,q ^d

v

j //S_p,q−1, with 0 ≤ i ≤ p and 0 ≤ j ≤ q respectively, such that, for all p and q, both S_p,∗ 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 one S^h : [p] 7→ S_p,∗and the vertical one S^v : [q] 7→ S_∗,q. Then, by taking realization, S gives rise to two simplicial spaces |S^h| : [p] 7→ |S_p,∗| and

|S^v| : [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→ S_n,n, whose face and degeneracy operators are given in terms of those of S by the formulas d_i = d^h_id^v_i and s_i = s^h_is^v_i, respectively. It is known (e.g. [13, Lemma in page 86]) that there are natural homeomorphisms

||S^h|| ∼= |diagS| ∼= ||S^v||. (5) We also point out (see [3, 1.2,4.3] for example) that if f : S → S⁰ is a bisimplicial morphism such that the induced maps |f_p,∗| : |S_p,∗| → |S_p,∗⁰ | (resp. |f_∗,q| : |S_∗,q| → |S_∗,q⁰ |) are homotopy equivalences for all p (resp.

q), then so is the map |diagf | : |diagS| → |diagS⁰|.

Nerves of categories and their classifying spaces. Any or- dered 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 cate- gory C, is the simplicial set defined as the restriction of the Yoneda

embedding Func(−, C) : Cat^op → Set to the category ∆∠ ^op; thus N_nC = 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) = x_i ^{xi,j //}x_j, is equivalent to giving a string of composable arrows in C

x₀ ^{x0,1 //}x₁ ^{x1,2 //}x₂ → · · · x_n−1^xn−1,n// x_n. (6) Therefore we can make the identification

N_nC = ^a

x0,...,xn∈Ob(C)

C(x₀, x₁) × . . . C(x_n−1, x_n)

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 x_i, putting x_j at place j − 1 for all j > i and using composition, if needed, to complete the new (n − 1)-simplex (resp. replacing the object x_i by the arrow identity Id_xi : x_i → x_i).

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 cel- lular 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(−) : Cat^op→ Cat to the category ∆∠ ^op; thus

N_nC = 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 N_nC (i.e., a lax natural transformation relative to the objects of [n]) is equivalent to giving the string of deformations in C,

x₀

x0,1

&&

x⁰_0,1

88⇓ f01 x₁

x1,2

&&

x⁰_1,2

88⇓ f12 x₂ x_n−1

xn−1,n

((

x⁰_n−1,n

66⇓ fn−1n x_n.

Therefore we can make the identification (cf. [12]) N_nC = ^a

x0,...,xn∈C

C(x₀, x₁) × C(x₁, x₂) × · · · × C(x_n−1, x_n)

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, BN_nC = |NN_nC|. Then, we have a natural homeomorphism

BC ∼= |diagNNC|, (7)

where NNC : ([p], [q]) 7→ N_pN_qC 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 = {x_i, x_i,j, x_i,j,k}_{0≤i≤j≤k≤n}

with x_i objects, x_i,j : x_i → x_j arrows and x_i,j,k : x_i,k ⇒ x_j,kx_i,j deformations in C, respectively, such that

• x_i,i = Id_xi, 0 ≤ i ≤ n,

• x_i,j,j = Id_xi,j = x_i,i,j, 0 ≤ i ≤ j ≤ n, and

• x_{k,`</sub>x<sub>i,j,k</sub>◦ x<sub>i,k,`} = x_{j,k,`</sub>x<sub>i,j</sub>◦ x<sub>i,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 d_m = δ^∗_m : ∆_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, (d_mx)_i= x_δm(i), (d_mx)_i,j = x_{δm(i),δm(j)} and (d_mx)_i,j,k = x_{δm(i),δm(j),δm(k)}. Analogously, the degeneracy operators are s_m = σ_m^∗ :

∆_nC → ∆_n+1C.

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,

diagNNC

i²²

Z //T^NC

T(i)

²² _η ¸¸

diagN∆C ^{Z0 //}T^∆C

∆C bbFFF ψ

FFFFF

φ

hh

(8)

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 T^NC and the maps Z and η

There is another way of associating a simplicial set T^NC 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

T_n(S) = (

(x₀, x₁, . . . , x_n) ∈ Yn

i=0

S_i,n−i; d^v₀x_i = d^h_i+1x_i+1, 0 ≤ i < n )

,

n ≥ 0, with face and degeneracy operators given by T_n+1(S)^{oo Si} T_n(S) ^{Di //}T_n−1(S) ,

D_i(x₀, . . . , x_n) = (d^v_ix₀, d^v_i−1x₁, . . . , d^v₁x_i−1, d^h_ix_i+1, d^h_ix_i+2, . . . , d^h_ix_n), S_i(x₀, . . . , x_n) = (s^v_ix₀, s^v_i−1x₁, . . . , s^v₀x_i, s^h_ix_i, s^h_ix_i+1, . . . , s^h_ix_n).

Also recall that Zisman proves that there is a natural comparison map

Z = Z_S : diagS → T(S), which takes a n-simplex x ∈ diagS to

Z(x) =^³(d^h₁)ⁿx, (d^h₂)ⁿ⁻¹d^v₀x, . . . , (d^h_i+1)ⁿ⁻ⁱ(d^v₀)ⁱx, . . . , (d^v₀)ⁿx^´. This simplicial map induces a homotopy equivalence on realizations

|Z| : |diag S|−→|T (S)| (cf. [8]).^∼

The simplicial set T^NC is defined as T^NC = T(NNC). We then have the canonical map

Z = Z_NNC: diagNNC −→ T^NC, (9) which induces a homotopy equivalence

|Z| : BC ^{∼ //}|T^NC|,

therefore the simplicial set T^NC also realizes the classifying space of the 2-category C.

The simplices of T^NC have the following geometric description: the 0- and 1-simplices of T^NC are the objects and arrows of C respectively, its 2-simplices are diagrams in C of the form

x₀ ^{x01 //}x₁

x¹₂

&&

x⁰₂

88⇑¯x¹₂ x₂ (10)

with ¯x¹₂: x⁰₂⇒ x¹₂ a deformation in C, its 3-simplices are diagrams in C of the form

x₀ ^{x01 //}x₁

x¹₂

&&

x⁰₂

88⇑¯x¹₂ x₂ x¹₃ //

x²₃

⇑x¯²₃ ¿¿

x⁰₃

⇑x¯¹₃ xBB ₃

and so on. More precisely, an n-simplex x of T^NC is a family

x = {x_i, xⁱ_j, ¯xⁱ_j}, (11) consisting of objects x_i ∈ C, for 0 ≤ i ≤ n, arrows xⁱ_j : x_j−1 → x_j, for 0 ≤ i < j ≤ n, and deformations ¯xⁱ_j : xⁱ⁻¹_j ⇒ xⁱ_j, for 0 < i < j ≤ n.

The face operators D_m : T^N_nC → T^N_n−1C, 0 ≤ m ≤ n, are given by the formulas

(D_mx)_i= x_δm(i), (D_mx)ⁱ_j =





x^δ_δ^m_m⁽ⁱ⁾_(j) j 6= m xⁱ_m+1xⁱ_m j = m, and

(D_mx)ⁱ_j =











¯

x^δ_δ^m_m⁽ⁱ⁾_(j) i 6= m 6= j

¯

xⁱ_m+1x¯ⁱ_m j = m

¯

x^m+1_j+1 ◦ ¯x^m_j+1 i = m,

and the degeneracy operators S_m : T^N_mC → T^N_m+1C, 0 ≤ m ≤ n, are given by

(S_mx)_i = x_σm(i), (S_mx)ⁱ_j =



 x^σ_σ^m(i)

m(j) j 6= m + 1 Id_xm j = m + 1, and

(S_mx)ⁱ_j =









¯ x^σ_σ^m(i)

m(j) i 6= m + 1 6= j Id_Idxm j = m + 1 Id_x^m_m+1 i = m + 1.

We are now ready to define the simplicial map

η : T^NC → ∆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

⇑x¯¹₂x⁰₁

x₁

x¹₂

C!!C CC CC CC

x₀

x{⁰₁{{{{{=={

{

x⁰₂x⁰₁

//x₂

and, in general, η takes an n-simplex x ∈ T^NC as (11) to the geometric n-simplex ηx given by

(ηx)_i= x_i,

(ηx)_i,j = xⁱ_j. . . xⁱ_i+1

(ηx)_i,j,k= (¯x^j_k◦ . . . ◦ ¯xⁱ⁺¹_k ) . . . (¯x^j_j+1◦ . . . ◦ ¯xⁱ⁺¹_j+1)xⁱ_j. . . xⁱ_i+1

⇑(ηx)i,j,k

x_j

(ηx)j,k=x^j_k...x^j_j+1

C!!C CC CC CC CC

x_i

(ηx)i,j=xⁱ_j...x}ⁱ_i+1}}}}}}}>>}

xⁱ_k...xⁱ_j+1xⁱ_j...xⁱ_i+1

//x_k .

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

∆C = `Func(−, C)_relOb[−]: ∆∠ ^op−→ 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 have an arrow f : x → x⁰ in ∆_nC we need x_i = x⁰_i, for all 0 ≤ i ≤ n, and in this case f consists of a family f = {f_i,j : x_i,j ⇒ x⁰_i,j}_{0≤i≤j≤n}of deformations in C, which satisfy the equations

• f_i,i = Id_xi, 0 ≤ i ≤ n and

• f_j,kf_i,j ◦ x_i,j,k= x⁰_i,j,k◦ f_i,k, 0 ≤ i ≤ j ≤ k ≤ n.

The face and degeneracy functors are d_m = δ^∗_m : ∆_nC → ∆_n−1C and s_m = σ_m^∗ : ∆_nC → ∆_n+1C respectively, so that they act on objects as in the geometric nerve of C, and for arrows f : x → x⁰ in ∆_nC,

(d_mf )_i,j = f_{δm(i),δm(j)} and (s_mf )_i,j= f_{σm(i),σm(j)}.

This simplicial category ∆C defines, by taking the nerve of each category ∆_nC, a bisimplicial set

N∆C : ([p], [q]) 7→ N_p∆_qC.

More explicitly, an element χ ∈ N_p∆_qC can be described as a string χ = ( x^{0 f1//}x¹ → · · · → x^{p−1 f}

p//x^p) ,

of p composable arrows in ∆_qC. The vertical face and degeneracy op- erators

N_p∆_q+1C^{oo svm} N_p∆_qC ^{dvm //}N_p∆_q−1C , 0 ≤ m ≤ q are induced by those of ∆C, that is

d^v_m(χ) = ( d_mx^{0 dmf}

1//d_mx¹→ · · · → d_mx^{p−1 dmf}

p//d_mx^p) and

s^v_m(χ) = ( smx^{0 smf}

1//s_mx¹ → · · · → s_mx^{p−1 smf}

p//s_mx^p) . The horizontal face and degeneracy operators

N_p+1∆_qC^{oo shm} N_p∆_qC ^{dhm //}N_p−1∆_qC , 0 ≤ m ≤ q

are those of the nerve N∆_qC, that is, d^h_m acts by deleting x^m in the string χ, by putting x^j at place j − 1 for all j > m and by using composition, if needed, to form the new string d_mχ. In the same way, s^h_m duplicates x^m at place m + 1 and introduces Id_x^m on the string.

We note now that the simplicial category NC is a simplicial subcat- egory of ∆C via the inclusion functors

i_q: N_qC = 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 i_q has a right adjoint. Therefore, the induced map on classifying spaces

Bi_q : BN_qC = |NN_qC| ^{∼ //}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 simpli- cial set T^NC coincides with ∆C (and also with T^NC). Thus, the 0- and 1-simplices of T^NC are the objects and arrows of C, respectively. For n ≥ 2, an n-simplex ζ of T^NC is a list

ζ = (xⁿ, d₀x^{n fn−1//}xⁿ⁻¹, · · · , d₀x^{2 f1 //}x¹), (15)