Theory and Applications of Categories, Vol. 7, No. 14, 2000, pp. 263–283.

A SIMPLICIAL DESCRIPTION OF THE HOMOTOPY CATEGORY OF SIMPLICIAL GROUPOIDS

A. R. GARZON, J. G. MIRANDA AND R. OSORIO

Transmitted by Ieke Moerdijk

ABSTRACT. In this paper we use Quillen’s model structure given by Dwyer-Kan for the category of simplicial groupoids (with discrete object of objects) to describe in this category, in the simplicial language, the fundamental homotopy theoretical constructions of path and cylinder objects. We then characterize the associated left and right homotopy relations in terms of simplicial identities and give a simplicial description of the homotopy category of simplicial groupoids. Finally, we show loop and suspension functors in the pointed case.

1. Introduction

1.1. Summary. A well-known and quite powerful context in which an abstract homotopy theory can be developed is supplied by a category with a closed model structure in the sense of Quillen [16]. The category Simp(Gp) of simplicial groups is a remarkable example of what a closed model category is, and the homotopy theory in Simp(Gp) developed by Kan [12] occurs as the homotopy theory associated to this closed model structure. According to the terminology used by Quillen, we have that the homotopy theory in Simp(Gp) is equivalent to the homotopy theory in the category of reduced simplicial sets and this last is equivalent to the homotopy theory in the category of pointed connected topological spaces.

If Y is an object of a closed model category C, a path object for Y is a factorization of the diagonal morphism

[figure: Y; σ; Y^{I}; (∂_{0},∂_{1}); Y × Y ,]

where (∂_{0}, ∂_{1}) is a fibration and σ is a weak equivalence.

If f, g ∈ Hom\underline{C}(X, Y), a right homotopy from f to g is defined as a morphism h : X → Y^{I} such that ∂_{0}h = g and ∂_{1}h = f. The morphism f is said to be right homotopic to g if such a right homotopy exists. When Y is fibrant “is right homotopic to” is an equivalence relation on Hom_{C}(X, Y). The notions of cylinder object and left homotopy are defined in a dual manner. Moreover, if X is cofibrant and Y is fibrant, then the left and right homotopy relations on Hom_{C}(X, Y) coincide. If [X, Y] denotes the set of equivalence classes, the category πC_{cf}, whose objects are the objects of C that are both fibrant and cofibrant, and Hom_{πC_{cf}} (X, Y) = [X, Y], with composition induced from that of C, is equivalent to the homotopy category of C, Ho(C), which is defined to be the localization of C, [5], with respect to the class of weak equivalences.

This paper has been financially supported by DGES PB97-0829 and by NATO PST.CLG 975316

Received by the editors 1999 November 2 and, in revised form, 2000 October 18.

Published on 2000 October 25.

2000 Mathematics Subject Classification: 18G30, 55U35.

Key words and phrases: closed model category, path object, cylinder object, homotopy relation.

\begin{matrix}c \\ ◯ \end{matrix} A. R. Garzon, J. G. Miranda and R. Osorio, 2000. Permission to copy for private use granted.

Moreover, if C is a closed simplicial model category (see [16]), X is cofibrant and Y is fibrant then the left and the right homotopy relations on Hom_{C}(X, Y) coincide with the simplicial homotopy relation. This is the case in Simp(Gp) when G_{•} is a free simplicial group in which case, for any simplicial group H_{•}, Hom_{Ho(Simp(Gp))}(G_{•}, H_{•}) = [G_{•}, H_{•}], the set of simplicial homotopy classes of simplicial morphisms from G_{•} to H_{•}.

In [3] Dwyer-Kan demonstrated that Simp(Gpd)_{∗}, the category of simplicial groupoids (with discrete object of objects), admits a closed model structure and the associated homotopy theory was then shown to be equivalent to the (unpointed) homotopy theory in the category Simp(Set) of simplicial sets and therefore to that one in the category of topological spaces. This was done by extending the well-known adjoint situation, [14],

G : Simp(set)_{red} ↔ Simp(Gp) : \overline{W}

to a pair of adjoint functors

G : Simp(Set) ↔ Simp(Gpd)_{∗} : \overline{W} ,

which induces the equivalence of homotopy theories. In particular, their homotopy categories are equivalent and there is a 1-1 correspondence of homotopy classes of maps. Moreover, analogously to the case of simplicial groups, in the model category Simp(Gpd)_{∗} every object is fibrant and the cofibrant objects are the free simplicial groupoids and their retracts.

The first aim of the present paper is to give explicit constructions of path space and cylinder object in the closed model category of simplicial groupoids and then, although Simp(Gpd)_{∗} is not a closed simplicial model category, we characterize the associated right and left homotopy relations in terms of simplicial identities, which correspond by the functor \overline{W} to the simplicial homotopy identities in simplicial sets. Thus, Simp(Gpd)_{∗} behaves as it was a closed simplicial model category, that is, there is a notion of simplicial homotopy when the source is cofibrant ant the target fibrant and it coincides with the axiomatic left homotopy and right homotopy that Quillen develops via cylinder or path objects. The surprise of this fact can be clarified if one looks at the objects of Simp(Gpd)_{∗} inside of the category of all simplicial groupoids (i.e., with objects not necessarily discrete) and then one restricts the notion of simplicial homotopy here to those simplicial groupoids. Recall that there are true Quillen model structures on the category of all simplicial groupoids, one created by the Moerdijk’s model structure on bisimplicial sets [13] and the dimensionwise nerve N : Simp(Gpd) → BiSsets, and the other by Joyal-Tierney in [10], obtained as a particular case (when E = Sets) of the model structure on the category Gpd(S(E)) of simplicial groupoids in any Grothendieck topos E. Both model structures should yield to equivalent homotopy theories (they have same weak equivalences and cofibrations ordered by inclusion). Besides, as it is deduced from ([10] Theorem 10), the inclusion I : Simp(Gpd)_{∗} → Simp(Gpd) induces an equivalence of homotopy categories and there is a 1-1 correspondence of homotopy classes of maps [G, H] ≅ [I(G), I(H)] for any objects G, H ∈ Simp(Gpd)_{∗}.

The fact that the axiomatic right and left homotopy in Simp(Gpd)_{∗} can be expressed in the simplicial language gives a more manageable version of the set of homotopy classes of maps between two simplicial groupoids and thus we are able of giving, as in simplicial groups, a simplicial description of the homotopy category of simplicial groupoids. Also we think that, in this way and in order to study non-connected n-types , it is possible to give truncated versions of the fundamental homotopy constructions and the homotopy relation, following the same approach as in simplicial groups [1], [7] and in simplicial sets [4].

Finally, we show pointed versions of the homotopy constructions of path and cylinder objects that allow us to give the corresponding constructions of loop and suspension functors.

The authors would like to thank the referee for several useful comments and suggestions.

1.2. Notation and terminology. We will freely use the results of [14], [16] and [3]. We denote by Simp(Gp) the category of simplicial groups and by Simp(Gpd)_{∗} the category of simplicial groupoids with discrete object of objects. An object X ∈ Simp(Gpd)_{∗} can be represented by means of a diagram

[figure: …; X_{n}; X_{n−1}; …; X_{2}; X_{1}; X_{0}; …; O; O; …; O; O; O; s_{0}; s_{n−1}; d_{n}; d_{0}; s_{1}; s_{0}; s_{0}; d_{2}; d_{0}; d_{1}; d_{0}; s; t; s; t; s; t; s; t; s; t]

where each [figure: X_{n}; O; t; s; I] is a groupoid and d_{i} and s_{j} are groupoid morphisms satisfying

the usual simplicial identities (see [14]). Note that if O = ∗ is a one-point set, then X can be seen as a simplicial group and so Simp(Gp) is a full subcategory of Simp(Gpd)_{∗}. For any two objects p, q ∈ O in X, X(p, q) will denote the simplicial set of morphisms in X from p to q. If p = q, X(p) = X(p, p) is a simplicial group called the simplicial group of automorphisms in p. Both categories Simp(Gp) and Simp(Gpd)_{∗} are closed model categories in the sense of Quillen [16]. Below we recall the structure in Simp(Gpd)_{∗} [3]:

-A morphism f : X → Y in Simp(Gpd)_{∗} is a fibration if [figure: X_{0}; f_{0}; s; t; Y_{0}; s; t; O; f; O′] is a fibration of

groupoids and, for any p ∈ Obj(X), the induced morphism X(p) → Y (f(p)) is a fibration of simplicial groups.

-The morphism f is a weak equivalence if f induces a 1-1 correspondence between the components of X and those of Y and, for any p ∈ Obj(X), the induced morphism X(p) → Y (f(p)) is a weak equivalence of simplicial groups.

-The morphism f is a cofibration if it is a retract of a free map.

In any groupoid [figure: X_{n}; O; t; s; I] , the composition of two morphisms x, y ∈ X_{n} such

that s(x) = t(y) will be denoted by x ∘ y.

2. Homotopy groupoids and simplicial homotopy identities

Given X ∈ Simp(Gpd)_{∗}, the Moore complex of X, NX, is defined as the following chain complex of groupoids:

N_{∗}X = ( [figure: N_{n}X; ^{\overline{d}^{n}}; N_{n−1}X; N_{2}X; ^{\overline{d}^{2}}; N_{1}X; ^{\overline{d}^{1}}; N_{0}X; O; O; O; O; O] )

where N_{0}X = X_{0}, [figure: N_{n}X; s; t; O] = Ker(d_{0}) ∩ ⋯ ∩ Ker(d_{n−1}) and d̄_{n} is the restriction of

d_{n}.

Note that, as the face and degeneracy morphisms of X are the identity on objects, the

groupoid [figure: N_{n}X; s; t; O] is, for n ≥ 1, the disjoint union of the vertex groups Kerd_{i}(p),

p ∈ O; in particular, \overline{d}_{n}\overline{d}_{n+1} is always an identity so that (NX, \overline{d}) is, indeed, a chain complex of groupoids over O. Then, this construction (see [6]) gives a functor from Simp(Gpd)_{∗} to the category of chain complexes of groupoids and, if O has only one element, it is clear that this construction reduces to the well-known Moore complex functor defined in Simp(Gp) (see [14]).

Recall that the homotopy groups of the underlying simplicial set of a simplicial group G_{•} (pointed by the identity element) can be obtained as the homology groups of the Moore complex of G_{•}. In the same way, we can consider homotopy groupoids of any X ∈ Simp(Gpd)_{∗}, defined as follows:

Let us consider [figure: B_{n}(X); s; t; O] , the groupoid whose set of morphisms is \overline{d_{n+1}}(N_{n+1}(X)),

and [figure: Z_{n}(X); s; t; O] , the groupoid whose set of morphisms is Ker(\overline{d_{n}}). It is clear that

[figure: B_{n}(X); s; t; O] is a normal subgroupoid of [figure: Z_{n}(X); s; t; O] [9], and then we define, for all

n ≥ 1, the n-th homotopy groupoid of X, denoted by π_{n}(X), as the quotient groupoid of

[figure: Z_{n}(X); s; t; O] by [figure: B_{n}(X); s; t; O] . Thus, π_{n}(X) has O as the set of objects and \frac{Z_{n}(X)}{≡} as

the set of morphisms, where x ≡ x′ if there exist y, y′ ∈ B_{n}(X) such that y′ ∘ x = x′ ∘ y.

This construction determines a functor from Simp(Gpd)_{∗} to the category of groupoids. Note that the group of automorphisms in p ∈ O of the groupoid π_{n}(X) is π_{n}(X(p)), which is the n-th homotopy group of the simplicial group of automorphisms in p, and, for n ≥ 1, π_{n}(X) is the disjoint union of the homotopy groups π_{n}(X(p)), p ∈ O.

Next we establish the notion of simplicial homotopy between morphisms in the category Simp(Gpd)_{∗}.

2.1. Definition. Let f, g : X → Y be two morphisms in Simp(Gpd)_{∗}. A simplicial homotopy from f to g, denoted by β : f ≃ g, consists of a map β : O → Y_{0} such that sβ = f and tβ = g together with a family of maps β_{n}^{j} : X_{n} → Y_{n}, 1 ≤ j ≤ n, satisfying the following relations:

a) d_{0}β_{n}^{1}(x) = g_{n−1}d_{0}(x) ∘ s_{0}^{n−1}βs(x) ; d_{n}β_{n}^{n}(x) = s_{0}^{n−1}βt(x) ∘ f_{n−1}d_{n}(x), ∀x ∈ X_{n}.

b) d_{i}β_{n}^{j} = \{ \begin{matrix}β_{n−1}^{j−1}d_{i} & i < j \\ β_{n−1}^{j}d_{i} & i ≥ j \end{matrix} ; s_{i}β_{n}^{j} = \{ \begin{matrix}β_{n+1}^{j+1}s_{i} & i < j \\ β_{n+1}^{j}s_{i} & i ≥ j \end{matrix} .

c) Given [figure: p; x; q; y; r] ∈ X_{n}, β_{n}^{j}(y ∘ x) = β_{n}^{j}(y) ∘ (s_{0}^{n}βt(x))^{−1} ∘ β_{n}^{j}(x).

2.2. Proposition. Let f, g : X → Y be two morphisms in Simp(Gpd)_{∗}. Then, giving a simplicial homotopy from f to g is equivalent to giving a map α : O → Y_{0} such that sα = f and tα = g, together with a family of maps α_{n}^{j} : X_{n} → Y_{n+1}, 0 ≤ j ≤ n, satisfying the following relations:

a) sα_{n}^{j} = fs ; tα_{n}^{j} = gt.

b) d_{0}α_{n}^{0}(x) = g_{n}(x) ∘ s_{0}^{n}αs(x) ; d_{n+1}α_{n}^{n}(x) = s_{0}^{n}αt(x) ∘ f_{n}(x), ∀x ∈ X_{n}.

c) d_{i}α_{n}^{j} = \{ \begin{matrix}α_{n−1}^{j−1}d_{i} & i < j \\ d_{i}α_{n}^{j+1} & i = j + 1 \\ α_{n−1}^{j}d_{i−1} & i > j + 1 \end{matrix} ; s_{i}α_{n−1}^{j} = \{ \begin{matrix}α_{n}^{j+1}s_{i} & i ≤ j \\ α_{n}^{j}s_{i−1} & i > j \end{matrix} .

d) Given [figure: p; x; q; y; r] ∈ X_{n}, α_{n}^{j}(y ∘ x) = α_{n}^{j}(y) ∘ (s_{0}^{n+1}αt(x))^{−1} ∘ α_{n}^{j}(x).

Proof. Let us suppose that α and α_{n}^{j} satisfy the above relations. Then, by putting β = α and, for each 1 ≤ j ≤ n, β_{n}^{j} = d_{j}α_{n}^{j−1} : X_{n} → Y_{n}, it is straightforward to see that β is a simplicial homotopy from f to g. Conversely, if β : f ≃ g is a simplicial homotopy, then α = β and α_{n}^{j} = β_{n+1}^{j+1}s_{j}, 0 ≤ j ≤ n, satisfy the above relations.

It is not very difficult to see that the functor \overline{W} preserves simplicial homotopies.

Now, given f, g : X → Y two morphisms in Simp(Gpd)_{∗} and β : f ≃ g a simplicial homotopy, there is, for each p ∈ O, a morphism β(p) : f(p) → g(p), and then we can consider the morphism g_{β} : X → Y defined by g_{β}(p) = f(p) and g_{β}(x) = s_{0}^{n}βt(x)^{−1} ∘ g(x) ∘ s_{0}^{n}βs(x), x ∈ X_{n}. The next proposition shows the relationship between the induced morphisms in the homotopy groupoids by two morphisms that are simplicially homotopic.

2.3. Proposition. Let f, g : X → Y be two morphisms in Simp(Gpd)_{∗}. If there exists a simplicial homotopy β : f ≃ g, then π_{n}(f) = π_{n}(g_{β}), n ≥ 0.

Proof. Let p ∈ O and consider the family of group morphisms β̄_{n}^{j}, 1 ≤ j ≤ n, where each β̄_{n}^{j} : X_{n}(p) → Y_{n}(f(p)) is given by β̄_{n}^{j} = s_{0}^{n}β(p)^{−1} ∘ β_{n}^{j}. It is straightforward to check that this family of morphisms determines a simplicial homotopy β̄ between the simplicial group morphisms f_{p}, (g_{β})_{p} : X(p) → Y (p). Thus, for each n ≥ 0, π_{n}(f_{p}) = π_{n}((g_{β})_{p}), and so it is clear that π_{n}(f) = π_{n}(g_{β}), for any n ≥ 1.

In addition, π_{0}(f) = π_{0}(g_{β}), since for any x : p → q ∈ X_{0} there exist y, y′ ∈ B_{0}(Y) such that y′ ∘f(x) = g_{β} ∘y. In fact, if we consider ȳ = s_{0}β(p)^{−1} ∘ s_{0}g(x)^{−1} ∘ β_{1}^{1}(s_{0}x) ∈ Y_{1}(f(p)) then d_{0}(ȳ) = Id_{f(p)} and d_{1}(ȳ) = β(p)^{−1} ∘ g(x)^{−1} ∘ β(q) ∘ f(x) and so the morphisms y = β(p)^{−1} ∘ g(x)^{−1} ∘ β(q) ∘ f(x) and y′ = Id_{f(q)} satisfy the required condition.

3. Path and cylinder constructions in simplicial groupoids

Let us start by recalling that in the closed model category Simp(Gp) of simplicial groups [16], given H_{•} ∈ Simp(Gp), the simplicial group H_{•}^{I}, whose n-simplices are:

(H_{•}^{I})_{n} = Hom_{Simp(Set)}(Δ[1] × Δ[n], H_{•}) ≅

≅ {(x_{0}, …, x_{n}) ∈ (H_{n+1})^{n+1} / d_{i}x_{i} = d_{i}x_{i−1}, 1 ≤ i ≤ n}

and the face and degeneracy operators are:

d_{i}(x_{0}, …, x_{n}) = (d_{i+1}x_{0}, …, d_{i+1}x_{i−1}, d_{i}x_{i+1}, …, d_{i}x_{n}), 0 ≤ i ≤ n,
s_{i}(x_{0}, …, x_{n}) = (s_{i+1}x_{0}, …, s_{i+1}x_{i}, s_{i}x_{i}, …, s_{i}x_{n}), 0 ≤ i ≤ n ,

is a path space for H_{•} since there is a factorization of the diagonal morphism

[figure: H_{•}; β_{•}; H_{•}^{I}; (∂_{0},∂_{1}); H_{•} × H_{•} ,]

where β_{•}, given by β_{n}(x) = (s_{0}x, …, s_{n}x), is a weak equivalence and (∂_{0}, ∂_{1}), the morphism induced by (∂_{0})_{n}(x_{0}, …, x_{n}) = d_{n+1}x_{n} and (∂_{1})_{n}(x_{0}, …, x_{n}) = d_{0}x_{0}, is a fibration (see [7], [8]).

3.1. Definition. Given H ∈ Simp(Gpd)_{∗}, consider the simplicial groupoid H^{I} whose set of objects is H_{0} and the set of morphisms in dimension n is the set of (n + 1)-uples of commutative squares:

(H^{I})_{n} = \{χ = ( [figure: p; q; p′; q′; s_{0}^{n+1}a; x_{0}; x′_{0}; s_{0}^{n+1}b], …, [figure: p; q; p′; q′; s_{0}^{n+1}a; x_{n}; x′_{n}; s_{0}^{n+1}b]) / \begin{matrix}a, b ∈ H_{0} & d_{i}x_{i} = d_{i}x_{i−1} \\ x_{i}, x′_{i} ∈ H_{n+1} & d_{i}x′_{i} = d_{i}x′_{i−1} \\ p, q ∈ O \end{matrix} \}

where, in each dimension, the source and target of the groupoid [figure: (H^{I})_{n}; s; t; H_{0}] are defined

by s(χ) = a and t(χ) = b, and the face and degeneracy operators are given, in each dimension, by:

d_{i}(χ) = ( [figure: p; q; p′; q′; s_{0}^{n}a; d_{i+1}x_{0}; d_{i+1}x′_{0}; s_{0}^{n}b], ..., [figure: p; q; p′; q′; s_{0}^{n}a; d_{i+1}x_{i−1}; d_{i+1}x′_{i−1}; s_{0}^{n}b], [figure: p; q; p′; q′; s_{0}^{n}a; d_{i}x_{i+1}; d_{i}x′_{i+1}; s_{0}^{n}b], ..., [figure: p; q; p′; q′; s_{0}^{n}a; d_{i}x_{n}; d_{i}x′_{n}; s_{0}^{n}b])

s_{j}(χ) = ( [figure: p; q; p′; q′; s_{0}^{n+2}a; s_{j+1}x_{0}; s_{j+1}x′_{0}; s_{0}^{n+2}b], ..., [figure: p; q; p′; q′; s_{0}^{n+2}a; s_{j+1}x_{j}; s_{j+1}x′_{j}; s_{0}^{n+2}b], [figure: p; q; p′; q′; s_{0}^{n+2}a; s_{j}x_{j}; s_{j}x′_{j}; s_{0}^{n+2}b], ..., [figure: p; q; p′; q′; s_{0}^{n+2}a; s_{j}x_{n}; s_{j}x′_{n}; s_{0}^{n+2}b]) .

The above definition determines a functor

(−)^{I} : Simp(Gpd)_{∗} ⟶ Simp(Gpd)_{∗}

and we actually have the following:

3.2. Proposition. The simplicial groupoid H^{I} is a path space for any H in Simp(Gpd)_{∗}.

Proof. We must prove that there exists a factorization of the diagonal morphism

[figure: H; H^{I}; H × H; β; (∂_{0},∂_{1}); Δ]

such that morphism β is a weak equivalence and (∂_{0}, ∂_{1}) is a fibration.

Let us define β : H → H^{I} as follows:

- On objects, β : O → H_{0} is given by β(p) = Id_{p} ∀p ∈ O.

- On morphisms, β : H_{n} → (H^{I})_{n} is given, for each x : p → q ∈ H_{n}, by

β(x) = ( [figure: p; p; q; q; s_{0}^{n+1}Id_{p}; s_{0}x; s_{0}x; s_{0}^{n+1}Id_{q}] , …, [figure: p; p; q; q; s_{0}^{n+1}Id_{p}; s_{n}x; s_{n}x; s_{0}^{n+1}Id_{q}] ) .

It is straightforward to check that β is a morphism of simplicial groupoids. To verify that it is a weak equivalence, we first show that it induces a bijection between the components of H and those of H^{I}. The injectivity is clear because, if β(p) and β(q) are in the same component of (H^{I})_{0}, then p y q are in the same component of H_{0}, since if the following square

[figure: p; p; q; q; x; x]

is a morphism connecting them in (H^{I})_{0}, then x : p → q is a morphism connecting p and q in H_{0}. To show the surjectivity it is enough to see that, given x : p → q with p ∈ O, then β(p) and x are in the same component in (H^{I})_{0}, but this is clear if we consider the following commutative square:

[figure: p; p; p; q .; s_{0}Id_{p}; s_{0}x; s_{0}x]

Second, we must show that the induced morphism β(p) : H(p) → H^{I}(β(p)) is a weak equivalence of simplicial groups for all p ∈ O and this is clear since we have:

(H^{I}(β(p)))_{n} = \{( [figure: p; p; p; p; s_{0}^{n+1}Id_{p}; x_{0}; x_{0}; s_{0}^{n+1}Id_{p}] , …, [figure: p; p; p; p; s_{0}^{n+1}Id_{p}; x_{n}; x_{n}; s_{0}^{n+1}Id_{p}] ) / \begin{matrix}d_{i}x_{i} = d_{i}x_{i−1} \\ x_{i} ∈ H_{n+1} \end{matrix} \} ≅

≅ \{(x_{0}, …, x_{n}) / \begin{matrix}d_{i}x_{i} = d_{i}x_{i−1} \\ x_{i} ∈ (H(p))_{n+1} \end{matrix} \} = (H(p))^{I} ,

and so morphism β(p) can be identified with the corresponding morphism in the construction of the path space in the category of simplicial groups (see the beginning of this section), and therefore it is a weak equivalence.

Next we define (∂_{0}, ∂_{1}) : H^{I} → H × H as follows:

- On objects, (∂_{0}, ∂_{1}) : H_{0} → O × O is given by (∂_{0}, ∂_{1})(a) = (s(a), t(a)), ∀a ∈ H_{0}.

- On morphisms, (∂_{0}, ∂_{1})_{n} : (H^{I})_{n} → H_{n} × H_{n} is given by

(∂_{0}, ∂_{1})_{n} ( [figure: p; q; p′; q′; s_{0}^{n+1}a; x_{0}; x′_{0}; s_{0}^{n+1}b], …, [figure: p; q; p′; q′; s_{0}^{n+1}a; x_{n}; x′_{n}; s_{0}^{n+1}b]) = (d_{n+1}x_{n}, d_{0}x′_{0}) .

The proof that (∂_{0}, ∂_{1}) is a morphism of simplicial groupoids is long and tedious but straightforward. For instance, in dimension 1, we have to prove that the following cube is commutative:

[figure: H_{0}; O × O; (H^{I})_{0}; H_{0} × H_{0}; H_{0}; O × O; (H^{I})_{1}; H_{1} × H_{1}; d_{0}; d_{1}; d_{0}; d_{1}; (∂_{0},∂_{1})_{0}; (∂_{0},∂_{1})_{1}; s; t]

and, for example, for the upper face and the morphism d_{0} this is true because:

(d_{0}(∂_{0}, ∂_{1})_{1}) ( [figure: p; q; p′; q′; s_{0}^{2}a; x_{0}; x′_{0}; s_{0}^{2}b], [figure: p; q; p′; q′; s_{0}^{2}a; x_{1}; x′_{1}; s_{0}^{2}b]) = d_{0}(d_{2}x_{1}, d_{0}x′_{0}) = (d_{0}d_{2}x_{1}, d_{0}d_{0}x′_{0})

((∂_{0}, ∂_{1})_{0}d_{0}) ( [figure: p; q; p′; q′; s_{0}^{2}a; x_{0}; x′_{0}; s_{0}^{2}b], [figure: p; q; p′; q′; s_{0}^{2}a; x_{1}; x′_{1}; s_{0}^{2}b]) = (∂_{0}, ∂_{1})_{0} ( [figure: p; q; p′; q′; d_{0}x_{1}; d_{0}x′_{1}] ) = (d_{1}d_{0}x_{1}, d_{0}d_{0}x′_{1})

and the simplicial identities and the relation d_{i}x_{i} = d_{i}x_{i−1} imply (d_{0}d_{2}x_{1}, d_{0}d_{0}x′_{0}) = (d_{1}d_{0}x_{1}, d_{0}d_{0}x′_{1}).

Finally, let us prove that the morphism (∂_{0}, ∂_{1}) is a fibration in Simp(Gpd)_{∗}. To do so, note first that the morphism of groupoids

[figure: (H^{I})_{0}; t; s; (∂_{0},∂_{1}); H_{0}; (∂_{0},∂_{1}); H_{0} × H_{0}; t; s; O × O]

is a fibration since, given a : p → q ∈ H_{0} and [figure: (p; x; p′, q; x′; q′)] ∈ H_{0} × H_{0}, the element of (H^{I})_{0} represented by the commutative square

[figure: p; s_{0}a; s_{0}x; q; s_{0}x′; p′; s_{0}(x′ax^{−1}); q′]

satisfies (∂_{0}, ∂_{1}) [figure: (; )] = (x, x′).

It remains to prove that, given a : p → q ∈ H_{0}, the morphism (∂_{0}, ∂_{1}) : H^{I}(a) → H(p) × H(q) is a fibration of simplicial groups. Now, since the objects p and q are in the same component of H_{0}, H(p) × H(q) ≅ H(p) × H(p). Moreover, it is easy to identify H^{I}(a) with H(p)^{I} because an element of (H^{I}(a))_{n} such as the following:

( [figure: p; q; p; q; s_{0}^{n+1}a; x_{0}; x′_{0}; s_{0}^{n+1}a] , [figure: p; q; p; q; s_{0}^{n+1}a; x_{1}; x′_{1}; s_{0}^{n+1}a] , ..., [figure: p; q; p; q; s_{0}^{n+1}a; x_{n}; x′_{n}; s_{0}^{n+1}a] )

determines (x_{0}, x_{1}, …, x_{n}) ∈ (H(p))_{n}^{I} and, conversely, given (x_{0}, x_{1}, …, x_{n}) ∈ (H(p))_{n}^{I}, the morphisms x′_{0}, …, x′_{n} are determined by composition. Thus, the morphism (∂_{0}, ∂_{1}) : H^{I}(a) → H(p) × H(q) can be identified with the morphism of simplicial groups H(p)^{I} → H(p) × H(p), which is a fibration since it is the corresponding morphism in the construction of the path space in the category Simp(Gp).

Finally, it is clear that (∂_{0}, ∂_{1})β = Δ.

Let us recall now that if G_{•} is a cofibrant (i.e., free) simplicial group, the simplicial

group G_{•} ⊗ I, whose n-simplices are (G_{•} ⊗ I)_{n} = ∐_{i=0}^{n+1}(G_{n})_{i}, where (G_{n})_{i} = G_{n}, 0 ≤ i ≤ n,

is a cylinder object for G_{•} in the closed model category Simp(Gp) since there is a factorization of the codiagonal morphism

[figure: G_{•} ∐ G_{•}; i_{0}+i_{1}; G_{•} ⊗ I; σ; G_{•}]

where σ, which is the morphism induced by the identities, is a weak equivalence, and the morphism i_{0} + i_{1}, which is induced by the first and last inclusions respectively, is a cofibration (see [7], [8]).

3.3. Definition. Given G ∈ Simp(Gpd)_{∗}, consider the simplicial groupoid G⊗I whose set of objects is O ∨ O = O × {0, 1} and the sets of morphisms are constructed as follows:

Given p ∈ O, we denote p_{0} = (p, 0) ∈ O × {0} and p_{1} = (p, 1) ∈ O × {1}. For any x ∈ G_{0}, x : p → q, we consider two morphisms, x^{0} : p_{0} → q_{0} and x^{1} : p_{0} → q_{0}, and the sets G_{0}^{0} = {x^{0} / x ∈ G_{0}} and G_{0}^{1} = {x^{1} / x ∈ G_{0}}. Also, for any p ∈ O we consider a morphism I_{p} : p_{0} → p_{1} and its inverse I_{p}^{−1} : p_{1} → p_{0}.

The set (G ⊗ I)_{0} of morphisms in dimension zero, denoted by G_{0}^{0} ∨ G_{0}^{1} ∨ O, consists of these three classes of morphisms and all the words α_{n} ∘ α_{n−1} ∘ … α_{0}, where each α_{i} is any of the morphisms described above, such that s(α_{i}) = t(α_{i−1}). All these morphisms are subject to the relations x^{0} ∘ y^{0} = (x ∘ y)^{0}, x^{1} ∘ y^{1} = (x ∘ y)^{1}, (Id_{p})^{0} = (Id_{p})^{1}.

Now, for any morphism x : p → q ∈ G_{1}, we consider three morphisms x^{00} : p_{0} → q_{0}, x^{01} : p_{0} → q_{0} and x^{11} : p_{0} → q_{0} and the sets G_{1}^{00} = {x^{00} / x ∈ G_{1}}, G_{1}^{01} = {x^{01} / x ∈ G_{1}} and G_{1}^{11} = {x^{11} / x ∈ G_{1}}. Also, for any p ∈ O, we consider the morphisms I_{p} : p_{0} → p_{1} and their inverses.

Then, the set (G⊗I)_{1} of morphisms in dimension one, denoted by G_{1}^{00} ∨G_{1}^{01} ∨G_{1}^{11} ∨O, consists of these four classes of morphisms and all the words α_{n} ∘α_{n−1} ∘… α_{0}, where each α_{i} is any of the morphisms described above, such that s(α_{i}) = t(α_{i−1}). All these morphisms are subject to the relations x^{00} ∘ y^{00} = (x ∘ y)^{00}, x^{01} ∘ y^{01} = (x ∘ y)^{01}, x^{11} ∘ y^{11} = (x ∘ y)^{11}, (Id_{p})^{00} = (Id_{p})^{01} = (Id_{p})^{11} = Id_{p_{0}}.

By iterating this process, we define (G ⊗ I)_{n} = ⋁_{τ∈(Δ[1])_{n}} G_{n}^{τ} ∨ O. Thus, given x ∈ G_{n} and τ ∈ (Δ[1])_{n}, we have x^{τ} ∈ G_{n}^{τ} ⊂ (G ⊗ I)_{n}.

Note that we have a groupoid

[figure: (G ⊗ I)_{0}; O ∨ O ,; Id; s; t]

where Id_{p_{0}} = (Id_{p})^{0} = (Id_{p})^{1}; Id_{p_{1}} = I_{p}I_{p}^{−1}; s(x^{0}) = s(x^{1}) = s(I_{p}) = p_{0};d s(I_{p}^{−1}) = p_{1}; t(x^{0}) = t(x^{1}) = t(I_{q}^{−1}) = q_{0} and t(I_{q}) = q_{1}. In general, we have, in each dimension n, a groupoid

[figure: (G ⊗ I)_{n}; O ∨ O .; Id; s; t]

Thus, we have constructed a simplicial groupoid where the face and degeneracy operators are defined by:

d_{i}(x^{τ}) = (d_{i}x)^{d_{i}τ}, x^{τ} ∈ G_{n}, 0 ≤ i ≤ n,

d_{i}(I_{p}) = I_{p} (in a smaller dimension),

s_{i}(x^{τ}) = (s_{i}x)^{s_{i}τ}, x^{τ} ∈ G_{n}, 0 ≤ i ≤ n − 1,

s_{i}(I_{p}) = I_{p} (in a higher dimension).

The above definition determines a functor

(−) ⊗ I : Simp(Gpd)_{∗} ⟶ Simp(Gpd)_{∗}

and we actually have the following:

3.4. Proposition. The simplicial groupoid G ⊗ I is a cylinder object for any cofibrant object G in Simp(Gpd)_{∗}.

Proof. We must prove that there exists a factorization of the codiagonal morphism

[figure: G ∐ G; G ⊗ I; G; i_{0}+i_{1}; σ; ∇]

such that i_{0} + i_{1} is a cofibration and σ is a weak equivalence.

We define σ : G ⊗ I → G by σ(p_{0}) = σ(p_{1}) = p, p ∈ O; σ(x^{τ}) = x, x^{τ} ∈ G_{n}^{τ}; σ(I_{p}) = Id_{p}, p ∈ O.

The morphisms i_{0}, i_{1} : G → G ⊗ I are defined by i_{0}(p) = p_{0}, p ∈ O; i_{0}(x) = x^{00...0}, x ∈ G_{n}; i_{1}(p) = p_{1}, p ∈ O; i_{1}(x) = I_{q}x^{1...11}I_{p}^{−1}, x ∈ G_{n}.

It is clear that σi_{0} = σi_{1} = 1_{G}. Moreover, if G is cofibrant, i_{0} + i_{1} is a free map, and thus a cofibration, since it is

clearly one-to-one on objects and maps and it is straightforward to see that, if V is a base of G, then W = {W_{n} / n ∈ ℕ} is a base of G ⊗ I where, with the same notation used in the construction of G ⊗ I, W_{n} = V_{n}^{00...01} ∪ V_{n}^{00...011} ∪ … ∪ V_{n}^{011...1} ∪ O.

To prove that σ is a weak equivalence it is enough to see that it is a homotopy equivalence (see [17], theorem 1.3). To do so, below we construct a right homotopy from i_{0}σ to 1_{G⊗I}. Note first that (i_{0}σ)(p_{0}) = p_{0}, (i_{0}σ)(p_{1}) = p_{1}, (i_{0}σ)(I_{p}) = (Id_{p})^{0}, p ∈ O; (i_{0}σ)(x^{τ}) = x^{00...0}, x^{τ} ∈ (G ⊗ I)_{n}.

The right homotopy is a morphism H : G ⊗ I → (G ⊗ I)^{I} such that ∂_{0}H = i_{0}σ, ∂_{1}H = 1_{G⊗I} and it is given as follows:

- On objects:

H(p_{0}) = ( p_{0} ^{Id} p_{0} ) H(p_{1}) = ( [figure: p_{0}; I_{p}; p_{1}] )

- In dimension 0:

H(x^{0}) = [figure: p_{0}; p_{0}; q_{0}; q_{0}; Id; (s_{0}x)^{00}; (s_{0}x)^{00}; Id] ; H(x^{1}) = [figure: p_{0}; p_{0}; q_{0}; q_{0}; Id; (s_{0}x)^{01}; (s_{0}x)^{01}; Id] ; H(I_{p}) = [figure: p_{0}; p_{0}; p_{0}; p_{1}; Id; Id; I_{p}; I_{p}]

- In dimension 1:

H(x^{00}) = ( [figure: p_{0}; p_{0}; q_{0}; q_{0}; (s_{0}x)^{000}; (s_{0}x)^{000}] , [figure: p_{0}; p_{0}; q_{0}; q_{0}; (s_{1}x)^{000}; (s_{1}x)^{000}] )

H(x^{01}) = ( [figure: p_{0}; p_{0}; q_{0}; q_{0}; (s_{0}x)^{001}; (s_{0}x)^{001}] , [figure: p_{0}; p_{0}; q_{0}; q_{0}; (s_{1}x)^{001}; (s_{1}x)^{001}] )

H(x^{11}) = ( [figure: p_{0}; p_{0}; q_{0}; q_{0}; (s_{0}x)^{011}; (s_{0}x)^{011}] , [figure: p_{0}; p_{0}; q_{0}; q_{0}; (s_{1}x)^{001}; (s_{1}x)^{001}] )

H(I_{p}) = ( [figure: p_{0}; p_{1}; p_{0}; p_{1}; I_{p}; Id; Id; I_{p}] , [figure: p_{0}; p_{1}; p_{0}; p_{1}; I_{p}; Id; Id; I_{p}] )

- In general, for any element x^{00 ...^{(n−j+1}011...^{(j}1} ∈ (G ⊗ I)_{n}, 0 ≤ j ≤ n + 1:

H(x^{00 ...^{(n−j+1}011...^{(j}1}) = (d_{0}, d_{1}, …, d_{k}, …, d_{n})

where each d_{k} is a diagram of the form:

[figure: p_{0}; p_{0}; q_{0}; q_{0}; (s_{k}x)^{00{(n−j+2 \atop ...}011{(j \atop ...}1}; (s_{k}x)^{00{(n−j+2 \atop ...}011{(j \atop ...}1}] if 0 ≤ k ≤ n − j + 1,

[figure: p_{0}; p_{0}; q_{0}; q_{0}; (s_{k}x)^{00{(k+1 \atop ...}011{(n−k+1 \atop ...}1}; (s_{k}x)^{00{(k+1 \atop ...}011{(n−k+1 \atop ...}1}] if n − j + 2 ≤ k ≤ n + 1.

Finally, it is straightforward to see that ∂_{0}H = i_{0}σ y ∂_{1}H = 1_{G⊗I}.

4. A simplicial description of Ho(Simp(Gpd)_{∗})

The construction of path space in Simp(Gpd)_{∗} given in Section 3, allows us to formulate the right homotopy relation between two morphisms in terms of the simplicial homotopy relations given in Proposition 2.2. In fact, if f, g : G → H are two morphisms in Simp(Gpd)_{∗}, a right homotopy from f to g is a morphism α : G → H^{I} such that ∂_{0}α = f and ∂_{1}α = g and this means to give:

- A map α : O → H_{0} such that (∂_{0}, ∂_{1})α = (f, g) and, since (∂_{0}, ∂_{1})α = (sα, tα), the map α must satisfy that sα = f and tα = g.

- A map α_{0} : G_{0} → H_{0}^{I} such that (∂_{0}, ∂_{1})α_{0} = (f_{0}, g_{0}) and the following diagram is commutative:

[figure: G_{0}; t; s; α_{0}; O; α; H_{0}^{I}; t; s; H_{0} .]

Now, if x : p → q ∈ G_{0}, α_{0}(x) : α(p) → α(q) ∈ H_{0}^{I} is a commutative square of the form

[figure: f(p); x_{0}; s_{0}α(p); g(p); x′_{0}; f(q); s_{0}α(q); g(q)] with d_{1}x_{0} = f_{0}(x) and d_{0}x′_{0} = g_{0}(x),

and so, giving α_{0} in these conditions is equivalent to giving a map

[figure: α_{0}^{0} : G_{0}; H_{1}; x; s_{0}αt(x) ∘ x_{0}]

such that: \{ \begin{matrix}sα_{0}^{0}(x) = fs(x), tα_{0}^{0}(x) = gt(x), \\ d_{1}α_{0}^{0}(x) = αt(x) ∘ f_{0}(x), d_{0}α_{0}^{0}(x) = g_{0}(x) ∘ αs(x) \end{matrix}

(note that d_{0}α_{0}^{0}(x) = αt(x) ∘ d_{0}x_{0} = αt(x) ∘ (αt(x))^{−1} ∘ d_{0}x′_{0} ∘ αs(x) = g_{0}(x) ∘ αs(x)).

The condition that the pair (α_{0}, α) is a morphism of groupoids is clearly equivalent to α_{0}^{0} satisfying the relation α_{0}^{0}(y ∘ x) = α_{0}^{0}(y) ∘ (s_{0}αt(x))^{−1} ∘ α_{0}^{0}(x), for any morphisms in G_{0}, [figure: p; x; q; y; r].

- A map α_{1} : G_{1} → (H^{I})_{1} such that (∂_{0}, ∂_{1})α_{1} = (f_{1}, g_{1}) and the following diagrams are commutative:

[figure: G_{1}; O; (H^{I})_{1}; H_{0}; t; s; t; s; α_{1}; α] , [figure: G_{1}; G_{0}; (H^{I})_{1}; (H^{I})_{0} .; s_{0}; d_{1}; d_{0}; s_{0}; d_{1}; d_{0}; α_{1}; α_{0}]

Now, if x : p → q ∈ G_{1}, α_{1}(x) : α(p) → α(q) ∈ (H^{I})_{1} and so it is of the form

( [figure: f(p); g(p); f(q); g(q); s_{0}^{2}α(p); x_{0}; x′_{0}; s_{0}^{2}α(q)] , [figure: f(p); g(p); f(q); g(q); s_{0}^{2}α(p); x_{1}; x′_{1}; s_{0}^{2}α(q)] / \begin{matrix}d_{1}x_{0} = d_{1}x_{1} \\ d_{1}x′_{0} = d_{1}x′_{1} \end{matrix} )

with d_{2}x_{1} = f_{1}(x), d_{0}x′_{0} = g_{1}(x), α_{0}(d_{0}x) = ( [figure: f(p); d_{0}x_{1}; s_{0}α(p); g(p); d_{0}x′_{1}; f(q); s_{0}α(q); g(q)] ) and

α_{0}(d_{1}x) = ( [figure: f(p); d_{2}x_{0}; s_{0}α(p); g(p); d_{2}x′_{0}; f(q); s_{0}α(q); g(q)] ).

Thus, giving α_{1} is equivalent to giving two maps [figure: α_{1}^{0}, α_{1}^{1} : G_{1}; H_{2}; x; x; s_{0}^{2}αt(x) ∘ x_{0}; s_{0}^{2}αt(x) ∘ x_{1}]

such that: \{ \begin{matrix}sα_{1}^{0}(x) = fs(x), tα_{1}^{0}(x) = gt(x), sα_{1}^{1}(x) = fs(x), tα_{1}^{1}(x) = gt(x) , \\ d_{0}α_{1}^{0}(x) = s_{0}αt(x) ∘ d_{0}x_{0} = d_{0}x′_{0} ∘ s_{0}αs(x) = g_{1}(x) ∘ s_{0}αs(x) , \\ d_{1}α_{1}^{0}(x) = s_{0}αt(x) ∘ d_{1}x_{0} = s_{0}αt(x) ∘ d_{1}x_{1} = d_{1}α_{1}^{1}(x) , \\ d_{2}α_{1}^{0}(x) = s_{0}αt(x) ∘ d_{2}x_{0} = α_{0}^{0}d_{1}(x) , \\ d_{0}α_{1}^{1}(x) = s_{0}αt(x) ∘ d_{0}x_{1} = α_{0}^{0}d_{0}(x) , \\ d_{2}α_{1}^{1}(x) = s_{0}αt(x) ∘ d_{2}x_{1} = s_{0}αt(x) ∘ f_{1}(x) . \end{matrix}

Moreover, if x : p → q ∈ G_{0}, then

s_{0}α_{0}(x) = s_{0} ( [figure: f(p); g(p); f(q); g(q); s_{0}α(p); x_{0}; x′_{0}; s_{0}α(q)] ) = ( [figure: f(p); g(p); f(q); g(q); s_{0}^{2}α(p); s_{1}x_{0}; s_{1}x′_{0}; s_{0}^{2}α(q)] , [figure: f(p); g(p); f(q); g(q); s_{0}^{2}α(p); s_{0}x_{0}; s_{0}x′_{0}; s_{0}^{2}α(q)] ) = α_{1}s_{0}(x)

and so s_{0}α_{0}^{0}(x) = s_{0}^{2}αt(x) ∘ s_{0}x_{0} = α_{1}^{1}s_{0}(x) and s_{1}α_{0}^{0}(x) = s_{1}s_{0}αt(x) ∘ s_{1}x_{0} = s_{0}^{2}αt(x) ∘ s_{1}x_{0} = α_{1}^{0}s_{0}(x).

Given [figure: p; x; q; y; r] ∈ G_{1}, the condition that (α_{1}, α) is a morphism of groupoids is equivalent to α_{1}^{0} and α_{1}^{1} satisfying the relations α_{1}^{0}(y ∘ x) = α_{1}^{0}(y) ∘ (s_{0}^{2}αt(x))^{−1} ∘ α_{1}^{0}(x) and α_{1}^{1}(y ∘ x) = α_{1}^{1}(y) ∘ (s_{0}^{2}αt(x))^{−1} ∘ α_{1}^{1}(x).

By iterating these calculations and recalling the characterization given in Proposition 2.2, we can state the following:

4.1. Proposition. Let G, H ∈ Simp(Gpd)_{∗} and f, g : G → H two morphisms. Then, giving a morphism α : G → H^{I} such that (∂_{0}, ∂_{1})α = (f, g) is equivalent to giving a simplicial homotopy α : f ≃ g.

In the same way as above, the construction of cylinder object in Simp(Gpd)_{∗} given in Section 3, allows us to formulate the left homotopy relation between two morphisms in terms of the simplicial homotopy relations given in Definition 2.1. In fact, if f, g : G → H are two morphisms in Simp(Gpd)_{∗}, giving a morphism β : G ⊗ I → H such that βi_{0} = f and βi_{1} = g is equivalent to giving the following data:

- A map β̄ : O ∨ O → O′ such that, for any p ∈ O, β̄(p_{0}) = f(p) and β̄(p_{1}) = g(p); thus, β̄ is completely determined by the maps f, g : O → O′.

- A map β_{0} : (G ⊗ I)_{0} → H_{0} such that β_{0}(i_{0} + i_{1}) = f_{0} + g_{0} and the following diagram

[figure: (G ⊗ I)_{0}; t; s; β_{0}; O ∨ O; β̄; H_{0}; t; s; O′]

is commutative.

Now, given x : p → q ∈ G_{0}, i_{0}(x) = x^{0} and β_{0}(x^{0}) = f_{0}(x) : f(p) → f(q), then β_{0} is determined on G_{0}^{0} by f_{0}. If p ∈ O, β_{0}(I_{p}) is a morphism in H_{0} such that sβ_{0}(I_{p}) = f(p) and tβ_{0}(I_{p}) = g(p). Thus, giving β_{0} on the morphisms I_{p} is equivalent to giving a map β : O → H_{0} such that sβ = f y tβ = g. This map determines β_{0} if we define β_{0}(x^{0}) = f_{0}(x); β_{0}(I_{p}) = β(p); β_{0}(x^{1}) = βs(x)^{−1} ∘ g_{0}(x) ∘ βs(x).

- A map β_{1} : (G ⊗ I)_{1} → H_{1} such that β_{1}(i_{0} + i_{1}) = f_{1} + g_{1} and the following diagrams are commutative:

[figure: (G ⊗ I)_{1}; O ∨ O; H_{1}; O′; t; s; t; s; β_{1}; β̄] , [figure: (G ⊗ I)_{1}; (G ⊗ I)_{0}; H_{1}; H_{0}; s_{0}; d_{1}; d_{0}; s_{0}; d_{1}; d_{0}; β_{1}; β_{0}]

Now, if p ∈ O, it is clear that β_{1}(I_{p}) = s_{0}β(p) and, given x : p → q ∈ G_{1}, β_{1}i_{0}(x) = β_{1}(x^{00}) = f_{1}(x) : f(p) → f(q) and β_{1}i_{1}(x) = s_{0}βt(x)∘β_{1}(x^{11})s_{0}∘βs(x)^{−1} = g_{1}(x) : g(p) → g(p). Thus, β_{1} is determined by giving β_{1}(x^{01}) : f(p) → f(q), satisfying that d_{0}β_{1}(x^{01}) = β_{0}(d_{0}(x^{01})) = β_{0}((d_{0}x)^{1}) = βt(x)^{−1} ∘ g_{0}d_{0}(x) ∘ βs(x) and d_{1}β_{1}(x^{01}) = β_{0}(d_{1}(x^{01})) = β_{0}((d_{1}x)^{0}) = f_{0}d_{1}(x), which is is equivalent to giving a map:

[figure: β_{1}^{1} : G_{1}; H_{1}; x; s_{0}βt(x) ∘ β_{1}(x^{01}) : f(p) → g(q)]

satisfying that \{ \begin{matrix}d_{0}β_{1}^{1}(x) = βt(x) ∘ βt(x)^{−1}(x) ∘ g_{0}d_{0}(x) ∘ βs(x) = g_{0}(x) ∘ d_{0}βs(x) \\ d_{1}β_{1}^{1}(x) = βt(x) ∘ f_{0}d_{1}(x) . \end{matrix}

Given [figure: p; x; q; y; r] ∈ G_{1}, the fact that (β_{1}, β̄) is a morphism of groupoids is equivalent to β_{1}^{1} satisfying the following relation:

β_{1}^{1}(y ∘ x) = s_{0}βt(y) ∘ β_{1}((y ∘ x)^{01}) = s_{0}βt(y) ∘ β_{1}(y^{01}x^{01}) = s_{0}βt(y) ∘ β_{1}(y^{01}) ∘ β_{1}(x^{01}) =

= s_{0}βt(y) ∘ β_{1}(y^{01}) ∘ s_{0}βt(x)^{−1} ∘ s_{0}βt(x) ∘ β_{1}(x^{01}) = β_{1}^{1}(y) ∘ (s_{0}βt(x))^{−1} ∘ β_{1}^{1}(x).

- A map β_{2} : (G ⊗ I)_{2} → H_{2} such that β_{2}(i_{0} + i_{1}) = f_{2} + g_{2} and the following diagrams are commutative:

[figure: (G ⊗ I)_{2}; O ∨ O; H_{2}; O′; t; s; t; s; β_{2}; β̄] , [figure: (G ⊗ I)_{2}; (G ⊗ I)_{1}; H_{2}; H_{1} .; s_{1}; s_{0}; d_{0}; d_{2}; s_{1}; s_{0}; d_{0}; d_{2}; β_{1}; β_{0}]

Now, given p ∈ O, β_{2}(I_{p}) = s_{0}^{2}β(p) and, if x : p → q ∈ G_{2}, β_{2}i_{0}(x) = β_{2}(x^{000}) = f_{2}(x) and β_{2}i_{1}(x) = β_{2}(I_{q}x^{111}I_{p}^{−1}) = s_{0}^{2}βt(x) ∘ β_{2}(x^{111}) ∘ s_{0}^{2}βs(x)^{−1} = g_{2}(x). Thus, β_{2} is determined by β_{2}(x^{001}) and β_{2}(x^{011}) satisfying that:

d_{0}β_{2}(x^{001}) = β_{1}d_{0}(x^{001}) = β_{1}((d_{0}x)^{01}) = s_{0}βt(x)^{−1} ∘ β_{1}^{1}d_{0}(x),

d_{1}β_{2}(x^{001}) = β_{1}d_{1}(x^{001}) = β_{1}((d_{1}x)^{01}) = s_{0}βt(x)^{−1} ∘ β_{1}^{1}d_{1}(x),

d_{2}β_{2}(x^{001}) = β_{1}d_{2}(x^{001}) = β_{1}((d_{2}x)^{00}) = f_{1}d_{2}(x) ∘ d_{0}β_{2}(x^{011})

Theory and Applications of Categories, Vol. 7, No. 14 279

= β_{1}d_{0}(x^{011}) = β_{1}((d_{0}x)^{11}) = s_{0}βt(x)^{−1} ∘ g_{1}d_{0}(x) ∘ s_{0}βs(x),

d_{1}β_{2}(x^{011}) = β_{1}d_{1}(x^{011}) = β_{1}((d_{1}x)^{01}) = s_{0}βt(x)^{−1} ∘ β_{1}^{1}d_{1}(x), d_{2}β_{2}(x^{011}) = β_{1}d_{2}(x^{011}) = β_{1}((d_{2}x)^{01}) = s_{0}βt(x)^{−1} ∘ β_{1}^{1}d_{2}(x),

which is equivalent to giving two maps:

[figure: β_{2}^{1} : G_{2}; H_{2}; x; s_{0}^{2}βt(x) ∘ β_{2}(x^{001}); and; β_{2}^{2} : G_{2}; H_{2}; x; s_{0}^{2}βt(x) ∘ β_{2}(x^{011})]

satisfying the following relations: d_{0}β_{2}^{1}(x) = d_{0}s_{0}^{2}βt(x) ∘ d_{0}β_{2}(x^{011}) = s_{0}βt(x) ∘ s_{0}βt(x)^{−1} ∘ g_{1}d_{0}(x) ∘ s_{0}βs(x) =

= g_{1}d_{0}(x) ∘ s_{0}βs(x), d_{1}β_{2}^{1}(x) = d_{1}s_{0}^{2}βt(x) ∘ d_{1}β_{2}(x^{011}) = s_{0}βt(x) ∘ s_{0}βt(x)^{−1} ∘ β_{1}^{1}d_{0}(x) = β_{1}^{1}d_{0}(x), d_{2}β_{2}^{1}(x) = d_{2}s_{0}^{2}βt(x) ∘ d_{2}β_{2}(x^{011}) = s_{0}d_{1}s_{0}βt(x) ∘ f_{1}d_{2}(x) = s_{0}βt(x) ∘ f_{1}d_{2}(x), d_{0}β_{2}^{2}(x) = d_{0}s_{0}^{2}βt(x) ∘ d_{0}β_{2}(x^{001}) = s_{0}βt(x)s_{0}βt(x)^{−1} ∘ β_{1}^{1}d_{0}(x) = β_{1}^{1}d_{0}(x), d_{1}β_{2}^{2}(x) = d_{1}s_{0}^{2}βt(x) ∘ d_{1}β_{2}(x^{001}) = s_{0}βt(x) ∘ s_{0}βt(x)^{−1} ∘ β_{1}^{1}d_{1}(x) = β_{1}^{1}d_{1}(x), d_{2}β_{2}^{2}(x) = d_{2}s_{0}^{2}βt(x) ∘ d_{2}β_{2}(x^{001}) = s_{0}d_{1}s_{0}βt(x) ∘ f_{1}d_{2}(x) = s_{0}βt(x)f_{1}d_{2}(x). Moreover, if x : p → q ∈ G_{1}, then d_{0}β_{2}^{2}s_{0}(x) = β_{1}^{1}(x) and so β_{2}^{2}s_{0}(x) = s_{0}β_{1}^{1}(x), and

d_{1}β_{2}^{1}s_{0}(x) = β_{1}^{1}(x) and so β_{2}^{1}s_{0}(x) = s_{1}β_{1}^{1}(x).

For any [figure: p; x; q; y; r] ∈ G_{2}, the fact that (β_{2}, β̄) is a morphism of groupoids is equivalent to β_{2}^{1} and β_{2}^{2} satisfying following relations: β_{2}^{1}(y ∘ x) = s_{0}^{2}βt(y) ∘ β_{2}((y ∘ x)^{011}) = s_{0}^{2}βt(y) ∘ β_{2}(y^{011}x^{011}) = s_{0}^{2}βt(y) ∘ β_{2}(y^{011}) ∘ β_{2}(x^{011})

= s_{0}^{2}βt(y) ∘ β_{2}(y^{011})(s_{0}^{2}βt(x))^{−1} ∘ s_{0}^{2}βt(x) ∘ β_{2}(x^{011}) = β_{2}^{1}(y) ∘ (s_{0}^{2}βt(x))^{−1} ∘ β_{2}^{1}(x), β_{2}^{2}(y ∘x) = s_{0}^{2}βt(y)∘β_{2}((y ∘x)^{001}) = s_{0}^{2}βt(y)∘β_{2}(y^{001}x^{001}) = s_{0}^{2}βt(y)∘β_{2}(y^{001})∘β_{2}(x^{001}) =

= s_{0}^{2}βt(y) ∘ β_{2}(y^{001}) ∘ (s_{0}^{2}βt(x))^{−1} ∘ s_{0}^{2}βt(x) ∘ β_{2}(x^{001}) =β_{2}^{2}(y) ∘ (s_{0}^{2}βt(x))^{−1} ∘ β_{2}^{2}(x). By iterating these calculations and recalling the definition of homotopy given in 2.1,

we can state that: 4.2. Proposition. Let f, g : G → H be two morphisms in Simp(Gpd)_{∗}. Then, giving a morphism β : G ⊗ I → H such that β(i_{0} + i_{1}) = f + g is equivalent to giving a simplicial homotopy β : f ≃ g.

Now, as a consequence of Propositions 4.1, 4.2 and 2.2, we have the following: 4.3. Corollary. The functor (−) ⊗ I : Simp(Gpd)_{∗} → Simp(Gpd)_{∗} is left adjoint to the functor (−)^{I} : Simp(Gpd)_{∗} → Simp(Gpd)_{∗}.

Lastly we have:

4.4. Theorem. The homotopy category Ho(Simp(Gpd)_{∗}) is equivalent to the category whose objects are the cofibrant (i.e. free) simplicial groupoids and whose morphisms are simplicial homotopy classes of simplicial groupoid morphisms.

Proof. If G is a cofibrant simplicial groupoid and H is any simplicial groupoid, Propositions 3.4 and 4.2 assure that the left homotopy relation in Hom_{Simp(Gpd)_{∗}}(G, H) coincides with the simplicial homotopy relation but this one coincides with the right homotopy relation according to Propositions 3.2 and 4.1. Since Simp(Gpd)_{∗} is a closed model category we have then that Hom_{Ho(Simp(Gpd)_{∗})}(G, H) = {left ≡ right homotopy classes of morphisms from G to H} = [G, H] the set of simplicial homotopy classes of simplicial morphisms from G to H.

5. Homotopy constructions in the pointed case

Let us denote by (Simp(Gpd)_{∗}, p) the category of pointed simplicial groups. This is the category whose objects are pairs (X, p), with X ∈ Simp(Gpd)_{∗} and p ∈ O a fixed object of X, and given (X, p), (Y, p′)∈ (Simp(Gpd)_{∗}, p), the morphisms f : (X, p) → (Y, p′) are morphisms f : X → Y ∈ Simp(Gpd)_{∗} such that f(p) = p′.

If ∗ denotes the zero object in (Simp(Gpd)_{∗}, p), it is clear that this category is isomorphic to the category (∗, Simp(Gpd)_{∗}) of objects under ∗. Thus, (Simp(Gpd)_{∗}, p) inherits a closed model structure from that of Simp(Gpd)_{∗} (see [16]) where f : (X, p) → (Y, p′) is a fibration (respectively cofibration or weak equivalence) if f : X → Y is a fibration (resp. cofibration or weak equivalence) in Simp(Gpd)_{∗}.

Below we see that path space and cylinder constructions in (Simp(Gpd)_{∗}, p) can be done by using the constructions of these objects in Simp(Gpd)_{∗} which we have shown in Section 3. This allows us to define loop and suspension functors in (Simp(Gpd)_{∗}, p).

5.1. Proposition. The pointed simplicial groupoid (H^{I}, Id_{p}) is a path space for any (H, p) in (Simp(Gpd)_{∗}, p).

Proof. We have the following factorization of the diagonal morphism

[figure: (H, p); β; (H^{I}, Id_{p}); (∂_{0},∂_{1}); (H × H, (p, p)) ,]

where the morphisms β and (∂_{0}, ∂_{1}), defined as in Proposition 3.2, are clearly a weak equivalence and a fibration in (Simp(Gpd)_{∗}, p) respectively.

Now we can consider the loop functor

\overline{Ω} : (Simp(Gpd)_{∗}, p) → (Simp(Gpd)_{∗}, p)

defined by [figure: \overline{Ω}((H, p)) = Ker((H^{I}, Id_{p}); (∂_{0},∂_{1}); (H × H, (p, p)).] This functor induces

the corresponding one in the homotopy category H_{0}((Simp(Gpd)_{∗}, p)).

Now, let us consider the groupoid interval I, which is the groupoid with only two objects 0 and 1 and one morphism between them. We also denote by I the simplicial groupoid constant I in any dimension, and let us consider, for any G ∈ Simp(Gpd)_{∗} and p ∈ O, the morphism of simplicial groupoids u : I → G ⊗ I defined, in each dimension, by: u(0) = p_{0}, u(1) = p_{1}, u(0 → 1) = I_{p}. This morphism induces another one in (Simp(Gpd)_{∗}, p), u : (I, 0) → (G ⊗ I, p_{0}), and we make the following:

5.2. Definition. Given (G, p) ∈ (Simp(Gpd)_{∗}, p), we define (\overline{G ⊗ I}, p_{0}) by means of the following pushout diagram:

[figure: (I, 0); u; f; (G ⊗ I, p_{0}); γ; ∗; i; (\overline{G ⊗ I}, p_{0})]

5.3. Proposition. For any cofibrant object G ∈ Simp(Gpd)_{∗} and any p ∈ O, the pointed simplicial groupoid (\overline{G ⊗ I}, p_{0}) is a cylinder object for (G, p).

Proof. Let us consider the morphisms σ : G ⊗ I → G and i_{0}, i_{1} : G → G ⊗ I defined in Proposition 3.4 and the morphisms they induce in (Simp(Gpd)_{∗}, p), σ : (G ⊗ I, p_{0}) → (G, p) and i_{0} : (G, p) → (G ⊗ I, p_{0}).

The morphisms σ and 0 : ∗ → (G, p) induce a morphism σ′ : (\overline{G ⊗ O}, p_{0}) → (G, p) according to the following commutative diagram:

[figure: (I, 0); u; 0; (G ⊗ I, p_{0}); γ; σ; ∗; 0; (\overline{G ⊗ I}, p_{0}); σ′; (G, p) .]

Moreover, we consider the morphism i′_{0} : (G, p) → (\overline{G ⊗ I}, p_{0}) given as the composition i′_{0} = γi_{0}, so that, in each dimension and for any q ∈ O, we have i′_{0}(q) = γ(i_{0}(q)) = γ(q_{0}) = q_{0}, and given x : q → r a morphism in G, we have i′_{0}(x) = γi_{0}(x) = γ(x^{00...0}) = x^{00...0}.

Now, considering the morphism γi_{1} : G → \overline{G ⊗ I}, we have the following induced morphism in (Simp(Gpd)_{∗}, p), i′_{1} : (G, p) → (\overline{G ⊗ I}, p_{0}). In each dimension, and for any q ∈ O, we have i′_{1}(q) = q_{0} and, given x : q → r a morphism in G,then i′_{1}(x) = I_{q}x^{11...1}I_{r}^{−1}.

Then we have the following factorization of the codiagonal morphism

[figure: (G, p) ∐(G, p); i′_{0}+i′_{1}; (\overline{G ⊗ I}, p_{0}); σ′; (G, p)]

since:

σ′i′_{0}(q) = σ′(q_{0}) = σ(q_{0}) = q, σ′i′_{0}(x) = σ′(x^{00...0}) = σ(x^{00...0}) = x, σ′i′_{1}(q) = σ′(q_{0}) = σ(q_{0}) = q, σ′i′_{1}(x) = σ′(I_{q}x^{11...1}I_{r}^{−1}) = σ(I_{q}x^{11...1}I_{r}^{−1}) = x. The proof that i′_{0} + i′_{1} is a cofibration is exactly similar to the proof of this fact given

in Proposition 3.4. Also, σ′ is a homotopy equivalence and therefore a weak equivalence. The homotopy H′ : (\overline{G ⊗ I}, p_{0}) → ((\overline{G ⊗ I})^{I}, Id_{p_{0}}) is defined similarly to the homotopy H : G ⊗ I → (G ⊗ I)^{I} given in the proof of Proposition 3.4 since we can identify the set of objects of the pointed simplicial groupoid ((\overline{G ⊗ I})^{I}, Id_{p_{0}}) with the set \{G_{0}^{0} ∨ G_{0}^{1} ∨ O/p_{0} = p_{1}; I_{p} = Id_{p_{0}} = Id_{p_{1}}\}, (see Definition 3.3) and, moreover, ((\overline{G ⊗ I})^{I})_{n} can be identified with the set

\{( [figure: q; r; q′; r′; s_{0}^{n+1}a; x_{0}; x′_{0}; s_{0}^{n+1}b], …, [figure: q; r; q′; r′; s_{0}^{n+1}a; x_{n}; x′_{n}; s_{0}^{n+1}b]) / \begin{matrix}a, b ∈ (\overline{G ⊗ I})_{0} & d_{i}x_{i} = d_{i}x_{i−1} \\ x_{i}, x′_{i} ∈ (\overline{G ⊗ I})_{n+1} & d_{i}x′_{i} = d_{i}x′_{i−1} \\ q, r ∈ O \end{matrix} \} .

The above construction allows us to define a suspension functor

\overline{Σ} : (Simp(Gpd)_{∗}, p) → (Simp(Gpd)_{∗}, p)

given by

[figure: \overline{Σ}((G, p)) = Coker((G, p) ∐(G, p); i′_{0}+i′_{1}; (\overline{G ⊗ I}, p_{0})) .]

This functor is left adjoint to the loop functor \overline{Ω} in (Simp(Gpd)_{∗}, p) and it induces the corresponding suspension functor in the homotopy category H_{0}((Simp(Gpd)_{∗}, p)).

References

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Departamento de Álgebra, Universidad de Granada 18071 Granada, Spain

Email: agarzon@ugr.es and jesusgm@ugr.es

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