Theory and Applications of Categories, Vol. 4, No. 4, 1998, pp. 73–81.

SIMPLICIAL AND CATEGORICAL DIAGRAMS, AND THEIR EQUIVARIANT APPLICATIONS

RUDOLF FRITSCH AND MAREK GOLASIŃSKI

Transmitted by Michael Barr

ABSTRACT. We show that the homotopy category of simplicial diagrams 𝕀-𝕊𝕊 indexed by a small category 𝕀 is equivalent to a homotopy category of 𝕊𝕊 ↓ N𝕀 simplicial sets over the nerve N𝕀. Then their equivalences, by means of the nerve functor N : ℂat → 𝕊𝕊 from the category ℂat of small categories, with respective homotopy categories associated to ℂat are established. Consequently, an equivariant simplicial version of the Whitehead Theorem is derived.

In his remarkable paper [14], Thomason shows the equivalence of the homotopy categories of ℂat, the category of small categories, and 𝕊𝕊, the category of simplicial sets, by means of the nerve functor N : ℂat → 𝕊𝕊 and one of its homotopy inverses (see [7, 9] for details). By [11], the homotopy structure on ℂat induces, for every small category 𝕀, homotopy structures on the category ℂat ↓ 𝕀 of small categories over 𝕀 and the category 𝕀-ℂat of contravariant functors from 𝕀 to ℂat. From [10], it follows that there is a pair of adjoint functors D : ℂat ↓ 𝕀 → 𝕀-ℂat and 𝕀 ∫ - : 𝕀-ℂat → ℂat ↓ 𝕀 which establishes an equivalence of respective homotopy categories. Similarly, by [5] the homotopy category of simplicial sets on which a fixed simplicial group G acts is equivalent to the homotopy category of simplicial sets over the classifying complex \overline{W} G. From this it follows the wellknown fact that the homotopy category of topological spaces on which a fixed discrete group G acts is equivalent to the homotopy category of spaces over the classifying space K(G, 1).

We were influenced by these papers to search for a link between the comma category 𝕊𝕊 ↓ N𝕀 and the category 𝕀-𝕊𝕊 of contravariant functors from 𝕀 to 𝕊𝕊. In Section 1 we define, by means of [3, p.327], a pair of adjoint functors A : 𝕊𝕊 ↓ N𝕀 → 𝕀-𝕊𝕊 and B : 𝕀-𝕊𝕊 → 𝕊𝕊 ↓ N𝕀, and examine in Proposition 1.4 their homotopy properties. Let \overline{N} : ℂat ↓ 𝕀 → 𝕊𝕊 ↓ N𝕀 and Ñ : 𝕀-ℂat → 𝕀-𝕊𝕊 be the associated functors to the nerve one N : ℂat → 𝕊𝕊. Then by Theorem 1.5, from the diagram of functors

[figure: ℂat ↓ 𝕀; \overline{N}; D; 𝕀-ℂat; 𝕀 ∫ -; _{N}; 𝕊𝕊 ↓ N𝕀; A; 𝕀-𝕊𝕊; B]

Received by the editors 1998 April 24.

Published on 1998 July 1.

1991 Mathematics Subject Classification: Primary 55P15, 55U10; secondary 18G30, 55P91.

Key words and phrases: comma category, Grothendieck construction, homotopy colimit, pull-back, simplicial set, small category.

\begin{matrix}c \\ ◯ \end{matrix} Rudolf Fritsch and Marek Golasiński 1998. Permission to copy for private use granted.

we derive equivalences of respective homotopy categories to generalize the result presented in [8].

Let G be a discrete group, O(G) the associated category of canonical orbits (see e.g., [1, 2]) and G-𝕊𝕊 the category of G-simplicial sets. Then, there is a canonical functor Φ : G-𝕊𝕊 → O(G)-𝕊𝕊 which establishes (cf. [4, 6]) an equivalence of respective homotopy categories. In Section 2, we apply our previous results to deduce an equivalence of the homotopy categories of G-𝕊𝕊 and 𝕊𝕊 ↓ NO(G). Furthermore, a G-simplicial version of the Whitehead Theorem is derived (cf. [2]).

The second author acknowledges the hospitality and financial support of the Mathematisches Institut, der Universität München during the time this work was done.

1. Main results

Let 𝕀 be a small category. As mentioned in the introduction, in [10] a pair of adjoint functors is given

[figure: ℂat ↓ 𝕀; D; 𝕀-ℂat; 𝕀 ∫ -]

such that the adjunction maps η : id → (𝕀 ∫ -)D and ξ : D(𝕀 ∫ -) → id are weak equivalences. The functor 𝕀 ∫ - is defined by the Grothendieck construction (see e.g., [13]). If F : 𝕀 → ℂat is a contravariant functor then objects in 𝕀 ∫ F are pairs (i, X) with i an object in 𝕀 and X an object in F (i), and with morphisms (α, x) : (i_{1}, X_{1}) → (i_{2}, X_{2}) consisting of a morphism α : i_{1} → i_{2} in 𝕀 and a morphism x : X_{1} → F (α)(X_{2}). Composition is defined by (α, x)(α′, x′) = (αα′, F (α′)(x)x′). For α : ℂ → 𝕀 (an object in ℂat ↓ 𝕀) and an object i in 𝕀, Dα(i) = i ↓ α is the comma category and the functor D is defined in an obvious way on morphisms in 𝕀.

By [11], the simplicial closed model category structure in the sense of Quillen on the category ℂat considered in [14] determines such a structure on the categories ℂat ↓ 𝕀 and 𝕀-ℂat, respectively. More precisely, from [11] one deduces immediately

1.1. Proposition. The categories ℂat ↓ 𝕀 and 𝕀-ℂat with fibrations, cofibrations and weak equivalences of small categories as defined below, are simplicial closed model categories in the sense of Quillen (see [11]) :

1) the model category structure on ℂat ↓ 𝕀 is the one induced by the usual one on the category of small categories ℂat (see [14]). In particular, the weak equivalences are the weak equivalences over 𝕀;

2) a map in 𝕀-ℂat is a fibration if it is an object-wise fibration (resp. weak equivalence) in ℂat (see [14]) and is a cofibration if it has the left lifting property with respect to maps which are simultaneously fibrations and weak equivalences.

Let now [1] denote the small category associated with the ordered set {0 < 1}. Then, any natural transformation ϕ : F_{0} → F_{1} of functors F_{0}, F_{1} : 𝔹 → ℂ determines a functor Θ : 𝔹 × [1] → ℂ. Therefore, for an object α : 𝔹 → 𝕀 in ℂat ↓ 𝕀 define the functor α×[1] : 𝔹×[1] → 𝕀 such that α×[1] = απ, where π : 𝔹×[1] → 𝔹 is the projection functor. Furthermore, let j_{0}, j_{1} : α → α × [1] denote the canonical inclusions and let α : 𝔹 → 𝕀 and β : ℂ → 𝕀 be objects in ℂat ↓ 𝕀. We say that maps F_{0}, F_{1} : α → β are homotopic if there is a map Θ : α × [1] → β such that Θj_{0} = F_{0} and Θj_{1} = F_{1}. Similarly, for an object G : 𝕀 → ℂat in the category 𝕀-ℂat let G × [1] : 𝕀 → ℂat denote the contravariant functor which is defined by (G × [1])(i) = G(i) × [1], for any object i in 𝕀 and in the obvious way on morphisms in 𝕀. For functors G, H : 𝕀 → ℂat, two maps ψ_{0}, ψ_{1} : G → H are homotopic if there is a map Ψ : G × [1] → H in the category 𝕀-ℂat such that Ψj_{0} = ψ_{0} and Ψj_{1} = ψ_{1}, where j_{0}, j_{1} : G → G × [1] are the canonical inclusions. Then, we may state the following result.

1.2. Proposition. 1) The functors D : ℂat ↓ 𝕀 ⟶ 𝕀-ℂat and 𝕀 ∫ - : 𝕀-ℂat ⟶ ℂat ↓ 𝕀 preserve weak equivalences.

2) If α : 𝔹 → 𝕀 and β : ℂ → 𝕀 are objects in ℂat ↓ 𝕀 and maps F_{0}, F_{1} : α → β are homotopic then the induced functors DF_{0}, DF_{1} : 𝕀 → ℂat are also homotopic. Conversely, for G, H : 𝕀 → ℂat and homotopic maps ψ_{0}, ψ_{1} : G → H the induced maps 𝕀 ∫ ψ_{0} : 𝕀 ∫ G → 𝕀 and 𝕀 ∫ ψ_{1} : 𝕀 ∫ H → 𝕀 are also homotopic.

Proof. 1) Let F_{0}, F_{1} : 𝕀 → ℂat be contravariant functors with a natural transformation F_{0} → F_{1} and such that the functors F_{0}(i) → F_{1}(i) are weak equivalences, for any object i in the category 𝕀 and consider the induced functor 𝕀 ∫ F_{0} → 𝕀 ∫ F_{1}. Put N for the nerve functor from the category ℂat to the category of simplicial sets 𝕊𝕊 and hocolim for the homotopy colimit functor (see [1]) on the category of diagrams of simplicial sets. Then, there is a commutative diagram

\xymatrix{hocolim NF_{0} \ar[r] \ar[d] & N(𝕀 ∫ F_{0}) \ar[d] \\ hocolim NF_{1} \ar[r] & N(𝕀 ∫ F_{1})}

with the horizontal maps as homotopy equivalences in the light of [13]. But by [3, p.335], the map hocolim NF_{0} → hocolim NF_{1} is a weak equivalence. Thus, the map N(𝕀 ∫ F_{0}) → N(𝕀 ∫ F_{1}) is a weak equivalence as well.

Let now

[figure: ℂ; α; F; 𝔻; β; 𝕀]

be a map in the category ℂat ↓ 𝕀 with F as a weak equivalence. Then by [12], the comma category d ↓ F is contractible, for any object d in 𝔻. For any object i in 𝕀, consider the induced functor

i ↓ α ⟶^{i↓F} i ↓ β.

Then, for any object d̄ = (d, i → β(d)) in i ↓ β we have d̄ ↓ (i ↓ F) = d ↓ F. Hence, the comma category d̄ ↓ (i ↓ F) is contractible, for any object d̄ in the category i ↓ β and again by [12] we get that the functor i ↓ F is a weak homotopy equivalence.

2) The proof is straightforward.

Denote by 𝕊𝕊 ↓ N𝕀 the category 𝕊𝕊 over the nerve N𝕀 of the small category 𝕀 and by 𝕀-𝕊𝕊 the category of contravariant functors from a small category 𝕀 to the category 𝕊𝕊 of simplicial sets. Then, the proposition below immediately follows from [11].

1.3. Proposition. The categories 𝕀-𝕊𝕊 and 𝕊𝕊 ↓ N𝕀 with fibrations, cofibrations and weak equivalences as defined below, are simplicial closed model categories in the sense of Quillen (see [11]).

1) The model category structure on the category 𝕊𝕊 ↓ N𝕀 is the one induced by the usual one on the category of simplicial sets (see [11]). In particular, the weak equivalences are the weak homotopy equivalences over N𝕀;

2) a map in 𝕀-𝕊𝕊 is a fibration (resp. weak equivalence) if it is an object-wise fibration (resp. weak equivalence) in 𝕊𝕊 (see [11]) and is a cofibration if it has the left lifting property with respect to maps which are simultaneously fibrations and weak equivalences.

Similarly as in the categories ℂat ↓ 𝕀 and 𝕀-ℂat, we define a homotopy notion in the categories 𝕊𝕊 ↓ N𝕀 and 𝕀-𝕊𝕊. Then, we have the following

1.4. Proposition. There is a pair of adjoint functors A : 𝕊𝕊 ↓ N𝕀 → 𝕀-𝕊𝕊 (the left adjoint) and B : 𝕀-𝕊𝕊 → 𝕊𝕊 ↓ N𝕀 (the right adjoint) such that in the terminology of the previous proposition:

1) both functors send weak equivalences into weak equivalences and homotopic maps into homotopic ones;

2) for every object Θ in 𝕊𝕊 ↓ N𝕀 and every object F in 𝕀-𝕊𝕊, the adjunction maps Θ → BA(Θ) and AB(F) → F are weak equivalences.

Proof. Given an object Θ : 𝕏 → N𝕀 in the category 𝕊𝕊 ↓ N𝕀, one defines the contravariant functor AΘ : 𝕀 → 𝕊𝕊 as follows. For an object i in 𝕀, let i ↓ 𝕀 be the category over i and i ↓ 𝕀 → 𝕀 the natural projection functor. Then, the simplicial set (AΘ)(i) is given by the pull-back

[figure: (AΘ)(i); N(i ↓ 𝕀); 𝕏; Θ; N𝕀.]

Moreover, for a map i → i′ in 𝕀, we get the induced simplicial map (AΘ)(i′) → (AΘ)(i). Given an object F in the category 𝕀-𝕊𝕊, one defines BF as follows. Let hocolim F be the homotopy colimit of F and define BF as the simplicial map hocolim F → N𝕀 given by the natural projection.

The adjunction of a map f : AΘ → F in the category 𝕀-𝕊𝕊 is determined as follows. Associate, with the induced map f̄: AΘ → F ×N(- ↓ 𝕀) by the maps f and AΘ → N(− ↓ 𝕀) given by the pull-back above, the simplicial map f̃ = colim f̄: colim AΘ → colim F × N(- ↓ 𝕀) over the nerve N𝕀. But, colim F × N(- ↓ 𝕀) = hocolim F by [3, Chap. XII] and colim AΘ = ∐_{i}(AΘ)(i)/ ∼, where i runs over all objects in 𝕀 and AΘ(α)x ∼ x for α : i′ → i and x ∈ AΘ(i). An n-simplex in 𝕏 determines an simplex Θ(x) = (i_{0} → ⋯ → i_{n}) in the nerve N𝕀. Then, Θ(x)^{}= (i_{0} = i_{0} → ⋯ → i_{n}) is a simplex in N(i_{0} ↓ 𝕀) and we get an injection η : 𝕏 → colim AΘ such that η(x) = (x, Θ(x)^{}). On the other hand, for any (x, σ) ∈ (AΘ)(i) we have (x, σ) ∼ (x, Θ(x)^{}). Thus, the map η : 𝕏 → colim AΘ is an isomorphism. Finally, we have a map f̃: 𝕏 → hocolim F such that the diagram

[figure: 𝕏; Θ; f̃; hocolim F; N𝕀]

commutes.

Given a map g : Θ → BF in the category 𝕊𝕊 ↓ N𝕀 one can also get an adjunction g̃ : AΘ → F. For an object i in 𝕀, we define a simplicial map g̃(i) : AΘ(i) → F (i). By [1, p.338], the simplicial set hocolim F is isomorphic to the diagonal of the double simplicial set ∐_{∗} F which in dimension n consists of the union ∐_{n} F = ∐_{i_{0}→···→i_{n}} F (i_{n}). Thus, for an n-simplex x in 𝕏 we have g(x) = y with y ∈ F (i_{n}) and Θ(x) = (i_{0} →^{α_{1}} ⋯ →^{α_{n}} i_{n}). Hence,

for (x, (i →^{α} i_{0} →^{α_{1}} ⋯ →^{α_{n}} i_{n})) in (AΘ)(i) we may define g̃(i)(x, (i →^{α} i_{0} →^{α_{0}} ⋯ →^{α_{n}} i_{n})) =

F (α_{n} ⋯ α_{0}α)(y).

1) Let now Θ : 𝕏 → N𝕀 and Θ′ : 𝕏′ → N𝕀 be two objects in the category 𝕊𝕊 ↓ 𝕀 and f : Θ → Θ′ a map such that the associated simplicial map 𝕏 → 𝕏′ is a weak homotopy equivalence. Then, for an object i in 𝕀 the induced maps AΘ(i) → 𝕏 and AΘ′(I) → 𝕏′ have the same homotopy fibre (up to homotopy). Hence, from the long homotopy exact sequences determined by AΘ(i) → 𝕏 and AΘ′(i) → 𝕏′, it follows that the map Af(i) : AΘ(i) → AΘ′(i) is a weak homotopy equivalence. Thus, the induced map Af : AΘ → AΘ′ in the category 𝕀-𝕊𝕊 is a weak homotopy equivalence.

On the other hand, if f : F → F′ is a map in the category 𝕀-𝕊𝕊 such that the simplicial maps f(i) : F (i) → F′(i) are weak homotopy equivalences for any object i in 𝕀 then by [3, p.335], one gets that the induced map hocolim f : hocolim F → hocolim F′ is a weak homotopy equivalence. Thus, the map Bf : BF → BF′ is a weak homotopy equivalence. The proof of the preservation of homotopic maps by functors A and B is straightforward.

2) Let |–| be the geometric realization functor. For an object Θ : 𝕏 → N𝕀 in the category 𝕊𝕊 ↓ N𝕀, the adjunction map τ_{Θ} : Θ → BAΘ is defined as follows. For n-simplex x in 𝕏 let Θ(x) = (i_{0} → ⋯ → i_{n}) be an n-simplex in the nerve N𝕀 and Θ(x)^{}= (i_{0} = i_{0} → ⋯ → i_{n}) an associated n-simplex in N(i_{0} ↓ 𝕀). Then, (x, Θ(x)^{}) is an n-simplex in AΘ and τ_{Θ}(x) = (x, Θ(x)^{}). Moreover, there is a natural projection π_{Θ} : hocolimAΘ → 𝕏 such that π_{Θ}τ_{Θ} = id_{Θ} and it is a routine to check that the map |τ_{Θ}||π_{Θ}| is homotopic to the identity map id_{|hocolimAΘ|}.

For an object F in the category 𝕀-𝕊𝕊, we define the adjunction map θ_{F} : ABF → F. Let Θ : hocolim F → N𝕀 be the natural projection and consider the simplicial map θ_{F} (i) : AΘ(i) → F (i), for an object i in 𝕀 defined as follows. If (x, (i →^{α} i_{0} →^{α_{1}} ⋯ →^{α_{n}} i_{n})) is an n-simplex in AΘ(i) then (x, (i_{0} →^{α_{1}} ⋯ →^{α_{n}} i_{n})) is an n-simplex in hocolim F with x in F (i_{n}). Then, θ_{F} (i)(x, (i →^{α} i_{0} →^{α_{1}} ⋯ →^{α_{n}} i_{n})) = F (α_{n} ⋯ α_{0}α)(x). Moreover, there is an injection σ(i) : F (i) → AΘ(i) given by σ(i)(x) = (x, ĩd_{i}), for an n-simplex x in F (i) with ĩd_{i} the degenerated n-simplex determined by an object i in 𝕀. Of course θ_{F} (i)σ(i) = id_{F (i)} and it is also easy to check that the map |σ(i)||θ_{F} (i)| is homotopic to the identity map id_{|AΘ(i)|} for any object i in 𝕀, and the proof is completed.

The nerve functor N : ℂat → 𝕊𝕊 determines functors \overline{N} : ℂat ↓ 𝕀 → 𝕊𝕊 ↓ N𝕀 and Ñ: 𝕀-ℂat → 𝕀-𝕊𝕊, respectively. Then, the diagram

[figure: ℂat ↓ 𝕀; \overline{N}; D; 𝕀-ℂat; _{N}; 𝕊𝕊 ↓ N𝕀; A; 𝕀-𝕊𝕊]

commutes and by [13], the diagram

[figure: ℂat ↓ 𝕀; \overline{N}; 𝕀-ℂat; 𝕀 ∫ -; _{N}; 𝕊𝕊 ↓ N𝕀; 𝕀-𝕊𝕊; B]

commutes up to homotopy. Moreover, by [7, 9] there are homotopy inverses Γ : 𝕊𝕊 → ℂat, for the nerve functor N : ℂat → 𝕊𝕊 such that for any small category 𝕀 there is a weak equivalence ΓN𝕀 → 𝕀. Thus, we get induced functors \overline{Γ} : 𝕊𝕊 ↓ N𝕀 → ℂat ↓ 𝕀 and Γ̃ : 𝕀-𝕊𝕊 → 𝕀-ℂat, respectively. Consequently, we may generalize the result presented in [8] and summarize this section with the following

1.5. Theorem. For any small category 𝕀, the diagram of above functors

[figure: ℂat ↓ 𝕀; N̄; D; 𝕀-ℂat; 𝕀 ∫ -; _{N}; 𝕊𝕊 ↓ N𝕀; Γ̄; A; 𝕀-𝕊𝕊; B; _{Γ}]

determines equivalences of respective homotopy categories.

In particular, if 𝕀 is the category associated with a discrete group G then the nerve N𝕀 is the classifying complex \overline{W} G and 𝕀-ℂat (resp. 𝕀-𝕊𝕊) is the category of right G-small categories (resp. right G-simplicial sets). Thus from the above, we may also deduce an equivalence of respective homotopy categories (cf. [5]).

2. Equivariant applications

Let G be a discrete group and O(G) the associated category of canonical orbits; its objects are the left cosets G/H as H ranges over all subgroups of G and morphisms are G-maps G/H → G/K with respect to left translation. If G-𝕊𝕊 denotes the category of G-simplicial sets then there is a canonical functor Φ : G-𝕊𝕊 → O(G)-𝕊𝕊 defined on objects by Φ(𝕏)(G/H) = 𝕏^{H} and in the obvious way on morphisms in G-𝕊𝕊, where 𝕏^{H} is the H-fixed point simplicial subset of 𝕏. Then (cf. [1]), the category G-𝕊𝕊 is a closed model category; a map f : 𝕏 → 𝕐 is a fibration (resp. weak equivalence) if the induced map Φ(f) : Φ(𝕏) → Φ(𝕐) is a fibration (resp. weak equivalence) in O(G)-𝕊𝕊. Cofibrations in G-𝕊𝕊 are determined by means of the left lifting property with respect to maps which are simultaneously fibrations and weak equivalences. From [4, 6], one may derive that the functor Φ : G-𝕊𝕊 → O(G)-𝕊𝕊 establishes an equivalence of respective homotopy categories. Therefore, Proposition 1.4 yields

2.1. Corollary. If G is a discrete group then the homotopy categories of G-𝕊𝕊 and 𝕊𝕊 ↓ NO(G) are equivalent.

Recall that a G-simplicial set 𝕏 is called fibrant (resp. cofibrant) if the unique map 𝕏 → ∗ (resp. ∅ → 𝕏) is a fibration (resp. cofibration) in the category G-𝕊𝕊, where ∅ is the empty simplicial set and ∗ a single point simplicial set with the trivial action of G.

2.2. Lemma. Every G-simplicial set 𝕏 is cofibrant in the category G-𝕊𝕊.

Proof. Let p : 𝔼 → 𝔹 be a fibration and a weak equivalence in the category G-𝕊𝕊, and f : 𝕏 → 𝔹 a G-map. The n-skeleton 𝕏^{(n)}, for n ≥ 0 is a G-simplicial subset of 𝕏 and 𝕏 = colim_{n} 𝕏^{(n)}. Let f^{(n)} denote the restriction of f to 𝕏^{(n)}; we proceed by induction to show an existence of a G-map f̃^{(n)} : 𝕏^{(n)} → 𝔼 with pf̃^{(n)} = f^{(n)}, for n ≥ 0.

Of course, there is a G-map f̃^{(0)} → 𝔼 such pf̃^{(0)} = f^{(0)}. Suppose there is a G-map f̃^{(n−1)} : 𝕏^{(n−1)} → 𝔼 with pf̃^{(n−1)} = f^{(n−1)} and put Δ[n] for the standard simplicial nsimplex, and Δ̇[n] for its boundary. Then, there is a pushout diagram

[figure: ∐_{x} G/H_{x} × Δ̇[n]; 𝕏^{(n−1)}; ∐_{x} G/H_{x} × Δ[n]; 𝕏^{(n)},]

where x runs over all non-degenerate n-simplexes in 𝕏 and H_{x} is the isotropy subgroup of an n-simplex x. By means of the right lifting property of the map p : 𝔼 → 𝔹 (see e.g., [11, p.2.1]) and the map f̃^{(n−1)} : 𝕏^{(n−1)} → 𝔼, we get a G-map f̃^{(n)} : 𝕏^{(n)} → 𝔼 such that pf̃^{(n)} = f^{(n)} and its restriction to 𝕏^{(n−1)} is equal to f̃^{(n−1)}. Therefore, the sequence of G-map f̃^{(n)} → 𝔼, for n ≥ 0 determines a G-map f̃: 𝕏 → 𝔼 with pf̃ = f and the proof is completed.

Let now f : 𝕏 → 𝕐 be a weak equivalence in the category G-𝕊𝕊. Then, from [11, p.3.15] it follows that the map [f, 𝔼]_{G} : [𝕐, 𝔼]_{G} → [𝕏, 𝔼]_{G} of G-homotopy classes is bijective, for any fibrant G-simplicial set 𝔼. We are now in a position to state the following G-simplicial version of the Whitehead Theorem (cf.[2]).

2.3. Corollary. Let f : 𝕏 → 𝕐 be a G-map of fibrant G-simplicial sets. Then the following conditions are equivalent:

1) the G-map f : 𝕏 → 𝕐 is a weak equivalence;

2) the G-map f : 𝕏 → 𝕐 is a G-homotopy equivalence;

3) the map in the category 𝕊𝕊 ↓ NO(G), induced by the commutative diagram

[figure: hocolim Φ(𝕏); hocolim Φ(f); hocolim Φ(𝕐); NO(G)]

is a homotopy equivalence.

Proof. Observe that by Proposition 1.4, we get 2) ⇒ 3) and 3) ⇒ 1). We show now the implication 1) ⇒ 2). By Lemma 2.2 and [11, p.3.15], the induced maps [f, 𝕏]_{G} : [𝕐, 𝕏]_{G} → [𝕏, 𝕏]_{G} and [f, 𝕐]_{G} : [𝕐, 𝕐]_{G} → [𝕏, 𝕐]_{G} of G-homotopy classes are bijective. Therefore, the G-map f : 𝕏 → 𝕐 is a G-homotopy equivalence, as claimed.

References

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[6] A.. Elmendorf, Systems of fixed point sets, Trans. Amer. Math. Soc. 277 (1983), 275-284.

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