Equivariant homotopy equivalence of homotopy colimits of $G$-functors

Abstract

Given a group G and a G-category ${\textbf{C}}$, we give a condition on a diagram of simplicial sets indexed by ${\textbf{C}}$ that allows us to define a natural action of G on its homotopy colimit, and some other simplicial sets defined in terms of the diagram. Well-known theorems on homeomorphisms and homotopy equivalences are generalized to equivariant versions.

1 Introduction

Let G be a group and ${\textbf{C}}$ a small category. Consider a ${\textbf{C}}$-diagram of simplicial sets, where the values of the diagram have a G-action. Then several structures defined in terms of the diagram, like the colimit and the homotopy colimit, have an induced structure of G-object. However, it is often the case that one has a diagram $F:{\textbf{C}}\rightarrow {\textbf{D}}$ where ${\textbf{C}}$ is a small G-category, ${\textbf{D}}$ is an arbitrary category and the values of F do not necessarily have a G-action, but nevertheless the homotopy colimit of F does have one. This situation was considered in [6], and independently, by this author in his Ph. D. thesis [11], where the concept of an action of a group G on a functor F by natural transformations is introduced. Here we define this concept formally in Sect. 3, after the basic definitions in Sect. 2.

We show that there are induced G-actions on colimits, coends, and bar constructions of G-functors. In Sect. 4 we consider more closely the homotopy colimit, and show some basic identities involving the constructions defined so far. One important feature of the homotopy colimit (and of the bar construction, which is its generalization) that the usual colimit lacks, is the fact that it induces a homotopy equivalence from a “pointwise” homotopy equivalence, that is, if $F,F'$ are two diagrams of simplicial sets, and $\eta$ is a natural transformation from F to $F'$ such that $\eta _{X}:FX\rightarrow F'X$ is a homotopy equivalence for all ${\textbf{C}}$-objects X, then ${{\,\textrm{hocolim}\,}}F$ and ${{\,\textrm{hocolim}\,}}F'$ are homotopy equivalent. In Sect. 5 we prove an equivariant version of the homotopy invariance of the bar construction, and this is the main result in this paper. Finally, in Sect. 6 we prove the equivariant versions of the four theorems listed in [2, p. 154] about the homotopy colimit. Some of them were noted in [6], however a mild additional hypothesis lets us obtain a more precise result.

2 Preliminaries

In this section, ${\textbf{C}}$ denotes a small category, and ${\textbf{D}}$ an arbitrary category.

Let G be a group. We will denote by ${\textbf{G}}$ the category with a single object $*$, in which $\hom _{\textbf{G}}(*,*)=G$ and the composition corresponds to group multiplication. For $n\ge 0$, let [n] be the small category with object set given by $\{0,1,\ldots ,n\}$, and with $\hom _{[n]}(i,j)$ a singleton whenever $i\le j$, otherwise empty.

An object $D\in {{\,\textrm{obj}\,}}{\textbf{D}}$ is called a G-object if there is a functor $J:{\textbf{G}}\rightarrow {\textbf{D}}$ such that $J(*)=D$. Clearly this is equivalent to the existence of a collection of ${\textbf{D}}$-morphisms $\{g:D\rightarrow D\}_{g\in G}$, such that the morphism corresponding to the identity element in G is the identity $1_{D}$, and composition of morphisms corresponds to group multiplication.

Let $\textbf{SCat}$ be the category of small categories and $\varvec{\Delta }$ be the subcategory of $\textbf{SCat}$ with objects ${{\,\textrm{obj}\,}}\varvec{\Delta } =\left\{ \,[n]\mid n\ge 0\,\right\}$ and morphisms given by the non-decreasing maps. We denote by ${\textbf{D}}^{\textbf{C}}$ the category of functors ${\textbf{C}}\rightarrow {\textbf{D}}$ ([7, page 40]). The category of simplicial sets (see [8]), denoted $s \textbf{Set}$, is equal to $\textbf{Set} ^{\varvec{\Delta } ^\textrm{op}}$. The category of small G-categories is defined as $\textbf{SCat} ^{{\textbf{G}}}$. We consider the nerve functor $N:\textbf{SCat} \rightarrow s \textbf{Set}$, given by ${\textbf{C}}\mapsto (\hom _{\textbf{SCat}}(-,{\textbf{C}}):{\varvec{\Delta } ^\textrm{op}}\rightarrow \textbf{Set})$. The nerve functor sends G-categories to G-simplicial sets. There is also a geometric realization functor $|\cdot |:s \textbf{Set} \rightarrow \textbf{Top}$, that sends G-simplicial sets to G-topological spaces. We denote $|N({\textbf{C}})|$ simply as $|{\textbf{C}}|$.

In the case that ${\textbf{C}}$ and ${\textbf{D}}$ are small G-categories, and $F:{\textbf{C}}\rightarrow {\textbf{D}}$ is a functor such that $F(gC)=gF(C)$, $F(g\phi )=gF(\phi )$ for all $g\in G$, $C\in {{\,\textrm{obj}\,}}{\textbf{C}}$ and all ${\textbf{C}}$-morphisms $\phi$, we will say that F is an equivariant functor.

If X and Y are G-topological spaces, a G-homotopy from X to Y is a continuous map $H:X\times [0,1]\rightarrow Y$ such that $H(gx,t)=gH(x,t)$ for all $g\in G$, $x\in X$ and $t\in [0,1]$. Two G-maps $f_1,f_2:X\rightarrow Y$ are G-homotopic if there is a G-homotopy H from X to Y such that $H(x,0)=f_1(x)$ and $H(x,1)=f_2(x)$. In this case we write $f_1\simeq _G f_2$. The G-topological spaces X and Y are G-homotopy equivalent if there are G-maps $f:X\rightarrow Y$ and $f':Y\rightarrow X$ such that $f'f\simeq _G 1_X$ and $ff'\simeq _G 1_Y$.

We say that two small G-categories ${\textbf{C}}_{1}$, ${\textbf{C}}_{2}$ are G-homotopy equivalent if the spaces $|{\textbf{C}}_{1}|$, $|{\textbf{C}}_{2}|$ are. It is not required that the map $|{\textbf{C}}_{1}|\rightarrow |{\textbf{C}}_{2}|$ defining the homotopy equivalence is induced from a functor ${\textbf{C}}_{1}\rightarrow {\textbf{C}}_{2}$. Similarly, we say that two G-simplicial sets are G-homotopic if their realizations are G-homotopy equivalent.

If $S:{\textbf{C}}^{\textrm{op}}\times {\textbf{C}}\rightarrow {\textbf{D}}$ is a functor and D is an object in ${\textbf{D}}$, a wedge from S to D is a dinatural transformation ([7, §IX.4]) $\zeta =\{\zeta _{A,B}:S(A,B)\rightarrow D\}$ from S to the constant functor D. A coend of S is a pair $({{\,\textrm{coend}\,}}S,\zeta )$, where ${{\,\textrm{coend}\,}}S$ is an object of ${\textbf{D}}$ and $\zeta$ is a wedge from S to ${{\,\textrm{coend}\,}}S$, which is universal among all wedges from S to a constant functor. If for any small category ${\textbf{C}}$ any functor $S:{\textbf{C}}^{\textrm{op}}\times {\textbf{C}}\rightarrow {\textbf{D}}$ has a coend, we say that ${\textbf{D}}$ has small coends. The category $s \textbf{Set}$ has small coends. If $F:{\textbf{C}}\rightarrow s \textbf{Set}$, $T:{\textbf{C}}^{\textrm{op}}\rightarrow s \textbf{Set}$ are functors, then ${{\,\textrm{coend}\,}}(T\times F)$ is denoted by $T\otimes _{{\textbf{C}}} F$ (as in [5], for the analogous case of diagrams of topological spaces).

Proposition 2.1

Let ${\textbf{D}}$ be a category with small coends. Then

• (1) Let ${\textbf{C}}_{1}$, ${\textbf{C}}_{2}$ be small categories, and $S_{i}:{\textbf{C}}_{i}^{\textrm{op}}\times {\textbf{C}}_{i}\rightarrow {\textbf{D}}$ be functors for $i=1,2$. Suppose we also have a functor $U:{\textbf{C}}_{1}\rightarrow {\textbf{C}}_{2}$ and $\eta$ a natural transformation $\eta :S_{1}\rightarrow S_{2}\circ (U^{\textrm{op}}\times U)$. Then there is a unique morphism ${{\,\textrm{coend}\,}}(U,\eta ):{{\,\textrm{coend}\,}}S_{1}\rightarrow {{\,\textrm{coend}\,}}S_{2}$ that makes the following diagram commute for all $C\in {{\,\textrm{obj}\,}}{\textbf{C}}$:

  

• (2) There is a category ${\mathbf {Bif({\textbf{D}})}}$, with class of objects:

  $$\begin{aligned} \left\{ \,({\textbf{C}},S)\mid {\textbf{C}}\text { is a small category, } S:{\textbf{C}}^{\textrm{op}}\times {\textbf{C}}\rightarrow {\textbf{D}}\,\right\} , \end{aligned}$$

  and a morphism $({\textbf{C}}_{1},S_{1})\rightarrow ({\textbf{C}}_{2},S_{2})$ is a pair $(U,\eta )$, as in (1).

• (3) There is a functor ${{\,\textrm{coend}\,}}:{\mathbf {Bif({\textbf{D}})}}\rightarrow {\textbf{D}}$, defined on objects by $({\textbf{C}},S)\mapsto {{\,\textrm{coend}\,}}S$, and on a morphism $(U,\eta )$ by the morphism ${{\,\textrm{coend}\,}}(U,\eta )$ of (1).

Proof

Statement (1) is a straightforward generalization of the dual of [7, Proposition IX.7.1]. Statements (2) and (3) are analogous to [7, Exercise V.2.5]. The composition in (2) is given by $(U',\eta ')\circ (U,\eta )=(U'\circ U,\eta '(U^{op}\times U)\circ \eta )$. $\square$

Following [7], we say that a functor $S:{\textbf{C}}^{\textrm{op}}\times {\textbf{C}}\rightarrow {\textbf{D}}$ is dummy on its first variable if its equal to a composition of the functor projection on the second factor $Q:{\textbf{C}}^{\textrm{op}}\times {\textbf{C}}\rightarrow {\textbf{C}}$ with a functor $F:{\textbf{C}}\rightarrow {\textbf{D}}$. In this case, we will identify the functor S with the functor F, since the latter determines the former. Note that ${\mathbf {Bif({\textbf{D}})}}$ has a subcategory isomorphic to the “super comma” category of [7], namely the full subcategory on the functors which are dummy in their first variable. The coend of such a functor, say the one determined by $F:{\textbf{C}}\rightarrow {\textbf{D}}$, can be identified with its colimit ${{\,\textrm{colim}\,}}F$.

Let $S:{\textbf{C}}^{\textrm{op}}\times {\textbf{C}}\rightarrow s \textbf{Set}$ be a functor. There is then a simplicial set $B({\textbf{C}},S)$, called the (simplicial) bar construction (see [9]), such that

$$\begin{aligned} B({\textbf{C}},S)_{n}&=\bigsqcup _{X_0\xrightarrow {\phi _1} X_1\xrightarrow {\phi _2}\cdots \xrightarrow {\phi _n} X_n\in N({\textbf{C}})_{n}} S(X_{n},X_{0})_{n}\end{aligned}$$

(1)

$$\begin{aligned}&=\left\{ \,(\phi _{1},\ldots ,\phi _{n};z)\mid z\in S(X_{n},X_{0})_{n}\,\right\} \end{aligned}$$

(2)

with boundaries and degeneracies given by:

$$\begin{aligned} d_{i}(\phi _{1},\ldots ,\phi _{n};z)&= {\left\{ \begin{array}{ll} \bigl (\phi _{2},\ldots ,\phi _{n};d_{0}(S(1_{X_n},\phi _{1})(z))\bigr ) &{} i=0,\\ (\phi _{1},\ldots ,\phi _{i+1}\phi _{i},\ldots ,\phi _{n};d_{i}z) &{} 1\le i\le n-1,\\ \bigl (\phi _{1},\ldots ,\phi _{n-1};d_{n}(S(\phi _{n},1_{X_0})(z))\bigr ) &{} i=n \end{array}\right. }\end{aligned}$$

(3)

$$\begin{aligned} s_{i}(\phi _{1},\ldots ,\phi _{n};z)&= (\phi _{1},\ldots ,\phi _{i},1_{X_i},\phi _{i+1},\ldots ,\phi _{n};s_{i}z),\qquad 0\le i\le n. \end{aligned}$$

(4)

If the functor S is of the form $T\times F$, with $T:{\textbf{C}}^{\textrm{op}}\rightarrow s \textbf{Set}$, $F:{\textbf{C}}\rightarrow s \textbf{Set}$, then we denote $B({\textbf{C}},S)$ as $B(T,{\textbf{C}},F)$.

Proposition 2.2

The bar construction can be interpreted as a functor

$$\begin{aligned} B:{\mathbf {Bif(s \textbf{Set} )}}\rightarrow s \textbf{Set} . \end{aligned}$$

(5)

Proof

Given a morphism $(U,\eta ):({\textbf{C}}_{1},S_{1})\rightarrow ({\textbf{C}}_{2},S_{2})$ in ${\mathbf {Bif(s \textbf{Set} )}}$, the morphism of simplicial sets $B(U,\eta ):B({\textbf{C}}_{1},S_{1})\rightarrow B({\textbf{C}}_{2},S_{2})$ is defined in n-simplices:

$$\begin{aligned} \bigsqcup _{X_0\rightarrow X_1\rightarrow \cdots \rightarrow X_n\in N({\textbf{C}}_{1})_{n}} S_{1}(X_{n},X_{0})_{n} \rightarrow \bigsqcup _{UX_0\rightarrow UX_1\rightarrow \cdots \rightarrow UX_n\in N({\textbf{C}}_{2})_{n}} S_{2}(UX_{n},UX_{0})_{n} \end{aligned}$$

(6)

by means of the coproduct of morphisms $\eta _{(X_{n},X_{0})}:S_{1}(X_{n},X_{0})\rightarrow S_{2}(UX_{n},UX_{0})$. $\square$

3 G-functors

We define a G-functor analogously to Definition 2.2 in [6].

Definition 3.1

Let G be a group, and ${\textbf{C}}$ be a small G-category, and ${\textbf{D}}$ be an arbitrary category. We say that $F:{\textbf{C}}\rightarrow {\textbf{D}}$ is a G-functor if for each $g\in G$ we have a natural transformation $\eta _{g}:F\rightarrow F\circ g$ in such a way that the components of $\eta _{1}$ are identities, and $\eta _{g_{1}g_{2}}=(\eta _{g_{2}}\circ g_{1})\circ \eta _{g_{1}}$ for all $g_{1},g_{2}\in G$.

Definition 3.2

Let $F_{1},F_{2}:{\textbf{C}}\rightarrow {\textbf{D}}$ be two G-functors, with collections of natural transformations indexed by G given by $\eta ^{1},\eta ^{2}$ respectively. A morphism of G-functors is a natural transformation $\epsilon :F_{1}\rightarrow F_{2}$ such that $(\epsilon \circ g)\circ \eta ^{1}_{g}=\eta ^{2}_{g}\circ \epsilon$ for all $g\in G$.

We have an equivalent definition of G-functor.

Proposition 3.3

Let ${\textbf{C}}$ be a small category, ${\textbf{D}}$ an arbitrary category and $F:{\textbf{C}}\rightarrow {\textbf{D}}$ be a functor, such that $({\textbf{C}},F)$ is a G-object in the category ${\mathbf {Bif({\textbf{D}})}}$. Then F has a structure of a G-functor.

Proof

If $({\textbf{C}},F)$ is a G-object in the category ${\mathbf {Bif({\textbf{D}})}}$, there is a functor $J:{\textbf{G}}\rightarrow {\mathbf {Bif({\textbf{D}})}}$ with $J(*)=({\textbf{C}},F)$. Then, for each $g\in G$ we obtain a morphism in the category ${\mathbf {Bif({\textbf{D}})}}$ corresponding to g, from $({\textbf{C}},F)$ to itself. Such a morphism has two components: a functor ${\textbf{C}}\rightarrow {\textbf{C}}$ which may be just denoted as g, and a natural transformation from F to $F\circ g$, which we can denote as $\eta _{g}$. The fact that J is a functor imply the properties required in the definition of G-functor. $\square$

Unraveling the definition, we see that if $F:{\textbf{C}}\rightarrow {\textbf{D}}$ is a G-functor, then ${\textbf{C}}$ has a structure of small G-category, and there is a family of ${\textbf{D}}$-morphisms $\eta =\{\eta _{g,X}:F(X)\rightarrow F(gX)\}$ indexed by $g\in G$ and $X\in {{\,\textrm{obj}\,}}{\textbf{C}}$ such that

• (1) $\eta _{1,X}=1_{F(X)}$ for all $X\in {{\,\textrm{obj}\,}}{\textbf{C}}$,

• (2) $\eta _{g_1,g_2X}\circ \eta _{g_2,X}=\eta _{g_1g_2,X}$ for any $X\in {{\,\textrm{obj}\,}}{\textbf{C}}$, $g_1,g_2\in G$,

• (3) $\eta _{g,Y}\circ F(f)=F(gf)\circ \eta _{g,X}$ for any $g\in G$ and $f:X\rightarrow Y$ a morphism in ${\textbf{C}}$.

Conversely, these conditions determine the G-object $({\textbf{C}},F)$ in ${\mathbf {Bif({\textbf{D}})}}$.

Note that, by Propositions 2.1.(3), 2.2 and 3.3, it follows that in the case where ${\textbf{D}}=s \textbf{Set}$, if S is an object in ${\mathbf {Bif({\textbf{D}})}}$, then both ${{\,\textrm{coend}\,}}S$ and $B({\textbf{C}},S)$ have defined a G-action on them.

It is often the case that we can derive a G-functor a G-object in some other category. We show some examples that will be of use later.

Example 3.4

Let ${\textbf{C}}$ and ${\textbf{D}}$ be G-categories, and $F:{\textbf{C}}\rightarrow {\textbf{D}}$ an equivariant functor. Then, for $D,D'\in {{\,\textrm{obj}\,}}{\textbf{D}}$, we have a category ${D}\backslash {F}/{D'}$ with objects

$$\begin{aligned} {{\,\textrm{obj}\,}}({D}\backslash {F}/{D'})=\left\{ \,(u,C,v)\mid C\in {{\,\textrm{obj}\,}}{\textbf{C}},D\xrightarrow {u} FC\xrightarrow {v} D'\,\right\} \end{aligned}$$

and a morphism $p:(u,C,v)\rightarrow (u',C',v')$ given by a ${\textbf{C}}$-morphism $p:C\rightarrow C'$ such that $F(p)\circ u=u'$ and $v'\circ F(p)=v$.

There is a functor ${\textbf{D}}^{\textrm{op}}\times {\textbf{D}}\rightarrow \textbf{SCat}$ defined on objects by $(D,D')\mapsto {D}\backslash {F}/{D'}$. If $(\phi ,\psi ):(D,D')\rightarrow (E,E')$ is a morphism in ${\textbf{D}}^{\textrm{op}}\times {\textbf{D}}$, the associated functor ${D}\backslash {F}/{D'}\rightarrow {E}\backslash {F}/{E'}$ sends (u, C, v) to $(u\phi ,C,\psi v)$.

Then ${\textbf{D}}^{\textrm{op}}$ and ${\textbf{D}}^{\textrm{op}}\times {\textbf{D}}$ have an obvious structure of G-categories, and there is an action of G on the functor ${\textbf{D}}^{\textrm{op}}\times {\textbf{D}}\rightarrow \textbf{SCat}$ we just defined: for $g\in G$, set $\eta _{g,(D,D')}$ as the functor ${D}\backslash {F}/{D'}\rightarrow {gD}\backslash {F}/{gD'}$ given by $(u,C,v)\mapsto (gu,gC,gv)$.

In this context, we can also consider the comma categories $D\backslash {{F}}$ and $F/{D}$ (see [7]) as functors ${\textbf{D}}^{\textrm{op}}\rightarrow \textbf{SCat}$, ${\textbf{D}}\rightarrow \textbf{SCat}$ with a G-action. If $\nu :F_{1}\rightarrow F_{2}$ is an equivariant natural transformation (that is, a natural transformation such that $g\nu _{C}=\nu _{gC}$), then there is an induced morphism of G-functors ${\bar{\nu }}:-\backslash F_{1}\rightarrow -\backslash F_{2}$, given by ${\bar{\nu }}_{D}:D\backslash F_{1}\rightarrow D\backslash F_{2}$, $(u,C)\mapsto (\nu _{C}u,C)$.

Example 3.5

Again, let ${\textbf{C}}$ and ${\textbf{D}}$ be G-categories, and $F:{\textbf{C}}\rightarrow {\textbf{D}}$ an equivariant functor. There is a functor ${\textbf{C}}^{\textrm{op}}\times {\textbf{C}}\rightarrow \textbf{Set}$ given on objects by $(X,Y)\mapsto \hom _{{\textbf{D}}}(FX,FY)$ and on morphisms by $(\phi ,\psi )\mapsto (f\mapsto F\psi \circ f\circ F\phi )$. It has a G-action defined by $\eta _{g,(X,Y)}:\hom _{{\textbf{D}}}(FX,FY)\rightarrow \hom _{{\textbf{D}}}(gFX,gFY)$, $f\mapsto gf$.

Since any set X can be considered as a simplicial set Y such that $Y_{n}=X$ for all n and all faces and degeneracies equal to the identity, we can as well consider the last functor as taking values in the category of simplicial sets.

Example 3.6

Consider G a group, $H\le G$ a subgroup, and let ${\textbf{C}}$ be the discrete small G-category with object set $G/\!/H=\{a_{1}H,a_{2}H,\ldots ,a_{n}H\}$, that is, the set of left cosets of H in G with the usual action by left translation, with $a_{1}=1$. Let Z be an H-simplicial set, and consider the constant functor $F:{\textbf{C}}\rightarrow s \textbf{Set}$ with value Z. We define a G-action $\eta$ on F as follows: Let $\eta _{g,H}:F(H)\rightarrow F(gH)$ be defined as $z\mapsto hz$, where $g=a_{i}h$, $h\in H$; and then $\eta _{g,aH}(z)=\eta _{ga,H}(z)$. It is straightforward to check that this defines an action of G on F, and so ${{\,\textrm{colim}\,}}F$ is a G-simplicial set. This construction is usually known as the induced action from H to G. We will denote ${{\,\textrm{colim}\,}}F$ in this case as ${Z}\!\uparrow _{H}^{G}$.

We finish this section by stating some basic and easily provable properties of G-functors.

Proposition 3.7

If $F:{\textbf{C}}\rightarrow {\textbf{D}}$ is a functor with a G-action given by $\eta$ and X is an object in ${\textbf{C}}$, then FX is a $G_{X}$-object, where $G_{X}$ is the stabilizer of X under the action of G on ${{\,\textrm{obj}\,}}{\textbf{C}}$. The action is defined by the morphisms $\eta _{g,X}:FX\rightarrow FX$. $\square$

Proposition 3.8

((2.3) from [6]) Let $F:{\textbf{C}}\rightarrow {\textbf{D}}$ be a G-functor, $U:{\textbf{C}}'\rightarrow {\textbf{C}}$ an equivariant functor, and $T:{\textbf{D}}\rightarrow {\textbf{E}}$ any functor. Then both $F\circ U$ and $T\circ F$ have induced structures of G-functors. $\square$

For example, for any G-category ${\textbf{C}}$, we have a G-functor

$$\begin{aligned} N(-\backslash {{\textbf {C}}}):{{\textbf {C}}}\rightarrow s {\textbf {Set}} . \end{aligned}$$

(7)

4 The homotopy colimit

Definition 4.1

If $F:{\textbf{C}}\rightarrow s \textbf{Set}$ is a functor, its homotopy colimit is defined as $B(*,{\textbf{C}},F)$, where $*$ is the constant functor ${\textbf{C}}^{\textrm{op}}\rightarrow s \textbf{Set}$ with value the simplicial set with exactly one simplex in each dimension.

Let ${\textbf{C}}$ be a G-category and $S:{\textbf{C}}^{\textrm{op}}\times {\textbf{C}}\rightarrow s \textbf{Set}$ a G-functor.

Proposition 4.2

We have an equivariant isomorphism:

$$\begin{aligned} N(-\backslash {{\textbf {C}}}/-)\otimes _{{{\textbf {C}}}^{\text {op}}\times {{\textbf {C}}}}S\cong _{G} B({{\textbf {C}}},S), \end{aligned}$$

(8)

Proof

This can be proven by showing that $B({\textbf{C}},S)$ satisfies the definition of coend of the functor $N({-}\backslash {{\textbf{C}}}/{-})\times S:({\textbf{C}}^{\textrm{op}}\times {\textbf{C}})^{\textrm{op}}\times ({\textbf{C}}^{\textrm{op}}\times {\textbf{C}})\rightarrow s \textbf{Set}$. In a much greater generality, this was shown in [9] in a non-equivariant setting, however, all the maps involved are equivariant. For example, a wedge from $N({-}\backslash {{\textbf{C}}}/{-})\times S$ to $B({\textbf{C}},S)$ is determined by $\alpha _{A,B}:N({B}\backslash {{\textbf{C}}}/{A})\times S(A,B)\rightarrow B({\textbf{C}},S)$ that is given on an n-simplex defined by $f_{0}:B\rightarrow A$, $\phi _{i}:C_{i}\rightarrow C_{i+1}$ for $i=0,1,\ldots ,n-1$, $g_{n}:C_{n}\rightarrow A$ and $z\in S(A,B)_{n}$, sending it to $\bigl ((\phi _{1},\phi _{2},\ldots ,\phi _{n}, S(g_{n},f_{0})(z)\bigr )$ $\square$

If it is the case that $S=T\times F$ where $F:{\textbf{C}}\rightarrow s \textbf{Set}$, $T:{\textbf{C}}^{\textrm{op}}\rightarrow s \textbf{Set}$ are G-functors, then using Fubini’s theorem for coends [7, p. 230], this leads to

$$\begin{aligned} T\otimes _{{\textbf{C}}} N({-}\backslash {{\textbf{C}}}/{-})\otimes _{{\textbf{C}}} F\cong _{G} B(T,{\textbf{C}},F). \end{aligned}$$

(9)

Using that, we can prove that for the G-functors $F:{\textbf{C}}^{\textrm{op}}\rightarrow s \textbf{Set}$, $T:{\textbf{C}}\times {\textbf{D}}^{\textrm{op}}\rightarrow s \textbf{Set}$ and $U:{\textbf{D}}\rightarrow s \textbf{Set}$, we have

$$\begin{aligned} B(B(F,{{\textbf {C}}},T),{{\textbf {D}}},U)&\cong _{G} B(F,{{\textbf {C}}},T)\otimes _{{{\textbf {D}}}}N(-\backslash {{{\textbf {D}}}})\otimes _{{{\textbf {D}}}}U\nonumber \\ {}&\cong _{G} F\otimes _{{{\textbf {C}}}}N(-\backslash {{{\textbf {C}}}}/-)\otimes _{{{\textbf {C}}}}T\otimes _{{{\textbf {D}}}}N(-\backslash {{{\textbf {D}}}}/-)\otimes _{{{\textbf {D}}}}U\nonumber \\ {}&\cong _{G} B(F,{{\textbf {C}}},B(T,{{\textbf {D}}},U)). \end{aligned}$$

(10)

whose non-equivariant version, and in the context of diagrams of topological spaces, is 3.1.3 from [5].

The constant functor $*$ clearly has a unique structure of G-functor. One has the isomorphism of G-functors:

$$\begin{aligned} *\otimes _{{{\textbf {C}}}} N(-\backslash {{{\textbf {C}}}}/-)\cong N(-\backslash {{\textbf {C}}}) \end{aligned}$$

(11)

which by (9), leads to the equivariant isomorphism, for a functor $F:{{\textbf {C}}}\rightarrow s {\textbf {Set}}$:

$$\begin{aligned} {{\,\textrm{hocolim}\,}}F=B(*,{\textbf{C}},F)\cong _{G}N(-{\textbf{C}})\otimes _{{\textbf{C}}} F, \end{aligned}$$

(12)

and so this last expression could also be taken as the definition of the homotopy colimit.

Note that the morphism of G-functors $N(-\backslash {{\textbf {C}}})\rightarrow *$ induces an equivariant map

$$\begin{aligned} {{\,\text {hocolim}\,}}F\cong N(-\backslash {{\textbf {C}}})\otimes _{{{\textbf {C}}}} F\rightarrow *\otimes _{{{\textbf {C}}}}F \cong {{\,\text {colim}\,}}F, \end{aligned}$$

(13)

and the morphism of G-functors $F\rightarrow *$ induces an equivariant map

$$\begin{aligned} {{\,\text {hocolim}\,}}F \cong N(-\backslash {{\textbf {C}}})\otimes _{{{\textbf {C}}}} F\rightarrow N(-\backslash {{\textbf {C}}})\otimes _{{{\textbf {C}}}}* \cong N({{\textbf {C}}}) \end{aligned}$$

(14)

Finally, we note that by similar categorical arguments, one can obtain:

Proposition 4.3

Let $U:{\textbf{D}}\rightarrow {\textbf{C}}$ be an equivariant functor between G-categories, and let $F:{\textbf{C}}\rightarrow s \textbf{Set}$ a G-functor. Then, with the induced G-functor structure on $F\circ U$, we have:

• (1) $N(-\backslash {\textbf{D}})\otimes _{{\textbf{D}}}\hom _{{\textbf{C}}}(C,U-) \cong B(*,{\textbf{D}},\hom _{{\textbf{C}}}(C,U-))\cong N(C\backslash U )$ as G-functors on C.

• (2) $(F\circ U)(D)\cong \hom _{{\textbf{C}}}(-,UD)\otimes _{{\textbf{C}}}F\cong B(\hom _{{\textbf{C}}}(-,UD),{\textbf{C}},F)$, as G-functors on D. $\square$

Here we consider a functor with values in the category of sets, such as $\hom _{{\textbf{C}}}(-,UD)$, as a functor with values in $s \textbf{Set}$, by identifying a set X with the simplicial set given by the constant functor ${\varvec{\Delta } ^\textrm{op}}\rightarrow \textbf{Set}$ with value X.

As a consequence of Proposition 4.3, if we take $U=1_{{\textbf{C}}}$ to be the identity functor, we obtain that for all $C\in {{\,\textrm{obj}\,}}{\textbf{C}}$,

$$\begin{aligned} B(*,{\textbf{C}},\hom _{{\textbf{C}}}(C,-))\cong _{G_C} N(C\backslash {C})\simeq _{G_C} *, \end{aligned}$$

(15)

given that $[\backslash {{F}}{\textbf{C}}]{C}$ has an initial object $1_{C}:C\rightarrow$ fixed by $G_{C}$ ([12, (4.3)]).

5 The homotopy invariance theorem

The proofs of the theorems of the next section are based on Theorem 5.2. The reader may refer to [10, Chapter I, Section 4.] for the basic properties of induced topological spaces. For completeness, we state here results needed from [12]. A morphism between G-simplicial sets is a weak G-homotopy equivalence if it induces a G-homotopy equivalence of topological spaces when passing to geometric realizations.

Theorem 5.1

(Theorem 3.5 from [12]) Let $\phi :X\rightarrow Y$ a map of bisimplicial G-sets. Suppose that for all n we have that $\phi _n:X_n\rightarrow Y_n$ is a weak G-homotopy equivalence. Then ${{\,\textrm{diag}\,}}{\phi }$ is a weak G-homotopy equivalence.

Sketch of proof

The non-equivariant version of this statement is [4, Chapter IV, Proposition 1.9.]. From there, one obtains the desired result using a result from Bredon ([1]), namely, that a G-equivariant map is a G-homotopy equivalence whenever the induced map between fixed point sets is a homotopy equivalence for every subgroup H of G. $\square$

Theorem 5.2

Let $S,S':{\textbf{C}}^{\textrm{op}}\times {\textbf{C}}\rightarrow s \textbf{Set}$ be G-functors. Let $\epsilon :S\rightarrow S'$ be a morphism of G-functors such that $\epsilon _{(X,Y)}:S(X,Y)\rightarrow S'(X,Y)$ is a $G_{(X,Y)}$-homotopy equivalence for all $X\in {{\,\textrm{obj}\,}}{\textbf{C}},Y\in {{\,\textrm{obj}\,}}{\textbf{C}}^{\textrm{op}}$. Then the map ${\bar{\epsilon }}$ induced by $\epsilon$:

$$\begin{aligned} {\bar{\epsilon }}:B({\textbf{C}},S)\rightarrow B({\textbf{C}},S') \end{aligned}$$

(16)

is a G-homotopy equivalence.

Proof

From [9], we know that $B({\textbf{C}},S)$ is the diagonal of a bisimplicial set ${{\tilde{B}}}({\textbf{C}},S)$ with (m, n)-simplices the set

$$\begin{aligned} \bigsqcup _{X_0\xrightarrow {\phi _1} X_1\xrightarrow {\phi _2}\cdots \xrightarrow {\phi _m} X_m} S(X_{m},X_{0})_{n}. \end{aligned}$$

(17)

This coproduct has an action of G defined by:

$$\begin{aligned} g(\phi _{1},\ldots ,\phi _{m};z)= \bigl (g\phi _{1},\ldots ,g\phi _{m};\eta _{g,(X_{m},X_0)}(z)\bigr ), \end{aligned}$$

(18)

which makes ${{\tilde{B}}}({\textbf{C}},S)$ a bisimplicial G-set. It follows that $\epsilon$ induces a map ${{\tilde{\epsilon }}}:{\tilde{B}}({\textbf{C}},S)\rightarrow {{\tilde{B}}}({\textbf{C}},S')$, sending

$$\begin{aligned} (\phi _{1},\ldots ,\phi _{m};z)\mapsto \bigl (\phi _{1},\ldots ,\phi _{m};\epsilon _{(X_m,X_0)}(z)\bigr ), \end{aligned}$$

(19)

The map ${{\tilde{\epsilon }}}$ is equivariant, and so if we define ${\bar{\epsilon }}$ as ${{\,\textrm{diag}\,}}{{\tilde{\epsilon }}}$, then ${\bar{\epsilon }}$ is equivariant as well.

Let us denote $X_0\xrightarrow {\phi _1} X_1\xrightarrow {\phi _2}\cdots \xrightarrow {\phi _m} X_m\in N({\textbf{C}})_{m}$ by ${\bar{X}}$. According to Theorem 5.1, to prove that ${\bar{\epsilon }}$ is a G-homotopy equivalence, it is sufficient to prove that

$$\begin{aligned} {{\tilde{\epsilon }}}_{m,-}:\bigsqcup _{{\bar{X}}\in N({\textbf{C}})_{m}} S(X_{m},X_{0}) \rightarrow \bigsqcup _{{\bar{X}}\in N({\textbf{C}})_{m}} S'(X_{m},X_{0}) \end{aligned}$$

(20)

is a G-homotopy equivalence for all m. Taking geometric realization on both sides of (20), since geometric realization commutes with coproducts, we obtain:

$$\begin{aligned} |{{\tilde{\epsilon }}}_{m,-}|:\bigsqcup _{{\bar{X}}\in N({\textbf{C}})_{m}} |S(X_{m},X_{0})| \rightarrow \bigsqcup _{{\bar{X}}\in N({\textbf{C}})_{m}} |S'(X_{m},X_{0})| \end{aligned}$$

(21)

Let $E_m$ be a set of representatives for the orbits of the action of G on $N({\textbf{C}})_m$. Then the map in (21) can be written as:

$$\begin{aligned} {|{\tilde{\epsilon }}_{m,-}|}\!\uparrow _{G_{{\bar{Y}}}}^{G}:\bigsqcup _{{\bar{Y}}\in E_m} {|S(X_{m},X_{0})|}\!\uparrow _{G_{{\bar{Y}}}}^{G}\rightarrow \bigsqcup _{{\bar{Y}}\in E_m} {|S'(X_{m},X_{0})|}\!\uparrow _{G_{{\bar{Y}}}}^{G} \end{aligned}$$

(22)

Since by hypothesis, each $\epsilon _{(X_{m},X_{0})}$ is a $G_{(X_{m},X_{0})}$-homotopy equivalence, given that $G_{{\bar{Y}}}\le G_{(X_{m},X_{0})}$, they are also $G_{{\bar{Y}}}$-homotopy equivalences, and so each map

$$\begin{aligned} {|S(X_{m},X_{0})|}\!\uparrow _{G_{{\bar{Y}}}}^{G}\rightarrow {|S'(X_{m},X_{0})|}\!\uparrow _{G_{{\bar{Y}}}}^{G} \end{aligned}$$

(23)

is a G-homotopy equivalence. Therefore the map in (22) is a coproduct of G-homotopy equivalences, hence a G-homotopy equivalence. $\square$

6 Equivariant cofinality and push-down theorems for $G$-functors

Some of the proofs in these section are as those in [5], adapted for the case of the group action, and in the context of diagrams of simplicial sets. We include more details than in the cited paper, given that homotopy colimit methods have been used by combinatorialists, see for example [13].

Theorem 6.1

(Equivariant Homotopy Invariance Of The Homotopy Colimit). Let $F,F':{\textbf{C}}\rightarrow s \textbf{Set}$ G-functors, and $\epsilon :F\rightarrow F'$ a morphism of G-functors such that each $\epsilon _{X}:FX\rightarrow F'X$ is a $G_{X}$-homotopy equivalence. Then the induced morphism ${\bar{\epsilon }}:{{\,\textrm{hocolim}\,}}F\rightarrow {{\,\textrm{hocolim}\,}}F'$ is a G-homotopy equivalence.

Proof

Straightforward from Theorem 5.2, since the homotopy colimit is a special case of a bar construction. $\square$

Theorem 6.2

(Reduction Theorem) Let $U:{\textbf{D}}\rightarrow {\textbf{C}}$ be an equivariant functor between G-categories, and let $F:{\textbf{C}}\rightarrow s \textbf{Set}$ a G-functor. Then we have the equivariant isomorphism.

$$\begin{aligned} {{\,\text {hocolim}\,}}F\circ U\cong _{G} N(-\backslash U)\otimes _{{{\textbf {C}}}}F \end{aligned}$$

(24)

Proof

$$\begin{aligned} {{\,\textrm{hocolim}\,}}F\circ U&\cong N(-\backslash {\textbf{D}})\otimes _{{\textbf{D}}} (F\circ U){} & {} \text {Equation}~(12)\\&\cong _{G} N(-\backslash {\textbf{D}})\otimes _{{\textbf{D}}} (\hom _{{\textbf{C}}}(-,UD)\otimes _{{\textbf{C}}}F){} & {} \text {Proposition}~4.3.(2)\\&\cong _{G} (N-\backslash {\textbf{D}})\otimes _{{\textbf{D}}} \hom _{{\textbf{C}}}(-,UD))\otimes _{{\textbf{C}}}F{} & {} \text {Fubini's theorem}\\&\cong _{G} N(-\backslash U)\otimes _{{\textbf{C}}}F{} & {} \text {Proposition}~4.3.(1)\square \end{aligned}$$

$\square$

In [6, (2.6)], this result is given as a homotopy equivalence. However, as noted in [5, 4.4] in the context of diagrams of topological spaces, this is even an isomorphism, which in this case is equivariant.

Theorem 6.3

(Cofinality Theorem) Let $U:{\textbf{D}}\rightarrow {\textbf{C}}$ be an equivariant functor between G-categories, and let $F:{\textbf{C}}\rightarrow s \textbf{Set}$ a G-functor. Consider the induced G-functor structure on $F\circ U$. If $N(C\backslash U)$ is $G_{C}$-contractible for all objects C in ${\textbf{C}}$, then ${{\,\textrm{hocolim}\,}}F\circ U\simeq _{G}{{\,\textrm{hocolim}\,}}F$.

Proof

$$\begin{aligned} {{\,\textrm{hocolim}\,}}F\circ U&= B(*,{\textbf{D}},F\circ U)\\&=B(*,{\textbf{D}},B(\hom _{{\textbf{C}}}(-,UD),{\textbf{C}},F)){} & {} \text {Proposition}~(4.3).(2)\\&\cong _{G} B(B(*,{\textbf{D}},\hom _{{\textbf{C}}}(C,U-)),{\textbf{C}},F){} & {} \text {Equation}~10\\&\cong _{G} B(N(-\backslash U),{\textbf{C}},F){} & {} \text {Proposition}~.(4.3).(1)\\&\simeq _{G} B(*,{\textbf{C}},F)={{\,\textrm{hocolim}\,}}F{} & {} \text {Hypothesis} \end{aligned}$$

$\square$

Here we also used the equivariant homotopy invariance (Theorem 5.2) of the bar construction in the last step, together with the $G_{C}$-contractibility of the fiber $N(C\backslash S)$ to obtain the equivariant homotopy. In [6, (2.7)], without assuming the equivariant contractibility, this result is given as a homotopy equivalence not necessarily equivariant.

Theorem 6.4

(Homotopy Pushdown Theorem) Let $U:{\textbf{D}}\rightarrow {\textbf{C}}$ be an equivariant functor and $F:{\textbf{D}}\rightarrow s \textbf{Set}$ a G-functor. Let $U_{h_*}(F):{\textbf{C}}\rightarrow s \textbf{Set}$ the functor given by $C\mapsto B(\hom _{{\textbf{C}}}(U-,C),{\textbf{D}},F)$. Then $U_{h_*}(F)$ is a G-functor and ${{\,\textrm{hocolim}\,}}U_{h_*}(F)\simeq _{G}{{\,\textrm{hocolim}\,}}F$.

Proof

$$\begin{aligned} {{\,\textrm{hocolim}\,}}U_{h_*}(F)&=B(*,{\textbf{C}},B(\hom _{{\textbf{C}}}(U-,C),{\textbf{D}},F))\\&\cong _{G} B(B(*,{\textbf{C}},\hom _{{\textbf{C}}}(UD,-)),{\textbf{D}},F){} & {} \text {Equation}~10\\&\simeq _{G} B(*,{\textbf{D}},F)={{\,\textrm{hocolim}\,}}F{} & {} \text {Equation}~15 \end{aligned}$$

$\square$

We again used Theorem 5.2 in the last step. Hence in [6, (2.5)] we do have a G-homotopy equivalence.

As an example of the application of Theorem 6.1, consider a simplicial set X that has two isomorphic proper subsimplicial sets $X_{1}$, $X_{2}$ that have union X and nonempty intersection $X_{1}\cap X_{2}$. The isomorphism $\phi :X_{1}\rightarrow X_{2}$ must be the identity when restricted to $X_{1}\cap X_{2}$. Let P be the partially ordered set with three points x, y, z, and let $x<y$, $x<z$ be the only nontrivial order relations. The poset P can be seen as a category $\textbf{P}$ ([7, Section 2 of Chapter I]), and in fact is a G-category when $G=C_{2}=\langle a\rangle$, the cyclic group of order 2, interchanges b with c and fixes a. We have a diagram $D:\textbf{P}\rightarrow s \textbf{Set}$ that sends x to $X_{1}\cap X_{2}$, y to $X_{1}$ and z to $X_{2}$. The morphism corresponding to $x<y$ is sent by D to the inclusion of $X_{1}\cap X_{2}$ into $X_{1}$, and similarly $x<z$. In this case, ${{\,\textrm{hocolim}\,}}D$ is equal to $X_{1}\cup X_{2}=X$ (see [3, Section 4]). To make D a G-functor, we define $\eta _{a,x}$ as the identity, $\eta _{a,y}:X_{1}\rightarrow X_{2}$ as $\phi$, and $\eta _{a,z}$ as $\phi ^{-1}$. Suppose there is another simplicial set Y that can be written as union of $Y_{1}, Y_{2}$, analogously to X, with an isomorphism $\psi :Y_{1}\rightarrow Y_{2}$, and there are homotopy equivalence $f_{i}:X_{i}\rightarrow Y_{i}$, for $i=1,2$, such that they coincide on $X_{1}\cap X_{2}$ and the common restriction gives a homotopy equivalence $X_{1}\cap X_{2}\rightarrow Y_{1}\cap Y_{2}$. We may then form a G-diagram $D'$ with ${{\,\textrm{hocolim}\,}}D'=Y$, and have a morphism of G-functors $\epsilon :D\rightarrow D'$ with $\epsilon _{y}=f_{1}$, $\epsilon _{z}=f_{2}$, and $\epsilon _{x}$ the restriction of $f_{1}$ to $X_{1}\cap X_{2}$. Since both stabilizers $G_{y}$ and $G_{z}$ are trivial, by Theorem 6.1 we conclude that X is homotopy equivalent (in fact, G-homotopy equivalent) to Y.