nLab Borel model structure



Model category theory

model category, model \infty -category



Universal constructions


Producing new model structures

Presentation of (,1)(\infty,1)-categories

Model structures

for \infty-groupoids

for ∞-groupoids

for equivariant \infty-groupoids

for rational \infty-groupoids

for rational equivariant \infty-groupoids

for nn-groupoids

for \infty-groups

for \infty-algebras

general \infty-algebras

specific \infty-algebras

for stable/spectrum objects

for (,1)(\infty,1)-categories

for stable (,1)(\infty,1)-categories

for (,1)(\infty,1)-operads

for (n,r)(n,r)-categories

for (,1)(\infty,1)-sheaves / \infty-stacks

Group Theory



Given a topological group GG, the Borel model structure is a model category structure on the category of topological G-spaces, hence of topological spaces equipped with continuous group actions.

Analogously, given a simplicial group G G_\bullet, the Borel model structure is a model category structure on the category of simplicial group actions, hence of simplicial sets equipped with GG-action

Both of these present the (∞,1)-category of ∞-actions of the ∞-group (see there) presented by GG.

In the context of equivariant homotopy theory this is also called the “coarse model structure” (e.g. Guillou 2006, section 5), since in general it has more weak equivalences than the fine model structure on topological G-spaces that enters Elmendorf's theorem.


In topological spaces

Throughout, write Top for the category of compactly generated weak Hausdorff spaces.


For GGrp(TopSp)G \,\in\, Grp(TopSp) a topological group, write

(1)BGTopCat \mathbf{B}G \;\in\; TopCat

for the Top-enriched category with a single object and GG as its unique hom-object.


There is an evident isomorphism of enriched categories

(2)GAct(TopSp)TopFnctr(BG,TopSp) G Act(TopSp) \;\simeq\; TopFnctr\big( \mathbf{B}G,\, TopSp \big)

between topological G-spaces and the Top-enriched functor category from BG\mathbf{B}G (1) to Top (topological presheves).


For GGrp(TopSp)G \in Grp(TopSp) a topological group there is a model category-structure

GAct(TopSp) projMdlCat G Act\big(TopSp\big)_{proj} \;\;\; \in \; MdlCat

on the category of topological G-spaces whose weak equivalences and fibrations are those morphisms whose underlying continuous functions are so in the classical model structure on topological spaces.

For discrete groups this may be argued as in Guillou 2006, Thm. 5.1. For general topological groups this follows as a special case of the projective model structure on Top-enriched functor (see this Thm.), under the identification GAct(TopSp)TopFun(BG,TopSp)G Act(TopSp) \,\simeq\, TopFun( \mathbf{B}G, TopSp ) (2).

In simplicial sets


For G G_\bullet a simplicial group write

This is the G G_\bullet Borel model structure, naturally a simplicial model category (DDK 80, Prop. 2.4, Goerss & Jardine 09, Chapter V, Thm. 2.3).

In combinatorial simplicial model categories

More generally, if C\mathbf{C} is an sSet-enriched category which is also tensored over sSet, via an enriched functor

sSet×CC, sSet \times \mathbf{C} \longrightarrow \mathbf{C} \,,

then for 𝒢Grp(sSet)\mathcal{G} \in Grp(sSet) the enriched hom-isomorphism of the tensoring

Hom(𝒢,C(𝒱,𝒱))Hom(𝒢𝒱,𝒱) Hom\big( \mathcal{G} ,\, \mathbf{C}(\mathscr{V},\,\mathscr{V}) \big) \;\;\simeq\;\; Hom\big( \mathcal{G} \cdot \mathscr{V} ,\, \mathscr{V} \big)

shows that sSet-enriched functors

F:B𝒢C F \,\colon\, \mathbf{B}\mathcal{G} \longrightarrow \mathbf{C}

may equivalently be thought of as simplicial group actions

𝒢𝒱ρ𝒱 \mathcal{G} \cdot \mathscr{V} \overset{\rho}{\longrightarrow} \mathscr{V}

of 𝒢\mathcal{G} acting on objects 𝒱F(*)\mathscr{V} \coloneqq F(\ast) via tensoring.

If now C\mathbf{C} is in fact a combinatorial simplicial model category, then the respective projective model structure on functors exists (that Prop.), which we may hence think of as the Borel model structure on 𝒢\mathcal{G}-actions on objects of C\mathbf{C}:

(3)𝒢Act(C) BorelsFunc(B𝒢,C) proj. \mathcal{G}Act(\mathbf{C})_{Borel} \,\coloneqq\, sFunc(\mathbf{B}\mathcal{G},\,\mathbf{C})_{proj} \,.


In topological spaces

Cofibrations and Cofibrant replacement


The model category GAct(TopSp Qu) projG Act\big(TopSp_{Qu}\big)_{proj} from Prop. is cofibrantly generated with generating cofibrations being (see this Def.) the product with GG (regarded with its free left multiplication action) of the generating cofibrations of TopSp Qu TopSp_{Qu} .

For discrete GG a statement along these lines appears as Guillou 2006, Prop. 5.3.

This is a special case of this Thm. about topological functor categories.

Alternatively, the statement is a special case of that for the fine model structure on topological G-spaces for the case of trivial family of closed subgroups (see there).


Since the universal principal space EGE G (the topological realization EG=|WG|E G \,=\, \big\vert W G\big\vert of the universal principal simplicial complex) is

Prop. implies that the product with EGE G in G Act(TopSp) (i.e. the product topological space with the induced diagonal action) serves as cofibrant replacement in GAct(TopSp)G Act(TopSp):

CofX×EGWFibpr 1XGAct(TopSp Qu) proj. \varnothing \underset{\;\in Cof\;}{\longrightarrow} X \times E G \underoverset{\;\in W \cap Fib\;}{pr_1}{\longrightarrow} X \;\;\; \in G Act\big(TopSp_{Qu}\big)_{proj} \,.

Hm, this is not a proper argument…

Topological Borel construction


There is a Quillen adjunction

TopSp Qutriv()/GGAct(TopSp Qu) proj TopSp_{Qu} \underoverset {\underset{triv}{\longrightarrow}} {\overset{(-)/G}{\longleftarrow}} {\bot} G Act\big(TopSp_{Qu}\big)_{proj}

between the classical model structure on topological spaces and the projective Borel model structure from Prop. , whose

(Guillou 2006, Ex. 5.5)

By definition of the weak equivalences and fibrations in Prop. , it is immediate that trivtriv preserves these classes of morphisms.


(Borel construction of free action is weak hom. equivalent to plain quotient space)
Let GG be a compact Lie group and let XX be a G-CW complex whose GG-action is free. Then the comparison morphism between the Borel construction and the plain quotient space of XX is a weak homotopy equivalence:

(X×EG)/G(id x×(EG*))/GWX/G. (X \times E G)/G \xrightarrow{ \; (id_x \times (E G \to \ast))/G \, \in\, W \; } X/G \,.


Since EGE G is a G-CW complex, the product X×EGX \times E G is cofibrant (by the analog of this Prop.) and XX is cofibrant by assumption and by Prop. .

Hence, by Prop. , the morphism in question is the image under a left Quillen functor, of a weak equivalence between cofibrant objects. Therefore the claim follows by Ken Brown's lemma (here).


The Borel construction exhibits the left derived functor of the quotient space-left Quillen functor in Prop. :

XAAct(TopSp)(𝕃()/G)(X)EG×XGHo(GAct(TopSp Qu) proj). X \,\in\, A Act(TopSp) \;\;\;\; \Rightarrow \;\;\;\; \big(\mathbb{L}(-)/G\big)(X) \;\simeq\; \frac{E G \times X}{G} \;\;\; \in \; Ho\Big( G Act\big( TopSp_{Qu} \big)_{proj} \Big) \,.


Since the left derived functor of a left Quillen functor is given by the application of the latter on any cofibrant replacement, the claim follows by Ex. .

or would follow, if that Example were argued properly

In simplicial sets

Cofibrant replacement and homotopy quotients/fixed points


(cofibrations of simplicial actions)
The cofibrations i:XYi \colon X \to Y in sSetCat(BG ,sSet) projsSetCat\big(\mathbf{B}G_\bullet, sSet\big)_{proj} (Def. ) are precisely those morphisms such that

  1. the underlying morphism of simplicial sets is a monomorphism;

  2. the G G_\bullet-action is a relatively free action, i.e. free on all simplices not in the image of ii.

This is (DDK 80, Prop. 2.2. (ii), Guillou 2006, Prop. 5.3, Goerss & Jardine 09, V Lem. 2.4 & Cor. 2.10).


In particular this means that an object is cofibrant in sSetCat(BG ,sSet) projsSetCat\big(\mathbf{B}G_\bullet, sSet\big)_{proj} if the G G_\bullet-action on it is free.

Hence cofibrant replacement is obtained by forming the product with the model WG W G_\bullet for the total space of the universal principal bundle over G G_\bullet (see at simplicial group for notation and more details).


It follows that for X,AsSetCat(BG ,sSet) projX, A \in sSetCat\big(\mathbf{B}G_\bullet, sSet\big)_{proj} the derived hom space

RHom G(X,A) R Hom_G(X,A)

models the Borel GG-equivariant cohomology of XX with coefficients in AA.

In particular, if AA is fibrant (the underlying simplicial set is a Kan complex) then:

  1. if the G G_\bullet-action on AA is trivial, then

    RHom G(X,A)Hom G(WG×X,A)Hom(WG× GX,A) R Hom_G(X,A) \simeq Hom_G(W G \times X , A) \simeq Hom(W G \times_G X, A)

    is equivalently maps of simplicial sets out of the Borel construction on XX;

  2. if X=*X = \ast is the point then

    RHom G(X,A)Hom G(WG,A)Hom(W¯G,A)A hG R Hom_G(X,A) \simeq Hom_G(W G, A) \simeq Hom(\overline{W} G , A) \simeq A^{h G}

    is the homotopy fixed points of AA.

Relation to the slice over the simplicial classifying space


For GG a simplicial group, there is a pair of adjoint functors

(4)GAct(sSet Qu) proj(()×WG)/G()× W¯GWG(sSet Qu) /W¯G G Act\big(sSet_{Qu}\big)_{proj} \underoverset {\underset{ \big((-) \times W G\big)/G }{\longrightarrow}} {\overset{ (-) \times_{\overline{W}G} W G }{\longleftarrow}} {\bot} \big(sSet_{Qu}\big)_{/\overline{W}G}

which constitute a simplicial Quillen equivalence between the Borel model structure (Def. ) and the slice model structure of the classical model structure on simplicial sets, sliced over the simplicial classifying space W¯G\overline{W}G.

(this is essentially the statement of DDK 80, Prop. 2.3, Prop. 2.4)


(This may also be understood as an instance of the “fundamental theorem of \infty -topos theory”, see there.)


Consider any morphism in sSet /W¯GsSet_{/\overline{W}G}:

X f Y. c X c Y W¯G \array{ X && \xrightarrow{\;\;f\;\;} && Y \mathrlap{\,.} \\ & {}_{\mathllap{c_X}}\searrow && \swarrow_{\mathrlap{c_Y}} \\ && \overline{W}G }

Its image under the left adjoint functor is, by definition, the top left arrow in the following commuting diagram:

Here the right square and the total rectangle are Cartesian squares (pullback squares), by defnition of the functor. It follows by the pasting law that also the square on the left is cartesian. Specifically, since fibrations are preserved under pullback, as shown, the top left morphism in question is the pullback of ff along a fibration.

It follows that (f)×W¯GWG(f) \underset{\overline{W}G}{\times} W G is:

  1. a weak equivalence if ff is a weak equivalence, because the classical model structure on simplicial sets is a right proper model category (see here);

  2. a monomorphism if ff is a monomorphism, since monomorphisms are preserved by pullback (see here).

    Moreover, since WGW¯GW G \to \overline{W}G is the universal principal bundle, it follows that c Y *(WG)Yc_Y^\ast(W G) \to Y is a simplicial principal bundle, so that, in particular, the action of GG on c Y *(WG)c_Y^\ast(W G) is free.

    By Prop. this means that (f)×W¯GWG(f) \underset{\overline{W}G}{\times} W G is a cofibration if ff is a cofibration.

In summary, the left adjoint functor in (4) preserves the classes of weak equivalences and of cofibrations, hence also that of acyclic cofibrations, and so it is a left Quillen functor.


In fact, these functors (4) are sSet-enriched functors which induce an equivalence of ( , 1 ) (\infty,1) -categories between the simplicial localizations L WsSetCat(BG ,sSet) projL WsSet /W¯HL_W sSetCat\big(\mathbf{B}G_\bullet, sSet\big)_{proj} \simeq L_W sSet_{/\overline{W}H} (DDK 80, Prop. 2.5).

This kind of relation is discussed in more detail at ∞-action.


(sSet-enrichement of the adjunction)
The statement that (4) is an sSet-enriched adjunction is not made explicit in DDK 80; there it only says that the functors form a plain adjunction (DDK 80, Prop. 2.3) and that they are each sSet-enriched functors (DDK 80, Prop. 2.4).

The remaining observation that we have a natural isomorphism of sSet-hom-objects

[X× W¯GWG,V][X,(V×WG)/G] \big[ X \times_{\overline{W}G} W G, \, V \big] \;\simeq\; \big[ X, \, (V \times W G)/G \big]


Hom((X× W¯GWG)×Δ[],V)Hom(X×Δ[],(V×WG)/G) Hom \Big( \big( X \times_{\overline{W}G} W G \big) \times \Delta[\bullet], \, V \Big) \;\simeq\; Hom \big( X \times \Delta[\bullet], \, (V \times W G)/G \big)

follows from the plain adjunction and the natural isomorphism

(X× W¯GWG)×Δ[](X×Δ[])× W¯GWG, (X \times_{\overline{W}G} W G) \times \Delta[\bullet] \;\simeq\; (X \times \Delta[\bullet]) \times_{\overline{W}G} W G \,,

which, in turn, follows, for instance, via the pasting law:

Relation to the model structure on plain simplicial sets

For 𝒢Groups(sSets)\mathcal{G} \,\in\, Groups(sSets) a simplicial group, write 𝒢Actions(sSets)\mathcal{G}Actions(sSets) for the category of 𝒢\mathcal{G}-actions on simplicial sets.


(underlying simplicial sets and cofree simplicial action)
The forgetful functor undrlundrl from 𝒢Actions\mathcal{G}Actions to underlying simplicial sets is a left Quillen functor from the Borel model structure (Def. ) to the classical model structure on simplicial sets.

Its right adjoint

sSet[𝒢,]undrl𝒢Actions(sSet) sSet \underoverset {\underset{ \;\;\; [\mathcal{G},-] \;\;\; }{\longrightarrow}} {\overset{ \;\;\; undrl \;\;\; }{\longleftarrow}} {\bot} \mathcal{G}Actions(sSet)

sends 𝒳sSet\mathcal{X} \in sSet to

  • the simplicial set

    [𝒢,𝒳]Hom sSet(𝒢×Δ[],𝒳)sSet [\mathcal{G},\mathcal{X}] \;\coloneqq\; Hom_{sSet}\big( \mathcal{G} \times \Delta[\bullet], \mathcal{X}\big) \;\;\; \in sSet
  • equipped with the 𝒢\mathcal{G}-action

    𝒢×[𝒢,𝒳]()()𝒢 \mathcal{G} \times [\mathcal{G},\mathcal{X}] \overset{ (-) \cdot (-) }{\longrightarrow} \mathcal{G}

    which in degree nn \in \mathbb{N} is the function

    (5)Hom(Δ[n],𝒢)×Hom(𝒢×Δ[n],𝒳)Hom(𝒢×Δ[n],𝒳) Hom(\Delta[n], \mathcal{G}) \,\times\, Hom \big( \mathcal{G} \times \Delta[n], \, \mathcal{X} \big) \longrightarrow Hom \big( \mathcal{G} \times \Delta[n], \, \mathcal{X} \big)

    that sends

    (6) (Δ[n]g n𝒢,𝒢×Δ[n]ϕ𝒳,) (𝒢×Δ[n]id×diag𝒢×Δ[n]×Δ[n]id×g n×id𝒢×𝒢×Δ[n]()()×id𝒢×Δ[n]ϕ𝒳) \begin{aligned} & \Big( \Delta[n] \overset{g_n}{\to} \mathcal{G}, \; \mathcal{G}\times \Delta[n] \overset{\phi}{\to} \mathcal{X}, \Big) \\ \;\;\mapsto\;\; & \Big( \mathcal{G} \times \Delta[n] \overset{id \times diag}{\longrightarrow} \mathcal{G} \times \Delta[n] \times \Delta[n] \overset{ id \times g_n \times id }{\longrightarrow} \mathcal{G} \times \mathcal{G} \times \Delta[n] \overset{(-)\cdot(-) \times id}{\to} \mathcal{G} \times \Delta[n] \overset{\phi}{\to} \mathcal{X} \Big) \end{aligned}

Here and in the following proof we make free use of the Yoneda lemma natural bijection

Hom sSet(Δ[n],𝒮)𝒮 n Hom_{sSet}(\Delta[n], \mathcal{S}) \;\simeq\; \mathcal{S}_n

for any simplicial set SS and for Δ[n]ΔysSet\Delta[n] \in \Delta \overset{y}{\hookrightarrow} sSet the simplicial n-simplex.


We already know from Def. that underlunderl preserves all weak equivalences and from Prop. that it preserves all cofibrations. Therefore it is a left Quillen functor as soon as it is a left adjoint at all.

The idea of the existence of the cofree right adjoint to undrlundrl is familiar from topological G-spaces (see the section on coinduced actions there), where it can be easily expressed point-wise in point-set topology. The formula (6) adapts this idea to simplicial sets. Its form makes manifest that this gives a simplicial homomorphism, and with this the adjointness follows the usual logic by focusing on the image of the non-degenerate top-degree cell in Δ[n]\Delta[n]:

To check that (6) really gives the right adjoint, it is sufficient to check the corresponding hom-isomorphism, hence to check for 𝒫𝒢Actions(sSet)\mathcal{P} \in \mathcal{G}Actions(sSet), and 𝒳sSet\mathcal{X} \in sSet, that we have a natural bijection of hom-sets of the form

{𝒫ϕ ()[𝒢,𝒳]}()˜{undrl(𝒫)ϕ˜ ()𝒳}. \big\{ \mathcal{P} \overset{\;\;\phi_{(-)}\;\;}{\longrightarrow} [\mathcal{G}, \mathcal{X}] \big\} \;\;\;\overset{ \;\; \widetilde{(-)} \;\; }{\leftrightarrow}\;\;\; \big\{ undrl(\mathcal{P}) \overset{\;\; {\widetilde \phi}_{(-)} \;\; }{\longrightarrow} \mathcal{X} \big\} \,.

So given

ϕ ():p n(ϕ p n:𝒢×Δ[n]𝒳) \phi_{(-)} \;\colon\; p_n \mapsto \big( \phi_{p_n} \;\colon\; \mathcal{G} \times \Delta[n] \to \mathcal{X} \big)

on the left, define

(7)ϕ˜ ():p nϕ p n(e n,σ n)𝒳 n, \widetilde \phi_{(-)} \;\colon\; p_n \mapsto \phi_{p_n}(e_n, \sigma_n) \;\in\; \mathcal{X}_n \,,

where e n𝒢 ne_n \in \mathcal{G}_n denotes the neutral element in degree nn \in \mathbb{N} and where σ n(Δ[n]) n\sigma_n \in (\Delta[n])_n denotes the unique non-degenerate element nn-cell in the n-simplex.

It is clear that this is a natural transformation in PP and XX. We need to show that ϕ˜ ():undrl(P)X{\widetilde \phi}_{(-)} \colon undrl(P) \to X uniquely determines all of ϕ ()\phi_{(-)}.

To that end, observe for any g n𝒢 ng_n \in \mathcal{G}_n the following sequence of identifications:

ϕ p n(g n,σ n) =ϕ p n(e ng n,σ n) =(g nϕ p n)(e n,σ n) =ϕ g np n(e n,σ n) =ϕ˜ g np n \begin{aligned} \phi_{p_n}(g_n, \sigma_n) & \;=\; \phi_{p_n}( e_n \cdot g_n, \sigma_n ) \\ & \;=\; \big( g_n \cdot \phi_{p_n} \big) ( e_n, \sigma_n ) \\ & \;=\; \phi_{ g_n \cdot p_n } (e_n, \sigma_n) \\ & \;=\; {\widetilde \phi}_{g_n \cdot p_n} \end{aligned}


  • the first step is the unit law in the component group 𝒢 n\mathcal{G}_n;

  • the second step uses the definition (6) of the cofree action;

  • the third step is the assumption that ϕ ()\phi_{(-)} is a homomorphism of 𝒢\mathcal{G}-actions (equivariance);

  • the fourth step is the definition (7).

These identifications show that ϕ ()\phi_{(-)} is uniquely determined by ϕ˜ (){\widetilde \phi_{(-)}}, and vice versa.


(B\mathbf{B}\mathbb{Z}-2-action on inertia groupoid)

  • GGroups(Sets)G \in Groups(Sets)

    be a discrete group,

  • XGActions(Sets)X \in G Actions(Sets)

    be a GG-action,

  • 𝒳XGN(X×GX)=X×G × sSet\mathcal{X} \;\coloneqq\; X \sslash G \;\coloneqq\; N( X \times G \rightrightarrows X ) \,=\, X \times G^{\times^\bullet} \in sSet

    the simplicial set which is the nerve of its action groupoid (a model for its homotopy quotient),

  • 𝒢BN(*) × Groups(sSet)\mathcal{G} \,\coloneqq\, \mathbf{B}\mathbb{Z} \,\coloneqq\, N(\mathbb{Z} \rightrightarrows \ast) \,\coloneqq\, \mathbb{Z}^{\times^\bullet} \,\in\, Groups(sSet)

    the simplicial group which is the nerve of the 2-group that is the delooping groupoid of the additive group of integers.

Then the functor groupoid

(8)Λ(XG) [B,XG] Func((*),(X×GX)) W[g]ConjCl(G)(X gC g) \begin{aligned} \Lambda(X \!\sslash\! G) & \;\coloneqq\; \big[ \mathbf{B}\mathbb{Z}, X \!\sslash\! G \big] \\ & \;\simeq\; Func \big( (\mathbb{Z} \rightrightarrows \ast), \, (X \times G \rightrightarrows X) \big) \\ & \;\underset{\in \mathrm{W}}{\leftarrow}\; \underset{ [g] \in ConjCl(G) }{\coprod} \Big( X^{g} \!\sslash\! C_g \Big) \end{aligned}

is known as the inertia groupoid of XGX \!\sslash\! G. Here

ConjCla(G)G/ adG,C g{hG|hg=gh} ConjCla(G) \;\coloneqq\; G/_{ad} G \,, \;\;\;\;\;\;\;\;\;\;\; C_g \;\coloneqq\; \big\{ h \in G \,\left\vert\, h \cdot g = g \cdot h \right. \big\}

denotes, respectively, the set of conjugacy classes of elements of GG, and the centralizer of {g}G\{g\} \subset G – this data serves to express the equivalent skeleton of the inertia groupoid in the last line of (8).

Now, by Prop. the inertia groupoid (8) carries a canonical 2-action of the 2-group B\mathbf{B}\mathbb{Z}:

By the formula (6), for nn \in \mathbb{Z} the 2-group element in degree 1

n:Δ[1]B {\color{purple}n} \;\colon\; \Delta[1] \longrightarrow \mathbf{B} \mathbb{Z}

acts on the morphisms

(x,g)h(hx,g)Λ(XG) (x,g) \overset{h}{\longrightarrow} (h\cdot x, g) \;\;\; \in \; \Lambda(X \!\sslash\! G)

of the inertia groupoid as follows (recall the nature of products of simplices):

Relation to the fine model structure of equivariant homotopy theory

The identity functor gives a Quillen adjunction between the Borel model structure and the final model structure on topological G-spaces? for equivariant homotopy theory (Guillou 2006, section 5).

The left adjoint is

L=id:G Act coarseG Act fine L = id \;\colon\; G_\bullet Act_{coarse} \longrightarrow G_\bullet Act_{fine}

from the Borel model structure to the genuine equivariant homotopy theory.


First of all, by (Guillou 2006, theorem 3.12, example 4.2) sSet BG sSet^{\mathbf{B}G_\bullet} does carry a fine model structure. By (Guillou 2006, last line of page 3) the fibrations and weak equivalences here are those maps which are ordinary fibrations and weak equivalences, respectively, on HH-fixed point simplicial sets, for all subgroups HH. This includes in particular the trivial subgroup and hence the identity functor

R=id:G Act fineG Act coarse R = id \;\colon\; G_\bullet Act_{fine} \longrightarrow G_\bullet Act_{coarse}

is right Quillen.

Generalization to simplicial presheaves

Since the universal simplicial principal complex-construction is functorial

SimplicialGroupsWSimplicialSets SimplicialGroups \xrightarrow{\;\; W \;\;} SimplicialSets

with natural transformations

𝒢iW𝒢pW¯𝒢 \mathcal{G} \xrightarrow{\;\; i \;\;} W\mathcal{G} \xrightarrow{\;\; p \;\;} \overline{W}\mathcal{G}

the pair of adjoint functors (4) extends to presheaves:


For 𝒞\mathcal{C} a small sSet-category with

sPSh(𝒞)sSetCat(𝒞 op,sSet) sPSh(\mathcal{C}) \;\coloneqq\; sSetCat( \mathcal{C}^{op}, \, sSet )

denoting its category of simplicial presheaves, and for

𝒢̲Groups(sPSh(𝒞)) \underline{\mathcal{G}} \;\in\; Groups \big( sPSh(\mathcal{C}) \big)

a group object internal to SimplicialPresheaves with

𝒢̲Acts(sPSh(𝒞)) \underline{\mathcal{G}} Acts \big( sPSh(\mathcal{C}) \big)

denoting its category of action objects internal to SimplicialPresheaves

we have an adjoint pair

𝒢̲Acts(sPSh(𝒞))(()×W𝒢̲)/𝒢̲()× W¯𝒢̲W𝒢̲sPSh(𝒞) /W¯𝒢̲ \underline{\mathcal{G}} Acts \big( sPSh(\mathcal{C}) \big) \underoverset { \underset{ \big( (-) \times W\underline{\mathcal{G}} \big) \big/ \underline{\mathcal{G}} } {\longrightarrow}} { \overset{ (-) \times_{\overline{W}\underline{\mathcal{G}}} W\underline{\mathcal{G}} }{\longleftarrow} } {\bot} sPSh(\mathcal{C})_{/\overline{W}\underline{\mathcal{G}}}


The required hom-isomorphism is the composite of the following sequence of natural bijections:

Hom((X̲,p),(Y̲×W𝒢̲)/𝒢̲) Hom(X̲,(Y̲×W𝒢̲)/𝒢̲)×Hom(X̲,W¯𝒢̲){p} cHom(X̲(c),(Y̲(c)×W𝒢(c)̲)/𝒢̲(c))× cHom(X̲(c),W¯𝒢̲(c)){p} c(Hom(X̲(c),(Y̲(c)×W𝒢̲(c))/𝒢̲(c))×Hom(X̲(c),W¯𝒢̲(c)){p(c)}) cHom /W¯𝒢̲(c)((X̲(c),p(c)),(Y̲(c)×W¯𝒢̲(c))/𝒢(c)) c(𝒢̲(c)Acts(sSet)(X̲(c)×W¯𝒢̲(c)W𝒢̲(c),Y̲(c))) 𝒢Acts(sPSh(𝒞))(X̲×W¯𝒢̲W𝒢̲,Y̲) \begin{aligned} Hom \Big( (\underline{X},p), \, \big( \underline{Y} \times W\underline{\mathcal{G}} \big) / \underline{\mathcal{G}} \Big) & \;\simeq\; Hom \Big( \underline{X}, \, \big( \underline{Y} \times W\underline{\mathcal{G}} \big) / \underline{\mathcal{G}} \Big) \underset{ Hom \Big( \underline{X}, \, \overline{W} \underline{\mathcal{G}} \Big) }{\times} \{p\} \\ & \;\simeq\; \int^c Hom \Big( \underline{X}(c), \, \big( \underline{Y}(c) \times W\underline{\mathcal{G}(c)} \big) / \underline{\mathcal{G}}(c) \Big) \underset{ \int^c Hom \Big( \underline{X}(c), \, \overline{W} \underline{\mathcal{G}}(c) \Big) }{\times} \{p\} \\ & \;\simeq\; \int^c \left( Hom \Big( \underline{X}(c), \, \big( \underline{Y}(c) \times W\underline{\mathcal{G}}(c) \big) / \underline{\mathcal{G}}(c) \Big) \underset{ Hom \Big( \underline{X}(c), \, \overline{W} \underline{\mathcal{G}}(c) \Big) }{\times} \{p(c)\} \right) \\ & \;\simeq\; \int^c Hom_{/\overline{W}\underline{\mathcal{G}}(c)} \Big( \big( \underline{X}(c), p(c)\big), \, \big( \underline{Y}(c) \times \overline{W} \underline{\mathcal{G}}(c) \big)\big/ \mathcal{G}(c) \Big) \\ & \;\simeq\; \int^c \left( \underline{\mathcal{G}}(c) Acts(sSet) \big( \underline{X}(c) \underset{ \overline{W}\underline{\mathcal{G}}(c) }{\times} W \underline{\mathcal{G}}(c), \, \underline{Y}(c) \big) \right) \\ & \;\simeq\; \mathcal{G}Acts(sPSh(\mathcal{C})) \big( \underline{X} \underset{\overline{W}\underline{\mathcal{G}}}{\times} W \underline{\mathcal{G}}, \, \underline{Y} \big) \end{aligned}


𝒢̲Acts(A̲,B̲) 𝒢(c 1)Acts(A̲(c 1),B̲(c 1)) 𝒢(c 2)Acts(A̲(c 2),B̲(c 2)) Hom(A̲(c 1),B̲(c 2)) \array{ \underline{\mathcal{G}}Acts \big( \underline{A}, \, \underline{B} \big) &\longrightarrow& \mathcal{G}(c_1)Acts \big( \underline{A}(c_1), \, \underline{B}(c_1) \big) \\ \big\downarrow && \big\downarrow \\ \mathcal{G}(c_2)Acts \big( \underline{A}(c_2), \, \underline{B}(c_2) \big) &\longrightarrow& Hom \big( \underline{A}(c_1), \, \underline{B}(c_2) \big) }

In combinatorial simplicial model categories

Base change


then its Borel model category (3)

𝒢Act(C) BorelsFunc(B𝒢,C) proj \mathcal{G}Act(\mathbf{C})_{Borel} \,\coloneqq\, sFunc(\mathbf{B}\mathcal{G},\,\mathbf{C})_{proj}

has, by this Prop., (acyclic) generating cofibrations given by tensoring these with 𝒢=(B𝒢)(*,*)\mathcal{G} = (\mathbf{B}\mathcal{G})(\ast, \ast) and understood as equipped with the 𝒢\mathcal{G}-action induced by the regular action of 𝒢\mathcal{G} on itself:

(9)I 𝒢Act(C) 𝒢(I C){𝒢i|iI} J 𝒢Act(C) 𝒢(J C){𝒢j|jJ}, \array{ I_{\mathcal{G}Act(\mathbf{C})} &\coloneqq& \mathcal{G}\cdot(I_{\mathbf{C}}) \;\equiv\; \big\{ \mathcal{G} \cdot i \,\big\vert\, i \in I \big\} \\ J_{\mathcal{G}Act(\mathbf{C})} &\coloneqq& \mathcal{G}\cdot(J_{\mathbf{C}}) \;\equiv\; \big\{ \mathcal{G} \cdot j \,\big\vert\, j \in J \big\} \mathrlap{\,,} }

where we noticed that these are just the images under the (acyclic) generating cofibrations of C\mathbf{C} under the left-induced action

(10)Cundrl𝒢()𝒢Act(C). \mathbf{C} \underoverset {\underset{undrl}{\longleftarrow}} {\overset{\mathcal{G}\cdot(-)}{\longrightarrow}} {\;\; \bot \;\;} \mathcal{G}Act(\mathbf{C}) \,.

But since

  1. the underlying simplicial set of 𝒢\mathcal{G} is necessarily a cofibrant object of sSet Qu sSet_{Qu}

  2. C\mathbf{C} is an sSetsSet-enriched model category

the pushout-product axiom satisfied by the tensoring implies that the underlying morphisms of these (acyclic) generating cofibration in the Borel structure, i.e. forgetting their equivariance under the simplicial group action, are still (acyclic) cofibrations in C\mathbf{C} itself, hence that we also have a Quillen adjunction of the form

(11)C QuRundrl𝒢Act(C) Borel \mathbf{C} \underoverset {\underset{R}{\longrightarrow}} {\overset{undrl}{\longleftarrow}} {\;\;\; \bot_{{}_{Qu}} \;\;\;} \mathcal{G}Act(\mathbf{C})_{Borel}

whose right adjoint may be thought of as the right induced action or equivalently as right Kan extension along the sSet-enriched functor *B𝒢\ast \to \mathbf{B}\mathcal{G}.

Monoidal model structure

We discuss (Prop. below) induced monoidal model category-structure on the Borel model structure, in the generality of coefficients any (cofibrantly generated) simplicial model structure (as above), which in addition carries compatible monoidal simplicial model structure.


For 𝒢Grp(sSet)\mathcal{G} \in Grp(sSet) and C\mathbf{C} a combinatorial simplicial model category with compatible closed monoidal enriched category-structure, i.e. with an sSet-enriched functor ()()(-)\otimes(-), the objectwise tensor product

𝒢Act(C)×𝒢Act(C)sFunc(B𝒢,C)×sFunc(B𝒢,C)sFunc(B𝒢,C×C)sFunc(B𝒢,)sFunc(B𝒢,C) \mathcal{G}Act(\mathbf{C}) \times \mathcal{G}Act(\mathbf{C}) \,\equiv\, sFunc(\mathbf{B}\mathcal{G},\,\mathbf{C}) \times sFunc(\mathbf{B}\mathcal{G},\,\mathbf{C}) \;\simeq\; sFunc(\mathbf{B}\mathcal{G},\,\mathbf{C} \times \mathbf{C}) \overset { sFunc(\mathbf{B}\mathcal{G},\, \otimes) } {\longrightarrow} sFunc(\mathbf{B}\mathcal{G},\,\mathbf{C})

and the objectwise internal hom

𝒢Act(C)×𝒢Act(C)sFunc(B𝒢,C op)×sFunc(B𝒢,C)sFunc(B𝒢,C×C)sFunc(B𝒢,[,])sFunc(B𝒢,C) \mathcal{G}Act(\mathbf{C}) \times \mathcal{G}Act(\mathbf{C}) \,\simeq\, sFunc\big(\mathbf{B}\mathcal{G},\,\mathbf{C}^{op}) \times sFunc(\mathbf{B}\mathcal{G},\,\mathbf{C}) \;\simeq\; sFunc(\mathbf{B}\mathcal{G},\,\mathbf{C} \times \mathbf{C}) \overset { sFunc(\mathbf{B}\mathcal{G},\, [-,-]) } {\longrightarrow} sFunc(\mathbf{B}\mathcal{G},\,\mathbf{C})

make the Borel model structure (3)

(𝒢Act(C),) \big(\mathcal{G}Act(\mathbf{C}), \otimes \big)

a monoidal model category, at least in that the pushout-product axiom holds. A sufficient condition that also the unit axiom hold is that all objects of C\mathbf{C} are cofibrant.

The following argument for the pushout-product axiom follows Berger & Moerdijk (2006), Lem. 2.5.2.

It is sufficient to check the pushout-product axiom on (acyclic) generating cofibrations (by this remark). To that end, given

  • generating cofibrationsAB\;A \to B and XYX \to Y in I 𝒢Act(C)I_{\mathcal{G}Act(\mathbf{C})}, we need to check that their pushout product is still a cofibration, which means that for any

  • acyclic fibrationZW\;Z \to W in WFib 𝒢Act(C)=undrl 1WFib CWFib_{\mathcal{G}Act(\mathbf{C})} = undrl^{-1} WFib_{\mathbf{C}}

we need to find a lift in any commuting square of the form

(AY)⨿AX(BX) Z BY W.in𝒢Act(C). \array{ (A \otimes Y) \overset{A \otimes X}{\amalg} (B \otimes X) &\longrightarrow& Z \\ \Big\downarrow && \Big\downarrow \\ B \otimes Y &\longrightarrow& W \mathrlap{\,.} } \;\;\;\text{in}\;\; \mathcal{G}Act(\mathbf{C}) \,.

By Joyal-Tierney calculus, this is equivalent to finding a lift in the internal hom-adjunct diagram:

A [X,Z] B [Y,W]×[X,W][X,Z]in𝒢Act(C). \array{ A &\longrightarrow& [X,Z] \\ \Big\downarrow && \Big\downarrow \\ B &\longrightarrow& [Y,W] \underset{[X,W]}{\times} [X,Z] } \;\;\;\text{in}\;\; \mathcal{G}Act(\mathbf{C}) \,.

Now observe that the (acyclic) Borel-cofibration on the left is, by (9), in the image of the left adjoint functor 𝒢()\mathcal{G}\cdot(-) whose right adjoint is the forgetful functor remembering just the underlying morphisms in C\mathbf{C}. Therefore a lift in the above diagram is equivalently a lift of its adjunct, which is just the underlying diagram in C\mathbf{C}:

A [Y,Z] B [Y,W]×[X,W][X,Z]inC. \array{ A &\longrightarrow& [Y,Z] \\ \Big\downarrow && \Big\downarrow \\ B &\longrightarrow& [Y,W] \underset{[X,W]}{\times} [X,Z] } \;\;\;\text{in}\;\; \mathbf{C} \,.

But now using

  1. that the underlying map of the previous acyclic fibration is still an acyclic fibration in C\mathbf{C}, by definition of the projective/Borel model structure,

  2. that the underlying maps of the previous cofibrations are still cofibrations in C\mathbf{C}, by (11),

  3. the pullback-power axiom satisfied in the monoidal model category C\mathbf{C}

it follows that the left map in this diagram is a cofibration and the right map is still an acyclic fibration, whence a lift exists by the model category axioms on C\mathbf{C}.

A directly analogous argument applies in the cases where ABA \to B or XYX \to Y are in addition weak equivalences. Hence the pushout-product axiom is verified.

Finally, to verify the unit axiom:

If 𝟙C\mathbb{1} \,\in\, \mathbf{C} denotes the given tensor unit, then the tensor unit in 𝒢Act(C)\mathcal{G}Act(\mathbf{C}) is the constant functor B𝒢*𝟙C\mathbf{B}\mathcal{G} \to \ast \overset{\mathbb{1}}{\longrightarrow} \mathbf{C}, corresponding to the trivial action on 𝟙\mathbb{1}. Writing p const𝟙:Qconst(𝟙)const𝟙p_{const \mathbb{1}} \colon Q const(\mathbb{1}) \to const \mathbb{1} for a cofibrant resolution, we need to see that given an object X𝒢Act(C)X \in \mathcal{G}Act(\mathbf{C}) the composite

(Qconst𝟙)Xp const𝕀id X(const𝟙)XlX (Q const \mathbb{1}) \otimes X \overset{ p_{const \mathbb{I}} \otimes id_X }{\longrightarrow} (const \mathbb{1}) \otimes X \underoverset{\sim}{l}{\longrightarrow} X

is a weak equivalence.

Since weak equivalence are just the underlying weak equivalences, for this it is sufficient (with the assumption that all objects in C\mathbf{C} are cofibrant), that ()X(-) \otimes X is a left Quillen functor on C\mathbf{C}, since as such it preserves all weak equivalences between cofibrant objects, by Ken Brown’s lemma.

But the underlying object of XX is still cofibrant in C\mathbf{C}, by (11), therefore ()X(-) \otimes X is left Quillen by the pushout-product axiom satisfied by the monoidal model category C\mathbf{C}.


In simplicial sets

The model structure, the characterization of its cofibrations, and its equivalence to the slice model structure of sSetsSet over W¯G\bar W G is due to

This Quillen equivalence also mentioned as:

  • William Dwyer, Exercise 4.2 in: Homotopy theory of classifying spaces, Lecture notes, Copenhagen 2008, (pdf, pdf)

Discussion in relation to the “fine” model structure of equivariant homotopy theory which appears in Elmendorf's theorem is in

Textbook account of (just) the Borel model structure:

Discussion with the model of ∞-groups by simplicial groups replaced by groupal Segal spaces is in

Discussion of monoidal model category-enhancement on the Borel model structure:

Discussion of a the integral model structure for actions of all simplicial groups:

In topological spaces

Last revised on November 2, 2023 at 08:25:58. See the history of this page for a list of all contributions to it.