nLab
Borel model structure

Contents

Context

Model category theory

model category

Definitions

Morphisms

Universal constructions

Refinements

Producing new model structures

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

Model structures

for \infty-groupoids

for ∞-groupoids

for rational \infty-groupoids

for nn-groupoids

for \infty-groups

for \infty-algebras

general

specific

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

Contents

Idea

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 which presents 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, section 5), since it is not equivalent to the “fine” homotopy theory of G-spaces which enters Elmendorf's theorem.

Definition

Definition

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).

Properties

Cofibrant replacement and homotopy quotients/fixed points

Proposition

(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, Prop. 5.3, Goerss & Jardine 09, V Lem. 2.4).

Remark

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).

Remark

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

Proposition

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

(1)G Acts(sSet) proj(()×WG)/G()× W¯GWGsSet /W¯G G_\bullet Acts(sSet)_{proj} \underoverset {\underset{ \big((-) \times W G\big)/G }{\longrightarrow}} {\overset{ (-) \times_{\overline{W}G} W G }{\longleftarrow}} {\bot} sSet_{/\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 slices over the simplicial classifying space W¯G\overline{W}G.,

(DDK 80, Prop. 2.3, Prop. 2.4) Here:

In fact, these are sSet-enriched functors which induced an equivalence of (infinity,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.

Remark

(sSet-enrichement of the adjunction)
The statement that (1) 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]

hence

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.

Proposition

(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

    (2)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

    (3) (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.

Proof

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 (3) 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 (3) 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

(4)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}

Here:

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

  • the second step uses the definition (3) 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 (4).

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

Example

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

  • 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

(5)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 (5).

Now, by Prop. the inertia groupoid (5) carries a canonical 2-action of the 2-group B\mathbf{B}\mathbb{Z}:

By the formula (3), 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 equivariant homotopy theory (Guillou, 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.

Because:

First of all, by (Guillou, theorem 3.12, example 4.2) sSet BG sSet^{\mathbf{B}G_\bullet} does carry a fine model structure. By (Guillou, 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 (1) extends to presheaves:

Proposition

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}}}

Proof

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}

Here:

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) }

References

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 a globalized model structure for actions of all simplicial groups is in

Last revised on July 4, 2021 at 14:39:01. See the history of this page for a list of all contributions to it.