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 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 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 homotopy theory of G-spaces which enters Elmendorf's theorem.

Definition

Writing BG \mathbf{B}G_\bullet for the one-object sSet-enriched category (simplicial groupoid) whose hom-object is G G_\bullet. Write sSet BG sSet^{\mathbf{B}G_\bullet} for the corresponding sSet-enriched functor category.

This carries the projective global model structure on functors (model structure on simplicial presheaves) (sSet BG ) proj(sSet^{\mathbf{B}G_\bullet})_{proj}. This is the Borel model structure (DDK 80).

Properties

Relation to slicing over W¯G\bar W G

The central theorem of (DDK 80) is that the Borel model structure is Quillen euqivalent to the slice model structure of the standard model structure on simplicial sets over the model W¯G \bar W G_\bullet (see at simplicial group for notation and details) for the delooping of G G_\bullet.

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

Cofibrant replacement and homotopy quotients/fixed points

The cofibrations i:XYi \colon X \to Y in (sSet BG ) proj(sSet^{\mathbf{B}G_\bullet})_{proj} are precisely those maps such that

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

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

This is part of (DDK 80, proposition 2.2. ii)). Also (Guillou, prop. 5.3).

In particular this means that an object is cofibrant if the G G_\bullet-action on it is free. Hence cofibrant replacement is in particular given 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,A(sSet BG ) projX,A\in (sSet^{\mathbf{B}G_\bullet})_{proj} the derived hom space

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

models the 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(\bar W G , A) \simeq A^{h G}

    is the homotopy fixed points of AA.

Relation to the fine model structure of equivariant homotopy theory

The identity functor 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.

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

  • E. Dror, William Dwyer, and Daniel Kan, Equivariant maps which are self homotopy equivalences, Proc. Amer. Math. Soc. 80 (1980), no. 4, 670–672 (JSTOR)

This is mentioned for instance as exercise 4.2in

  • William Dwyer, Homotopy theory of classifying spaces, Lecture notes Copenhagen (June, 2008) pdf

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

  • Bert Guillou, A short note on models for equivariant homotopy theory (pdf)

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

Created on April 15, 2014 at 01:41:15. See the history of this page for a list of all contributions to it.