Contents

model category

for ∞-groupoids

group theory

# Contents

## Idea

Given a simplicial group $G_\bullet$, the Borel model structure is a model category structure on the category of simplicial sets equipped with $G$-action which presents the (∞,1)-category of ∞-actions of the ∞-group (see there) presented by $G$.

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

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

## Properties

### Relation to slicing over $\bar W G$

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 $\bar W G_\bullet$ (see at simplicial group for notation and details) for the delooping of $G_\bullet$.

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

### Cofibrant replacement and homotopy quotients/fixed points

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

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

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

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 $G_\bullet$-action on it is free. Hence cofibrant replacement is in particular given by forming the product with the model $W G_\bullet$ for the total space of the universal principal bundle over $G_\bullet$ (see at simplicial group for notation and more details).

It follows that for $X,A\in (sSet^{\mathbf{B}G_\bullet})_{proj}$ the derived hom space

$R Hom_G(X,A)$

models the $G$-equivariant cohomology of $X$ with coefficients in $A$.

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

1. if the $G_\bullet$-action on $A$ is trivial, then

$R Hom_G(X,A) \simeq Hom_G(W G \times X , A) \simeq Hom(W G \times_G X, A)$

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

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

$R Hom_G(X,A) \simeq Hom_G(W G, A) \simeq Hom(\bar W G , A) \simeq A^{h G}$

is the homotopy fixed points of $A$.

### 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

$L = id \;\colon\; G_\bullet Act_{coarse} \longrightarrow G_\bullet Act_{fine}$

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

Because:

First of all, by (Guillou, theorem 3.12, example 4.2) $sSet^{\mathbf{B}G_\bullet}$ 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 $H$-fixed point simplicial sets, for all subgroups $H$. This includes in particular the trivial subgroup and hence the identity functor

$R = id \;\colon\; G_\bullet Act_{fine} \longrightarrow G_\bullet Act_{coarse}$

is right Quillen.

## References

The model structure, the characterization of its cofibrations, and its equivalence to the slice model structure of $sSet$ over $\bar W G$ 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.