nLab
fine model structure on topological G-spaces

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 equivariant \infty-groupoids

for rational \infty-groupoids

for rational equivariant \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

Representation theory

Contents

Idea

For GG a topological group, there exists a model category-structure on the category of topological G-spaces whose weak equivalences and fibrations are those morphisms whose underlying continuous functions between HH-fixed loci, for all closed subgroups H clsdGH \subset_{clsd} G, are weak equivalences or fibrations, respectively, in the classical model structure on topological spaces, hence weak homotopy equivalences or Serre fibrations, respectively.

In the case that GG is a compact Lie group, the corresponding homotopy theory coincides with that of G-CW complexes localized at GG-equivariant homotopy equivalences.

For general GG, Elmendorf's theorem asserts that the fine equivariant model structure is Quillen equivalent to the model category of simplicial presheaves on the orbit category of GG.

All this makes the fine model structure serve as a foundation for equivariant homotopy theory and for equivariant cohomology in its refined form subsuming Bredon cohomology.

This is in contrast to the “coarse” or Borel model structure whose weak equivalences are simply the underlying weak homotopy equivalences (which need not restrict to weak homotopy equivalences on all fixed loci). The coarse Borel model structure instead presents the slice homotopy theory over the classifying space BGB G. The intrinsic cohomology of this coarse equivariant homotopy theory is just “Borel equivariant”, hence computes cohomology of Borel constructions.

While one may, therefore, think of the fine model structure as exhibiting “genuine” equivariance (e.g. Guillou, May & Rubin 13, p. 14-15), beware that the term “genuine equivariant homotopy theory” has come to be adopted for something yet a little richer, namely to equivariant stable homotopy theory whose G-spectra are in addition equipped with “transfer maps”.

However, when the closed subgroups of GG that enter the definition of the fine model structure are taken to be compact groups, then it is not wrong to speak of proper equivariant homotopy theory (conflating two usages of the term “proper”, but in a sensible way).

Definition

Throughout, we write

Proposition

(fine model structure on GG-spaces)
There is a model category-structure GAct(TopSp Qu) fineG Act\big(TopSp_{Qu}\big)_{fine} on topological G-spaces whose weak equivalences and fibrations are those morphisms f:XYf \,\colon\, X \xrightarrow{\;} Y such that for each closed subgroup HclsdGH \,\underset{clsd}{\subset}\, G their (co-)restriction f H:X HY Hf^H \,\colon\, X^H \xrightarrow{\;} Y^H to the HH-fixed loci is, respectivelhy, a weak equivalence or fibration in the classical model structure on topological spaces, hence a weak homotopy equivalence or Serre fibration.

(Dwyer & Kan 1984, 1.2)

Properties

Cofibrant generation, enrichment and properness

Proposition

The model category GAct(TopSp Qu) fineG Act\big( TopSp_{Qu}\big)_{fine} (Prop. ) is:

  1. proper and cofibrantly generated model category with generating (acyclic) cofibrations the images under forming products (k-ified product topological spaces) of coset spaces G/HG/H with the classical generating cofibrations (here and here):

    (1)I GTop {G/H×S n1G/H×D n} n,HclsdG I GTop {G/H×D nG/H×D n×I} n,HclsdG \begin{aligned} I_{G Top} &\;\coloneqq\; \big\{ G/H \times S^{n-1} \xhookrightarrow{\;} G/H \times D^n \big\}_{n \in \mathbb{N}, H \underset{clsd}{\subset} G} \\ I_{G Top} &\;\coloneqq\; \big\{ G/H \times D^n \xhookrightarrow{\;} G/H \times D^n \times I \big\}_{n \in \mathbb{N}, H \underset{clsd}{\subset} G} \end{aligned}

    (Guillou 2006, Prop. 3.12; Fausk 2008, Prop. 2.11; Stephan 2013, Prop. 2.6)

  2. in addition an enriched model category over TopSp Qu TopSp_{Qu} with hom-objects given by the GG-fixed loci of the conjugation action on the mapping spaces, hence such that

    (2)Maps(,) G:GAct(TopSp Qu) fine op×GAct(TopSp Qu) fine opTopSp Qu Maps(-,-)^G \;\colon\; G Act\big(TopSp_{Qu}\big)_{fine}^{op} \times G Act\big(TopSp_{Qu}\big)_{fine}^{op} \xrightarrow{\;} TopSp_{Qu}

    is a Quillen bifunctor.

    (Guillou, May & Rubin 2013, Thm 3.7; Schwede 2018, Prop. B.7; DHLPS 2019, Prop. 1.1.3 (i-ii))

Remark

(specialization to Borel model structure)
The direct analog of Prop. , Prop. holds for any choice of family of closed subgroups of GG. In the case that the family contains only the trivial group 1G1 \subset G the result is the topological Borel model structure.

Corollary

Evey G-CW complex (being, by definition, a special cell complex in the generating cofibrations (1)) is a cofibrant object in the fine equivariant model structure.

The TopSp QuTopSp_{Qu} enrichment of Prop. in fact underlies a model enrichment of GAct(TopSp Qu) fineG Act(TopSp_{Qu})_{fine} over itself:

Proposition

(cartesian monoidal model category structure)
The model category GAct(TopSp Qu) fineG Act\big( TopSp_{Qu}\big)_{fine} (Prop. ) is a cartesian monoidal model category in that it satisfies the pushout-product axiom with respect to Cartesian product of (cgwh) GG-spaces.

(Observe that the unit axiom is automatic, by this Prop., since the tensor unit – which here is the point space equipped with the trivial action – is clearly a G-CW complex, *G/G\ast \simeq G/G, and hence cofibrant, by Cor. .)
Proof

Prop. seems to have been folklore statement, based on the fact that the equivariant triangulation theorem implies that products of coset spaces G/H 2×G/H 2G/H_2 \,\times\, G/H_2 admit an G-CW complex-structure (crucially using here that GG is assumed to be a Lie group, so that its coset spaces have the structure of smooth manifolds with smooth group actions.) The required argument to make this into a proof of monoidal model category structure is spelled out as DHLPS 2019, Prop. 1.1.3 (iii), there in the further generality of proper equivariant homotopy theory. (Under the above assumption that GG is not just a Lie group but a compact Lie group, the classes of “𝒞ℴ𝓂\mathcal{Com}-cofibrations” and of “GG-cofibrations” in DHLPS 19, Def. 1.1.2 agree, since closed subspaces of compact Hausdorff spaces are equivalently compact subspaces).

Prop. immediately implies (by this general Prop.):

Proposition

(internal hom Quillen adjunction)
For XGAct(TopSp Qu) fineX \,\in\, G Act\big(TopSp_{Qu}\big)_{fine} a cofibrant object, the functor which assigns mapping spaces out of XX equipped with the conjugation action, is a right Quillen functor, hence makes a Quillen adjunction together with the functor of taking the product with XX (the k-ified product topological space) equipped with the diagonal action:

GAct(TopSp Qu) fine QuMaps(X,)X×()GAct(TopSp Qu) fine. G Act\big( TopSp_{Qu}\big)_{fine} \underoverset {\underset{Maps(X,-)}{\longrightarrow}} {\overset{X \times (-)}{\longleftarrow}} {\bot_{\mathrlap{Qu}}} G Act\big( TopSp_{Qu}\big)_{fine} \mathrlap{\,.}

References

The model structure itself was first discussed in:

Further properties, such as cofibrant generation, properness, and topological enrichment and are established in:

In addition, the monoidal model category structure is made explicit (in the generality of proper equivariant homotopy theory) in:

For more see the references at Elmendorf's theorem.

Last revised on September 17, 2021 at 01:28:05. See the history of this page for a list of all contributions to it.