nLab fine model structure on topological G-spaces

Idea

For $G$ 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 $H$-fixed loci, for all closed subgroups $H \subset_{clsd} G$, are weak equivalences or fibrations, respectively, in the classical model structure on topological spaces, hence weak homotopy equivalences or Serre fibrations, respectively. Hence the weak equivalences are the equivariant weak homotopy equivalences.

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

For general $G$, 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 $G$.

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 $B G$. 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 $G$ 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

• TopSp for the convenient category of compactly generated weak Hausdorff spaces;

• $TopSp_{Qu}$ for its classical model structure on topological spaces (here);

• $G \,\in\, Grp(CptSmthMfd) \xrightarrow{\;} Grp(TopSp)$ for the underlying topological group of a compact Lie group

  (most of the following statements hold for general topological groups, at least if some further qualifications are added)

• $G Act(TopSp)$ for the category of continuous $G$-actions, hence for the category of topological G-spaces with continuous equivariant functions between them.

Proposition

(fine model structure on $G$-spaces)
There is a model category-structure $G Act\big(TopSp_{Qu}\big)_{fine}$ on topological G-spaces whose weak equivalences and fibrations are those morphisms $f \,\colon\, X \xrightarrow{\;} Y$ such that for each closed subgroup $H \,\underset{clsd}{\subset}\, G$ their (co-)restriction $f^H \,\colon\, X^H \xrightarrow{\;} Y^H$ to the $H$-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 $G Act\big( TopSp_{Qu}\big)_{fine}$ (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 $G/H$ with the classical generating cofibrations (here and here):

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

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

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

   (2)$$Maps(-,-)^G \;\colon\; G Act\big(TopSp_{Qu}\big)_{fine}^{op} \times G Act\big(TopSp_{Qu}\big)_{fine}^{op} \xrightarrow{\;} TopSp_{Qu}$$

   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 $G$. In the case that the family contains only the trivial group $1 \subset G$ the result is the topological Borel model structure.

Corollary

Every 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 $TopSp_{Qu}$ enrichment of Prop. in fact underlies a model enrichment of $G Act(TopSp_{Qu})_{fine}$ over itself:

Proposition

(cartesian monoidal model category structure)
The model category $G Act\big( TopSp_{Qu}\big)_{fine}$ (Prop. ) is a cartesian monoidal model category in that it satisfies the pushout-product axiom with respect to Cartesian product of (cgwh) $G$-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, $\ast \simeq G/G$, 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 $G/H_2 \,\times\, G/H_2$ admit an G-CW complex-structure (crucially using here that $G$ 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 $G$ is not just a Lie group but a compact Lie group, the classes of “$\mathcal{Com}$-cofibrations” and of “$G$-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 $X \,\in\, G Act\big(TopSp_{Qu}\big)_{fine}$ a cofibrant object, the functor which assigns mapping spaces out of $X$ equipped with the conjugation action, is a right Quillen functor, hence makes a Quillen adjunction together with the functor of taking the product with $X$ (the k-ified product topological space) equipped with the diagonal action:

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

Related concepts

• Borel model structure

• equivariant homotopy theory

• Elmendorf's theorem

References

The model structure itself was first discussed in:

• William Dwyer, Daniel Kan, Section 1.2 of: Singular functors and realization functors, Indagationes Mathematicae (Proceedings) Volume 87, Issue 2, 1984, Pages 147-153 (doi:10.1016/1385-7258(84)90016-7)

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

• Bert Guillou, Prop. 3.12 in: A short note on models for equivariant homotopy theory, 2006 (pdf, pdf)

• Halvard Fausk, Prop. 2.11 in: Equivariant homotopy theory for pro-spectra, Geom. Topol. 12 (2008) 103-176 (doi:10.2140/gt.2008.12.103, arXiv:math/0609635)

• Bertrand Guillou, Peter May, Jonathan Rubin, Theorem 1.6 in: Enriched model categories in equivariant contexts, Homology, Homotopy and Applications 21 (1), 2019 (arXiv:1307.4488, doi:10.4310/HHA.2019.v21.n1.a10)

• Marc Stephan, On equivariant homotopy theory for model categories, Homology Homotopy Appl. 18(2) (2016) 183-208 (arXiv:1308.0856, doi:10.4310/HHA.2016.v18.n2.a10)

• Stefan Schwede, Prop. B.7 in: Global homotopy theory, New Mathematical Monographs 34 Cambridge University Press, 2018 (doi:10.1017/9781108349161, arXiv:1802.09382)

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

• Dieter Degrijse, Markus Hausmann, Wolfgang Lück, Irakli Patchkoria, Stefan Schwede, Prop. 1.1.3 in: Proper equivariant stable homotopy theory, Memoirs of the AMS (arXiv:1908.00779)

For more see the references at Elmendorf's theorem.