nLab global equivariant homotopy theory

under construction (some more harmonization needed)

Idea

What is called global equivariant homotopy theory is a variant of equivariant cohomology in homotopy theory where pointed topological spaces/homotopy types are equipped with $G$-infinity-actions “for all compact Lie groups $G$ at once”, or more generally for a global family.

Sometimes this is referred to just as “global homotopy theory”, leaving the equivariance implicit. There is also a stable version involving spectra equipped with infinity-actions, see at global equivariant stable homotopy theory.

More precisely, the global equivariant homotopy category is the (∞,1)-category (or else its homotopy category) of (∞,1)-presheaves $PSh_\infty(Orb)$ on the global orbit category $Orb$ (Henriques-Gepner 07, section 1.3), regarded as an (∞,1)-category.

Here $Orb$ has as objects compact Lie groups and the (∞,1)-categorical hom-spaces $Orb(G,H) \coloneqq \Pi [\mathbf{B}G, \mathbf{B}H]$, where on the right we have the fundamental (∞,1)-groupoid of the topological groupoid of group homomorphisms and conjugations.

Definition

We follow (Rezk 14). Beware that the terminology there differs slightly but crucially in some places from (Henriques-Gepner 07). Whatever terminology one uses, the following are the key definitions.

The following is the global equivariant indexing category.

Definition

Write $Glo$ for the (∞,1)-category whose

• objects are compact Lie groups;

• (∞,1)-categorical hom-spaces$Glo(G,H)$ are the geometric realizations of the Lie groupoid of smooth functors and smooth natural transformations $Top\infty Grpd(\mathbf{B}G, \mathbf{B}H)$.

(Rezk 14, 2.1)

Remark

Equivalent models for the global indexing category, def. include the category “$O_{gl}$” of (May 90). Another variant is $\mathbf{O}_{gl}$ of (Schwede 13).

(Rezk 14, 2.4, 2.5)

The following is the global orbit category.

Definition

Write

$$Orb \longrightarrow Glo$$

for the non-full sub-(∞,1)-category of the global indexing category, def. , on the injective group homomorphisms.

(Rezk14, 4.5)

The following defines the global equivariant homotopy theory $PSh_\infty(Glo)$.

Definition

Write

$$Top_{Glo} \coloneqq PSh_\infty(Glo)$$

for the (∞,1)-category of (∞,1)-presheaves (an (∞,1)-topos) on the global indexing category $Glo$ of def. , and write

$$\mathbb{B} \;\colon\; Glo \longrightarrow PSh_\infty(Glo)$$

for the (∞,1)-Yoneda embedding.

Similarly write

$$Top_{Orb} \coloneqq PSh_\infty(Orb)$$

for the (∞,1)-category of (∞,1)-presheaves on the global orbit category $Orb$ of def. , and write again

$$\mathbb{B} \;\colon\; Orb \longrightarrow PSh_\infty(Orb)$$

for its (∞,1)-Yoneda embedding.

(Rezk 14, 3.1 and 4.5)

The following recovers the ordinary (“local”) equivariant homotopy theory of a given compact Lie group $G$ (“of $G$-spaces”).

Definition

For $G$ a compact Lie group, write

$$G Top \coloneqq PSh_\infty(Orb)/\mathbb{B}G$$

for the slice (∞,1)-topos of $PSh_\infty(Orb)$ over the image of $G$ under the (∞,1)-Yoneda embedding, as in def. .

This is (Rezk 14, 1.5). Depending on axiomatization this is either a definition or Elmendorf's theorem, see at equivariant homotopy theory for more on this.

Properties

Cohesion

For more see at cohesion of global- over G-equivariant homotopy theory.

Proposition

The global equivariant homotopy theory $PSh_\infty(Glo)$ of def. is a cohesive (∞,1)-topos over the canonical base (∞,1)-topos ∞Grpd:

the global section geometric morphism

$$(\Delta \dashv \Gamma) \;\colon\; PSh_\infty(Glo) \longrightarrow \infty Grpd$$

is given (as for all (∞,1)-presheaf (∞,1)-toposes) by the direct image/global section functor being the homotopy limit over the opposite (∞,1)-site

$$\Gamma X \simeq \underset{\leftarrow}{\lim}(Glo^{op}\stackrel{X}{\to} \infty Grpd)$$

and the inverse image/constant ∞-stack functor literally assigning constant presheaves:

$$\Delta S \colon G \mapsto S \,.$$

This is a full and faithful (∞,1)-functor.

Moreover, $\Delta$ has a further left adjoint $\Pi$ which preserves finite products, and $\Gamma$ has a further right adjoint $\nabla$.

(Rezk 14, 5.1)

More in detail, the shape modality, flat modality and sharp modality of this cohesion of the global equivariant homotopy theory has the following description.

Relation between global and local equivariant homotopy theory

Some aspects of the cohesion of global- over G-equivariant homotopy theory:

Definition

For $G$ a compact Lie group define an (∞,1)-functor

$$\delta_G \;\colon\; G Top \longrightarrow PSh_\infty(Glo)$$

sending a topological G-space to the he presheaf which sends a group $H$ to the geometric realization of the topological groupoid of maps from $\mathbf{B}H$ to the action groupoid $X//G$:

$$\delta_G(X)\;\colon\; H \mapsto \Pi( [\mathbf{B}H, X//G] ) \,.$$

Observe that by def. this gives $\delta_G(\ast) \simeq \mathbb{B}G$ and so $\delta_G$ induces a functor

$$\Delta_G \;\colon\; G Top \simeq G Top/\ast \simeq PSh_\infty(Orb)/\mathbb{B}G \stackrel{\delta_G}{\longrightarrow} PSh_\infty(Glo)/\mathbb{B}G \,.$$

(Rezk 14, 3.2)

Proposition

(ordinary quotient and homotopy quotient via equivariant cohesion)

On a $G$-space $X \in G Top$ included via def. into the global equivariant homotopy theory,

• the shape modality of def. produces the homotopy type of the ordinary quotient of the $G$-action

  $$\Pi(\delta_G(X)) \simeq \vert X/G \vert \,,$$

• the flat modality of def. produces the homotopy type of the homotopy quotient/homotopy coinvariants of the $G$-action (∞-action)

  $$\Gamma(\delta_G(X)) \simeq \vert X//G \vert \,,$$

In particular then the points-to-pieces transform of general cohesion yields the comparison map

$$\vert X//G \vert \longrightarrow \vert X/G \vert \,.$$

(Rezk 14, 5.1)

Proposition

For $G$ any compact Lie group, the cohesion of the global equivariant homotopy theory, prop. , descends to the slice (∞,1)-toposes

$$PSh_\infty(Glo)/\mathbb{B}G \longrightarrow PSh_\infty(Orb)/\mathbb{B}G \simeq G Top \,,$$

hence to cohesion over the “local” $G$-equivariant homotopy theory.

The inclusion $\Delta_G$ is that of def. .

(Rezk 14, 5.3)

Relation to topological stacks and orbispaces

under construction

By the main theorem of (Henriques-Gepner 07) the (∞,1)-presheaves on the global orbit category are equivalently “cellular” topological stacks/topological groupoids (“orbispaces”), we might write this as

$$ETopGrpd^{cell} = PSh_\infty(Orb) \,.$$

(As such the global equivariant homotopy theory should be similar to ETop∞Grpd. Observe that this is a cohesive (∞,1)-topos with $\Pi$ such that it sends a topological action groupoid of a topological group $G$ acting on a topological space $X$ to the homotopy quotient $\Pi(X)//\Pi(G)$.)

The central theorem of (Rezk 14) (using a slightly different definition than Henriques-Gepner 07) is that $PSh_\infty(Orb)$ is a cohesive (∞,1)-topos with $\Gamma$ producing homotopy quotients.

Relation to differentiable stacks

Let $\mathrm{SepStk}$ denote the $(2,1)$-category of separated differentiable stacks, i.e. those whose diagonal is a proper map. This $(2,1)$-category admits a topology by open covers of stacks; we let $\mathrm{Shv}(\mathrm{SepStk})$ be the corresponding ∞-category of sheaves of spaces, and we let the homotopy invariant sheaves $\mathrm{Shv}^{\mathrm{htp}}(\mathrm{SepStk}) \subset \mathrm{Shv}(\mathrm{SepStk})$ be the full subcategory spanned by those sheaves $\mathcal{F}$ such that the map $\mathcal{F}(\mathfrak{X}) \rightarrow \mathcal{F}(\mathfrak{X} \times \mathbb{R})$ induced by the projection $\mathfrak{X} \times \mathbb{R} \rightarrow \mathfrak{X}$ is an equivalence for every separated differentiable stack $\mathfrak{X}$.

Given $G$ a finite group, its action groupoid lifts to a separated differentiable stack $\mathbb{B}G$, which is homotopy-invariant by Clough, Cnossen & Linskens 2024. In general, we may probe elements of $\mathrm{Shv}^{\mathrm{htp}}(\mathrm{SepStk})$ by evaluating them on the localization $L_{\mathrm{htpy}} \mathbb{B}G$, producing a functor of ∞-categories

$$\mathrm{ev}_{\mathbb{B}-}:\mathrm{Shv}^{\mathrm{htp}}(\mathrm{SepStk}) \rightarrow \mathrm{Psh}(\mathrm{Glo}^{\mathrm{Stk}}),$$

where $\mathrm{Glo}^{\mathrm{stk}}$ denotes the image of the global indexing category in $\mathrm{Shv}^{\mathrm{htp}}(\mathrm{SepStk})$ under $L_{\mathrm{htp}} \mathbb{B}(-)$.

The main theorem of Clough, Cnossen & Linskens 2024 is the following.

Theorem

There are equivalences

$$\mathrm{Shv}^{\mathrm{htp}}(\mathrm{SepStk}) \xrightarrow{\mathrm{ev}} \mathrm{Psh}(\mathrm{Glo}^{\mathrm{Stk}}) \simeq \mathrm{Psh}(\mathrm{Glo}),$$

i.e. homotopy invariant sheaves on separated differentiable stacks are equivalent to global spaces.

This generalizes the discussion of Sati & Schreiber 2020, pp. 58.

Related concepts

• cohesion of global- over G-equivariant homotopy theory

• orbifold cohomology

• global equivariant stable homotopy theory

• proper equivariant homotopy theory

Rezk-global equivariant homotopy theory:

cohesive (∞,1)-topos	its (∞,1)-site	base (∞,1)-topos	its (∞,1)-site
global equivariant homotopy theory $PSh_\infty(Glo)$	global equivariant indexing category $Glo$	∞Grpd $\simeq PSh_\infty(\ast)$	point
… sliced over terminal orbispace: $PSh_\infty(Glo)_{/\mathcal{N}}$	$Glo_{/\mathcal{N}}$	orbispaces $PSh_\infty(Orb)$	global orbit category
… sliced over $\mathbf{B}G$: $PSh_\infty(Glo)_{/\mathbf{B}G}$	$Glo_{/\mathbf{B}G}$	$G$-equivariant homotopy theory of G-spaces $L_{we} G Top \simeq PSh_\infty(Orb_G)$	$G$-orbit category $Orb_{/\mathbf{B}G} = Orb_G$

representation theory and equivariant cohomology in terms of (∞,1)-topos theory/homotopy type theory (FSS 12 I, exmp. 4.4):

homotopy type theory	representation theory
pointed connected context $\mathbf{B}G$	∞-group $G$
dependent type on $\mathbf{B}G$	$G$-∞-action/∞-representation
dependent sum along $\mathbf{B}G \to \ast$	coinvariants/homotopy quotient
context extension along $\mathbf{B}G \to \ast$	trivial representation
dependent product along $\mathbf{B}G \to \ast$	homotopy invariants/∞-group cohomology
dependent product of internal hom along $\mathbf{B}G \to \ast$	equivariant cohomology
dependent sum along $\mathbf{B}G \to \mathbf{B}H$	induced representation
context extension along $\mathbf{B}G \to \mathbf{B}H$	restricted representation
dependent product along $\mathbf{B}G \to \mathbf{B}H$	coinduced representation
spectrum object in context $\mathbf{B}G$	spectrum with G-action (naive G-spectrum)

References

The global orbit category $Orb$ is considered in

• André Henriques, David Gepner, Homotopy Theory of Orbispaces (arXiv:math/0701916)

• Jacob Lurie, Section 3 of Elliptic cohomology III: Tempered Cohomology (pdf)

Global unstable equivariant homotopy theory is discussed as a localization of the category of “orthogonal spaces” (the unstable version of orthogonal spectra) in

• Stefan Schwede, Orbispaces, orthogonal spaces, and the universal compact Lie group (arXiv:1711.06019)

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

see also

• Peter May, Some remarks on equivariant bundles and classifying spaces, Asterisque 191 (1990), 7, 239-253. International Conference on Homotopy Theory (Marseille-Luminy, 1988).

Discussion of the global equivariant homotopy theory as a cohesive (∞,1)-topos is in

• Charles Rezk, Global Homotopy Theory and Cohesion (2014)

and further equipped also with smooth structure as required for differential orbifold cohomology:

• Hisham Sati, Urs Schreiber: Proper Orbifold Cohomology [arXiv:2008.01101]

• Adrian Clough, Bastiaan Cnossen, Sil Linskens: Global spaces and the homotopy theory of stacks [arXiv:2407.06877]

Relatedly, this is developed via partially lax limits in

• Sil Linskens, Denis Nardin, Luca Pol, Global homotopy theory via partially lax limits (arXiv:2206.01556)

Discussion of a model structure for global equivariance with respect to geometrically discrete simplicial groups/∞-group (globalizing the Borel model structure for ∞-actions) is in

• Yonatan Harpaz, Matan Prasma, The Grothendieck construction for model categories (arXiv:1404.1852)

Discussion from a perspective of homotopy type theory is in

• Mike Shulman, Univalence for inverse EI diagrams (arXiv:1508.02410)

The example of global equivariant algebraic K-theory:

• Stefan Schwede, Global algebraic K-theory (arXiv:1912.08872)

On orbifold cohomology seen in global equivariant homotopy theory:

• Branko Juran, Orbifolds, Orbispaces and Global Homotopy Theory (arXiv:2006.12374)

following the suggestion in Schwede 17, Intro, Schwede 18, p. ix-x.

An ongoing project to develop global equivariant higher category theory and associated universal properties for global spaces and spectra:

• Bastiaan Cnossen, Tobias Lenz, Sil Linskens, Parametrized stability and the universal property of global spectra (arXiv:2301.08240)

• Bastiaan Cnossen, Tobias Lenz, Sil Linskens, Partial parametrized presentability and the universal property of equivariant spectra (arXiv:2307.11001)

• Bastiaan Cnossen, Tobias Lenz, Sil Linskens, The Adams isomorphism revisited (arXiv:2311.04884)

• Bastiaan Cnossen, Tobias Lenz, Sil Linskens, Parametrized higher semiadditivity and the universality of spans (arXiv:2403.07676)