under construction (some more harmonization needed)
homotopy theory, (∞,1)-category theory, homotopy type theory
flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed…
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Be?linson-Bernstein localization?
group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
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.
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.
Write $Glo$ for the (∞,1)-category whose
(∞,1)-categorical hom-spaces$Glo(G,H)$ are the geometric realizations of the Lie groupoid of smooth functors and smooth natural transformations $Top\infty Grpd(\mathbf{B}G, \mathbf{B}H)$.
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).
The following is the global orbit category.
Write
for the non-full sub-(∞,1)-category of the global indexing category, def. , on the injective group homomorphisms.
The following defines the global equivariant homotopy theory $PSh_\infty(Glo)$.
Write
for the (∞,1)-category of (∞,1)-presheaves (an (∞,1)-topos) on the global indexing category $Glo$ of def. , and write
for the (∞,1)-Yoneda embedding.
Similarly write
for the (∞,1)-category of (∞,1)-presheaves on the global orbit category $Orb$ of def. , and write again
for its (∞,1)-Yoneda embedding.
The following recovers the ordinary (“local”) equivariant homotopy theory of a given compact Lie group $G$ (“of $G$-spaces”).
For $G$ a compact Lie group, write
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.
For more see at cohesion of global- over G-equivariant homotopy theory.
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
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
and the inverse image/constant ∞-stack functor literally assigning constant presheaves:
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 $\nabla$.
More in detail, the shape modality, flat modality and sharp modality of this cohesion of the global equivariant homotopy theory has the following description.
Some aspects of the cohesion of global- over G-equivariant homotopy theory:
For $G$ a compact Lie group define an (∞,1)-functor
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 $X//G$:
Observe that by def. this gives $\delta_G(\ast) \simeq \mathbb{B}G$ and so $\delta_G$ induces a functor
(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
the flat modality of def. produces the homotopy type of the homotopy quotient/homotopy coinvariants of the $G$-action (∞-action)
In particular then the points-to-pieces transform of general cohesion yields the comparison map
For $G$ any compact Lie group, the cohesion of the global equivariant homotopy theory, prop. , descends to the slice (∞,1)-toposes
hence to cohesion over the “local” $G$-equivariant homotopy theory.
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
(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.
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
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.
There are equivalences
i.e. homotopy invariant sheaves on separated differentiable stacks are equivalent to global spaces.
This generalizes the discussion of Sati & Schreiber 2020, pp. 58.
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) |
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
Discussion of the global equivariant homotopy theory as a cohesive (∞,1)-topos is in
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
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
Discussion from a perspective of homotopy type theory is in
The example of global equivariant algebraic K-theory:
On orbifold cohomology seen in global equivariant homotopy theory:
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)
Last revised on July 30, 2024 at 20:20:05. See the history of this page for a list of all contributions to it.