under construction (some more harmonization needed)
homotopy theory, (∞,1)-category theory, homotopy type theory
flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed…
models: topological, simplicial, localic, …
see also algebraic topology
Introductions
Definitions
Paths and cylinders
Homotopy groups
Basic facts
Theorems
geometric representation theory
representation, 2-representation, ∞-representation
Grothendieck group, lambda-ring, symmetric function, formal group
principal bundle, torsor, vector bundle, Atiyah Lie algebroid
Eilenberg-Moore category, algebra over an operad, actegory, crossed module
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 topological group $G$ (“of $G$-spaces”).
For $G$ a topological 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.
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.
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.
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, chapter I of Global homotopy theory, 2013 (pdf)
Stefan Schwede, Orbispaces, orthogonal spaces, and the universal compact Lie group (arXiv:1711.06019)
see also
Discussion of the global equivariant homotopy theory as a cohesive (∞,1)-topos is 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
Last revised on April 18, 2019 at 21:26:03. See the history of this page for a list of all contributions to it.