nLab global orbit category

Idea

The global orbit category is the unification of all $G$-orbit categories as the groups $G$ are allowed to vary.

The (∞,1)-presheaves over the global orbit category provide the base (∞,1)-topos over which the global equivariant homotopy theory (see there) is a cohesive (∞,1)-topos (Rezk 14).

Definition

The following defines the global equivariant indexing category $Glo$.

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.

(Rezk 14, 4.5)

Properties

Relation to the local orbit category

The slice (∞,1)-category of the global orbit category $Orb$ (the version with faithful functors as morphisms) over $\mathbf{B}G$ is the local orbit category of $G$

$$Orb_{/\mathbf{B}G} \simeq Orb_G \,.$$

Relation to orbispaces and $G$-spaces

The (∞,1)-category of (∞,1)-presheaves over the global orbit category is that of orbispaces.

Accordingly, by the discussion here, the slice (∞,1)-topos of orbispaces over $\mathbf{B}G$ is that of G-spaces

$$PSh_\infty(Orb)_{/\mathbf{B}G} \simeq PSh_\infty(Orb_{/\mathbf{B}G}) \simeq PSh_\infty(Orb_G) \simeq G Space$$

(where the last step is Elmendorf's theorem).

Relation to equivariant homotopy theory

The (∞,1)-category of (∞,1)-presheaves on the global equivariant indexing category is the global equivariant homotopy theory and under the canonical projection is a cohesive (∞,1)-topos over ∞Grpd. Its slice (∞,1)-topos over the terminal orbispace is cohesive over orbispaces

$$PSh_\infty(Glo)_{/\mathcal{N}} \to PSh_\infty(Orb) \,.$$

Related concepts

• orbifold cohomology

• cohesive relation between global- and G-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 $Orb_{/\mathbf{B}G} = Orb_G$

References

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

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

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

See also

• Stefan Schwede, Global homotopy theory, 2013 (pdf)

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