nLab global orbit category

Contents

Context

Representation theory

representation theory

geometric representation theory

Contents

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

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).

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.

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

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

Rezk-global equivariant homotopy theory:

cohesive (∞,1)-toposits (∞,1)-sitebase (∞,1)-toposits (∞,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 $Orb_{/\mathbf{B}G} = Orb_G$