# nLab global equivariant indexing category

Contents

### Context

#### Representation theory

representation theory

geometric representation theory

cohomology

# Contents

## Idea

The global equivariant indexing category is the full subcategory of topological ∞-groupoids on those which are deloopings of compact Lie groups, with hom-spaces being the geometric realization of the internal homs there.

The (∞,1)-presheaves over the global equivariant indexing category is the global equivariant homotopy theory. This is a cohesive (∞,1)-topos over ∞Grpd (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 over $\mathbf{B}G$ is the local orbit category of $G$

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

### 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 oribispace is cohesive over orbispaces

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