# nLab orbit category

## Theorems

#### Representation theory

representation theory

geometric representation theory

cohomology

# Contents

## Idea

The orbit category of a group $G$ is the category of “all kinds” of orbits of $G$, namely of all suitable coset spaces regarded as G-spaces.

## Definition

###### Definition

Given a topological group $G$ the orbit category $\operatorname{Or}G$ (denoted also $\mathcal{O}_G$) is the category whose

• objects are the homogeneous spaces ($G$-orbit types) $G/H$, where $H$ is a closed subgroup of $G$,

• and whose morphisms are $G$-equivariant maps.

###### Remark

For suitable continuous actions of $G$ on a topological space $X$, every orbit of the action is isomorphic to one of the homogeneous spaces $G/H$ (the stabilizer group of any point in the orbit is conjugate to $H$). This is the sense in which def. 1 gives “the category of all $G$-orbits”.

###### Remark

Def. 1 yields a small topologically enriched category (though of course if $G$ is a discrete group, the enrichment of $\operatorname{Or}G$ is likewise discrete).

Of course, like any category, it has a skeleton, but as usually defined it is not itself skeletal, since there can exist distinct subgroups $H$ and $K$ such that $G/H\cong G/K$.

###### Remark

Warning: This should not be confused with the situation where a group $G$ acts on a groupoid $\Gamma$ so that one obtains the orbit groupoid.

More generally, given a family $F$ of subgroups of $G$ which is closed under conjugation and taking subgroups one looks at the full subcategory $\mathrm{Or}_F\,G \subset \operatorname{Or}G$ whose objects are those $G/H$ for which $H\in F$.

## Variants

Sometimes a family, $\mathcal{W}$, of subgroups is specified, and then a subcategory of $\operatorname{Or}G$ consisting of the $G/H$ where $H\in \mathcal{W}$ will be considered. If the trivial subgroup is in $\mathcal{W}$ then many of the considerations of results such as Elmendorf's theorem will go across to the restricted setting.

## Properties

### Relation to $G$-spaces and Elmendorf’s theorem

Elmendorf's theorem (see there for details) states that the (∞,1)-category of (∞,1)-presheaves on the orbit category $Orb_G$ are equivalent to the localization of topological spaces with $G$-action at the “fixed point weak equivalences”.

$L_{we} G Top \simeq PSh_\infty(Orb_G) \,.$

### Relation toequivariant homotopy theory

The $G$-orbit category is the slice (∞,1)-category of the global orbit category $Orb$ over the delooping $\mathbf{B}G$:

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

This means that in the general context of global equivariant homotopy theory, the orbit category appears as follows.

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 $Orb_{/\mathbf{B}G} = Orb_G$

### Relation to Mackey functors

Orbit categories are used often in the treatment of Mackey functors from the theory of locally compact groups and in the definition of Bredon cohomology.

### Relation to Bredon equivariant cohomology

It appears in equivariant stable homotopy theory, where the $H$-fixed homotopy groups of a space form a presheaf on the homotopy category of the orbit category (e.g. page 8, 9 here).

## References

A very general setting for the use of orbit categories is described in

• W. G. Dwyer and D. M. Kan, Singular functors and realization functors , Nederl. Akad. Wetensch. Indag. Math., 87, (1984), 147 – 153.

For more on the relation to global equivariant homotopy theory see

Revised on April 14, 2014 01:23:36 by Urs Schreiber (185.37.147.12)