nLab
orbit category

Definition

Given a topological group $G$ the orbit category $\mathrm{Or}G$ (denoted also ${𝒪}_G$) 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.

It is a small topologically enriched category (though of course if $G$ is a discrete group, the enrichment of $\mathrm{Or}G$ 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 $G/H\cong G/K$.

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

Applications

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.

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).

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.