geometric representation theory
representation, 2-representation, ∞-representation
Grothendieck group, lambda-ring, symmetric function, formal group
principal bundle, torsor, vector bundle, Atiyah Lie algebroid
Eilenberg-Moore category, algebra over an operad, actegory, crossed module
group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
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.
Given a topological group $G$ the orbit category $\operatorname{Orb}_G$ (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.
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”.
Def. 1 yields a small topologically enriched category (though of course if $G$ is a discrete group, the enrichment of $\operatorname{Orb}_G$ 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 $G/H\cong G/K$.
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 $\mathrm{Orb}_F\,G \subset \operatorname{Orb}_G$ whose objects are those $G/H$ for which $H\in F$.
Sometimes a family, $\mathcal{W}$, of subgroups is specified, and then a subcategory of $\operatorname{Orb}_G$ 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.
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”.
The $G$-orbit category is the slice (∞,1)-category of the global orbit category $Orb$ over the delooping $\mathbf{B}G$:
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)-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 $Orb_{/\mathbf{B}G} = Orb_G$ |
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).
See at global equivariant homotopy theory.
A very general setting for the use of orbit categories is described in
For more on the relation to global equivariant homotopy theory see