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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 (coset spaces, $G$-orbit types) $G/H$, where $H$ is a closed subgroup of $G$,
morphisms are the $G$-equivariant continuous functions.
For suitable continuous actions of $G$ on a topological space $X$, every orbit of the action is isomorphic to one of the coset spaces $G/H$ (the stabilizer group of any point in the orbit is conjugate to $H$). This is the sense in which def. gives “the category of all $G$-orbits”.
Def. 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 $G/H \cong G/K$.
($G$-sets are the free coproduct completion of $G$-orbits)
Let $G \,\in\, Grp(Set)$ be a discrete group. Since every G-set $X$ decomposes as a disjoint union of transitive actions, namely of orbits of elements of $X$, the defining inclusion of the orbit category into $G Set$ exhibits the latter as its free coproduct completion (see also this Prop.).
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 still hold.
(orbit category of Z/2Z)
For equivariance group the cyclic group of order 2:
the orbit category looks like this:
i.e.:
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 weak homotopy equivalences on fixed point spaces.
The $G$-orbit category is the slice (∞,1)-category of the global orbit category $Orb$ (the version with faithful functors as morphisms) 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.
The notion of the orbit category (for use in equivariant cohomology/Bredon cohomology) is due to
announced in
and there considered specifically for finite groups.
Discussion for any topological group (and further generalization) is considered (together with the model category theoretic proof of Elmendorf's theorem) in:
Textbook accounts:
Tammo tom Dieck, Section I.10 of: Transformation Groups, de Gruyter 1987 (doi:10.1515/9783110858372)
Peter May, Section I.4 of: Equivariant homotopy and cohomology theory CBMS Regional Conference Series in Mathematics, vol. 91, Published for the Conference Board of the Mathematical Sciences, Washington, DC, 1996 (ISBN:978-0-8218-0319-6, pdf, pdf)
Lecture notes:
For more on the relation to global equivariant homotopy theory see
Last revised on October 12, 2021 at 12:44:02. See the history of this page for a list of all contributions to it.