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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.
(e.g. tom Dieck 1987, I (10.1))
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.
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.).
(families of subgroups)
More generally, given a family $\mathcal{F}$ 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 $H\in F$. For such families, many of the considerations of results such as Elmendorf's theorem will still hold.
Given a topological G-space, there is (following tom Dieck 1987, Sec. I.10) a sequence of variants (enhancements) of the $G$-orbit category which mixes transitions between the fixed loci with n-truncations of their homotopy types.
(…)
(…)
For $G \,\in\, Grp(Set)$ a discrete group and $G \curvearrowright X \,\in\, G Act(TopSp)$ a topological $G$-space, its fundamental category (tom Dieck 1987 (10.7), Lück 1989, 8.15) or equivariant fundamental groupoid (Pronk & Scull 2021, Def. 3.1)
is the small category whose
objects are pairs consisting of a subgroup $H \subset G$ and a $G$-equivariant function $\; x_X \,\colon\, G/H \to X$, hence, equivalently, a point in the fixed locus $X^H$;
morphisms$x_H \xrightarrow{ (\phi,[\gamma]) } x'_{H'}$ are pairs consisting of a $G$-equivariant function $G/H \xrightarrow{ \phi } G/H'$ and a relative $G$-homotopy class $[\gamma]$ of a continuous path $\gamma \,\colon\, G/H \times [0,1] \xrightarrow{\;} X$ from $\gamma(0) = x_H$ to $\gamma_1 = \phi^\ast x'_X'$.
For $G \curvearrowright X \,=\, G \curvearrowright \ast$ the point, its fundamental category (equivariant fundamental groupoid) according to Def. is the plain orbit category $G Orbts$ (Def. ).
(equivariant fundamental groupoid under Grothendieck construction)
If we write
for the fundamental groupoid of $X$ in its incarnation as a presheaf of fixed loci over the orbit category, hence for the (2,1)-functor
which sends $G/H$ to the ordinary fundamental groupoid of the fixed locus $X^H$, then the equivariant fundamental groupoid of $G \curvearrowright X$ (Def. ) is equivalently the Grothendieck construction
on (1).
This follows by direct unwinding of the definitions; see also Pronk & Scull 2021, below Def. 3.1.
(equivariant universal covering)
For $G \in Grp(Set)$ a discrete group and $G \curvearrowright X \,\in\, G Act(TopSp)$ a well-behaved topological $G$-space, its equivariant universal cover (tom Dieck 1987 (10.13)) “is” the functor from the fundamental equivariant groupoid (2) of $G \curvearrowright X$ to TopSp
which sends $x_H \,\in\, X^H$ to the universal covering $\widehat{X_x}$ of the connected component of $x_H \,\in\, X$ given by relative-homotopy-equivalences classes of paths based at $x_H$.
Write
for the $\infty$-category of $\infty$-presheaves on the $G$-orbit category, hence for the proper $G$-equivariant homotopy theory.
For $G \curvearrowright X \,\in\, G Act(TopSp)$ a topological $G$-space, its equivariant homotopy type is
For $G \curvearrowright X \,\in\, G Act(TopSp)$ a topological $G$-space, its equivariant universal covering according to Def. is, under Prop. and the (infinity,1)-Grothendieck construction equivalent to the 1-truncation unit
in that
bottom right corner not quite right, will fix…
(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.
For more on this see also at cohesion of global- over G-equivariant homotopy theory the section the adjunction of sites.
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).
(left base change along covering of equivariance group)
Let $\widehat{G} \overset{p}{\twoheadrightarrow} G$ be a surjective homomorphism of discrete groups. For $H \subset G$ a subgroup, write $\widehat{H} \subset \widehat{G}$ for its pullback
Then there is a reflective subcategory-inclusion of orbit categories
where
$R(G/H) \,=\, \widehat{G}/\widehat{H}$
$L(\widehat{G}/K) \,=\, G/p(K)$.
A transparent way to see this is to identify (as above) $G$-orbits with the 0-truncated objects in the $B G$-slice of $Singlrt \,\coloneqq\, Grpd_1^{cn}$:
Under this identification, the adjunction is the canonical left base change on slices composed with the 0-truncation reflection (here):
That the right adjoint $R$ is fully faithful is most readily seen by direct inspection: A map between $\widehat{G}$-sets on both of which $\widehat{G}$ acts only through $G$ is evidently $\widehat{G}$-equivariant if and only if it is $G$-equivariant.
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)
Wolfgang Lück, (8.16) in Transformation Groups and Algebraic K-Theory, Lecture Notes in Mathematics 1408 (Springer 1989) (doi:10.1007/BFb0083681)
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 in relation to global equivariant homotopy theory see
The notion of fundamental category of a $G$-space is due to
with further discussion in
Last revised on December 28, 2021 at 09:31:19. See the history of this page for a list of all contributions to it.