nLab
orbit category

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 $\operatorname{Orb}_G$ (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.

Remark

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”.

Remark

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 $G/H\cong G/K$.

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 $\mathrm{Orb}_F\,G \subset \operatorname{Orb}_G$ 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 $\operatorname{Orb}_G$ 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.

Examples

Example

(orbit category of Z/2Z)

For equivariance group the cyclic group of order 2:

$$G \;\coloneqq\; \mathbb{Z}_2 \;\coloneqq\; \mathbb{Z}/2\mathbb{Z} \,.$$

the orbit category looks like this:

(1)$$\mathbb{Z}_2 Orbits \;=\; \left\{ \array{ \mathbb{Z}_2/1 & \overset{ \phantom{AAAAA} }{ \longrightarrow } & \mathbb{Z}_2/\mathbb{Z}_2 \\ Aut = \mathbb{Z}_2 && Aut = 1 } \right\}$$

i.e.:

$$\begin{aligned} \mathbb{Z}_2 Orbits \big( \mathbb{Z}_2/\mathbb{Z}_2 \,,\, \mathbb{Z}_2/\mathbb{Z}_2 \big) \;\simeq\; 1 \\ \mathbb{Z}_2 Orbits \big( \mathbb{Z}_2/1 \,,\, \mathbb{Z}_2/\mathbb{Z}_2 \big) \;\simeq\; \ast \\ \mathbb{Z}_2 Orbits \big( \mathbb{Z}_2/\mathbb{Z}_2 \,,\, \mathbb{Z}_2/1 \big) \;\simeq\; \varnothing \\ \mathbb{Z}_2 Orbits \big( \mathbb{Z}_2/1 \,,\, \mathbb{Z}_2/1 \big) \;\simeq\; \mathbb{Z}_2 \end{aligned}$$

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 weak homotopy equivalences on fixed point spaces.

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

Relation to equivariant 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)-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 $Orb_{/\mathbf{B}G} = Orb_G$

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).

Relation to the category of groups, homomorphisms and conjugations

See at global equivariant homotopy theory.

Related concepts

• orbit, orbit type

• equivariant homotopy theory

• global orbit category, global equivariant stable homotopy theory

• Bredon cohomology, equivariant cohomology

• topological G-space

• vector G-space

• equivariant dgc-algebra

• equivariant rational homotopy theory

References

The notion of the orbit category (for use in equivariant cohomology/Bredon cohomology) is due to

• Glen Bredon, Section I.3. of Equivariant cohomology theories, Springer Lecture Notes in Mathematics Vol. 34. 1967 (doi:10.1007/BFb0082690)

announced in

• Glen Bredon, Equivariant cohomology theories, Bull. Amer. Math. Soc. Volume 73, Number 2 (1967), 266-268. (educlid:1183528794)

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:

• William Dwyer, Daniel Kan, Sec. 2.1 and Thm. 3.1 of: Singular functors and realization functors, Indagationes Mathematicae (Proceedings) Volume 87, Issue 2, 1984, Pages 147-153 (doi:10.1016/1385-7258(84)90016-7)

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:

• Tammo tom Dieck, Section 1.3 of: Representation theory, 2009 (pdf)

For more on the relation to global equivariant homotopy theory see

• Charles Rezk, Global Homotopy Theory and Cohesion