nLab
orbit category

Contents

Context

Homotopy theory

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

models: topological, simplicial, localic, …

see also algebraic topology

Introductions

Definitions

Paths and cylinders

Homotopy groups

Basic facts

Theorems

Representation theory

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Contents

Idea

The orbit category of a group GG is the category of “all kinds” of orbits of GG, namely of all suitable coset spaces regarded as G-spaces.

Definition

Definition

Given a topological group GG the orbit category Orb G\operatorname{Orb}_G (denoted also 𝒪 G\mathcal{O}_G) is the category whose

Remark

For suitable continuous actions of GG on a topological space XX, every orbit of the action is isomorphic to one of the coset spaces G/HG/H (the stabilizer group of any point in the orbit is conjugate to HH). This is the sense in which def. gives “the category of all GG-orbits”.

Remark

Def. yields a small topologically enriched category (though of course if GG is a discrete group, the enrichment of Orb G\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 HH and KK such that G/HG/KG/H \cong G/K.

Remark

( G G -sets are the free coproduct completion of G G -orbits)
Let GGrp(Set)G \,\in\, Grp(Set) be a discrete group. Since every G-set XX decomposes as a disjoint union of transitive actions, namely of orbits of elements of XX, the defining inclusion of the orbit category into G Set G Set exhibits the latter as its free coproduct completion (see also this Prop.).

Remark

Warning: This should not be confused with the situation where a group GG acts on a groupoid Γ\Gamma so that one obtains the orbit groupoid.

More generally, given a family FF of subgroups of GG which is closed under conjugation and taking subgroups one looks at the full subcategory Orb FGOrb G\mathrm{Orb}_F\,G \subset \operatorname{Orb}_G whose objects are those G/HG/H for which HFH\in F.

Variants

Sometimes a family, 𝒲\mathcal{W}, of subgroups is specified, and then a subcategory of Orb G\operatorname{Orb}_G consisting of the G/HG/H where H𝒲H\in \mathcal{W} will be considered. If the trivial subgroup is in 𝒲\mathcal{W} then many of the considerations of results such as Elmendorf's theorem will still hold.

Examples

Example

(orbit category of Z/2Z)

For equivariance group the cyclic group of order 2:

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

the orbit category looks like this:

(1) 2Orbits={ 2/1 AAAAA 2/ 2 Aut= 2 Aut=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.:

2Orbits( 2/ 2, 2/ 2)1 2Orbits( 2/1, 2/ 2)* 2Orbits( 2/ 2, 2/1) 2Orbits( 2/1, 2/1) 2 \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}

proProperties

Relation to GG-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 GOrb_G are equivalent to the localization of topological spaces with GG-action at the weak homotopy equivalences on fixed point spaces.

L weGTopPSh (Orb G). L_{we} G Top \simeq PSh_\infty(Orb_G) \,.

Relation to global equivariant homotopy theory

The GG-orbit category is the slice (∞,1)-category of the global orbit category OrbOrb (the version with faithful functors as morphisms) over the delooping BG\mathbf{B}G:

Orb GOrb /BG. 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)-toposits (∞,1)-sitebase (∞,1)-toposits (∞,1)-site
global equivariant homotopy theory PSh (Glo)PSh_\infty(Glo)global equivariant indexing category GloGlo∞Grpd PSh (*) \simeq PSh_\infty(\ast)point
sliced over terminal orbispace: PSh (Glo) /𝒩PSh_\infty(Glo)_{/\mathcal{N}}Glo /𝒩Glo_{/\mathcal{N}}orbispaces PSh (Orb)PSh_\infty(Orb)global orbit category
sliced over BG\mathbf{B}G: PSh (Glo) /BGPSh_\infty(Glo)_{/\mathbf{B}G}Glo /BGGlo_{/\mathbf{B}G}GG-equivariant homotopy theory of G-spaces L weGTopPSh (Orb G)L_{we} G Top \simeq PSh_\infty(Orb_G)GG-orbit category Orb /BG=Orb GOrb_{/\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 HH-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.

References

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:

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.