orbit category



Homotopy theory

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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.



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

  • objects are the homogeneous spaces (GG-orbit types) G/HG/H, where HH is a closed subgroup of GG,

  • and whose morphisms are GG-equivariant maps.


For suitable continuous actions of GG on a topological space XX, every orbit of the action is isomorphic to one of the homogeneous 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”.


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.


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.


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 go across to the restricted setting.



(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\}


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}


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 equivariant homotopy theory

The GG-orbit category is the slice (∞,1)-category of the global orbit category OrbOrb 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.


Lecture notes include

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

  • 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. With contributions by M. Cole, G. Comezana, S. Costenoble, A. D. Elmenddorf, J. P. C. Greenlees, L. G. Lewis, Jr., R. J. Piacenza, G. Triantafillou, and S. Waner. (pdf)

A very general setting for the use of orbit categories is described in

  • W. G. Dwyer and D. M. Kan, Singular functors and realization functors , Nederl. Akad. Wetensch. Indag. Math., 87, (1984), 147 – 153.

For more on the relation to global equivariant homotopy theory see

Last revised on October 24, 2020 at 06:18:11. See the history of this page for a list of all contributions to it.