nLab
orbit category

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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

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Representation theory

Cohomology

cohomology

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Special notions

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Extra structure

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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

(e.g. tom Dieck 1987, I (10.1))

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.

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

(families of subgroups)
More generally, given a family \mathcal{F} 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. For such families, many of the considerations of results such as Elmendorf's theorem will still hold.

Equivariant Postnikov tower

Given a topological G-space, there is (following tom Dieck 1987, Sec. I.10) a sequence of variants (enhancements) of the GG-orbit category which mixes transitions between the fixed loci with n-truncations of their homotopy types.

Equivariant set of connected components

(…)

tom Dieck 1987, Sec. I.10

(…)

Equivariant fundamental groupoid

Definition

For GGrp(Set)G \,\in\, Grp(Set) a discrete group and GXGAct(TopSp)G \curvearrowright X \,\in\, G Act(TopSp) a topological G 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 HGH \subset G and a GG-equivariant function x X:G/HX\; x_X \,\colon\, G/H \to X, hence, equivalently, a point in the fixed locus X HX^H;

  • morphismsx H(ϕ,[γ])x Hx_H \xrightarrow{ (\phi,[\gamma]) } x'_{H'} are pairs consisting of a GG-equivariant function G/HϕG/HG/H \xrightarrow{ \phi } G/H' and a relative GG-homotopy class [γ][\gamma] of a continuous path γ:G/H×[0,1]X\gamma \,\colon\, G/H \times [0,1] \xrightarrow{\;} X from γ(0)=x H\gamma(0) = x_H to γ 1=ϕ *x X\gamma_1 = \phi^\ast x'_X'.

Example

For GX=G*G \curvearrowright X \,=\, G \curvearrowright \ast the point, its fundamental category (equivariant fundamental groupoid) according to Def. is the plain orbit category GOrbtsG Orbts (Def. ).

Proposition

(equivariant fundamental groupoid under Grothendieck construction)
If we write

τ 1ʃ(X ())Sh(GOrbt,Grpd 1) \tau_1 \esh \big( X^{(-)}\big) \;\in\; Sh\big( G Orbt, Grpd_1 \big)

for the fundamental groupoid of XX in its incarnation as a presheaf of fixed loci over the orbit category, hence for the (2,1)-functor

(1)GOrbt op τ 1ʃ(X ()) Grpd 1 G/H τ 1ʃ(X H), \array{ G Orbt^{op} &\xrightarrow{ \tau_1 \esh \big( X^{(-)}\big) }& Grpd_1 \\ G/H &\mapsto& \tau_1 \esh (X^H) \mathrlap{\,,} }

which sends G/HG/H to the ordinary fundamental groupoid of the fixed locus X HX^H, then the equivariant fundamental groupoid of GXG \curvearrowright X (Def. ) is equivalently the Grothendieck construction

(2)G/HGOrbtτ 1ʃ(X H)Grpd 1 \underset {G/H \in G Orbt} {\int} \, \tau_1 \, \esh (X^{H}) \;\;\; \in \; Grpd_1

on (1).

This follows by direct unwinding of the definitions; see also Pronk & Scull 2021, below Def. 3.1.

Equivariant universal cover

Definition

(equivariant universal covering)
For GGrp(Set)G \in Grp(Set) a discrete group and GXGAct(TopSp)G \curvearrowright X \,\in\, G Act(TopSp) a well-behaved topological G G -space, its equivariant universal cover (tom Dieck 1987 (10.13)) “is” the functor from the fundamental equivariant groupoid (2) of GXG \curvearrowright X to TopSp

G/HGOrbtτ 1ʃ(X H) TopSp ʃ Grpd x H X x H^ \array{ \underset {G/H \in G Orbt} {\int} \, \tau_1 \, \esh (X^{H}) &\xrightarrow{\;}& TopSp &\xrightarrow{\esh}& Grpd_{\infty} \\ x_H &\mapsto& \widehat{X^H_x} }

which sends x HX Hx_H \,\in\, X^H to the universal covering X x^\widehat{X_x} of the connected component of x HXx_H \,\in\, X given by relative-homotopy-equivalences classes of paths based at x Hx_H.

Write

(3)GGrpd Sh(GOrbt,Grpd ) G Grpd_\infty \;\coloneqq\; Sh\big( G Orbt, \, Grpd_\infty \big)

for the \infty -category of \infty -presheaves on the GG-orbit category, hence for the proper GG-equivariant homotopy theory.

For GXGAct(TopSp)G \curvearrowright X \,\in\, G Act(TopSp) a topological G G -space, its equivariant homotopy type is

ʃ(X ())GGrpd . \esh \big(X^{(-)}\big) \;\; \in \; G Grpd_\infty \,.

Proposition

For GXGAct(TopSp)G \curvearrowright X \,\in\, G Act(TopSp) a topological G G -space, its equivariant universal covering according to Def. is, under Prop. and the (infinity,1)-Grothendieck construction equivalent to the 1-truncation unit

ʃ(X ()) η τ 1 τ 1ʃ(X ())(GGrpd ) /τ 1ʃ(X ()), \array{ \esh \big( X^{(-)}\big) \\ \big\downarrow {}^{\mathrlap{ \eta^{\tau_1} }} \\ \tau_1 \, \esh \big( X^{(-)}\big) } \;\;\; \in \; \big( G Grpd_\infty\big)_{/ \tau_1 \, \esh \big( X^{(-)}\big)} \,,

in that

G/HGOrbtʃ(X H) Grpd^ G/HGOrbtτ 1ʃ(X H) X () ()^ Grpd . \array{ \underset{G/H \in G Orbt}{\int} \esh \big( X^{H}\big) &\xrightarrow{\;}& \widehat{Grpd}_\infty \\ \big\downarrow &\swArrow& \big\downarrow \\ \underset{G/H \in G Orbt}{\int} \tau_1 \, \esh \big( X^{H}\big) & \xrightarrow{ \widehat{ X^{(-)}_{(-)} } } & Grpd_\infty \,. }

bottom right corner not quite right, will fix…


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:

(4) 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}

Properties

General properties

  • Orbit categories as well as fundamental categories of GG-spaces are EI-categories.

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.

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)-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).

Base change of equivariance groups

Example

(left base change along covering of equivariance group)
Let G^pG\widehat{G} \overset{p}{\twoheadrightarrow} G be a surjective homomorphism of discrete groups. For HGH \subset G a subgroup, write H^G^\widehat{H} \subset \widehat{G} for its pullback

Then there is a reflective subcategory-inclusion of orbit categories

where

  • R(G/H)=G^/H^R(G/H) \,=\, \widehat{G}/\widehat{H}

  • L(G^/K)=G/p(K)L(\widehat{G}/K) \,=\, G/p(K).

A transparent way to see this is to identify (as above) GG-orbits with the 0-truncated objects in the BGB G-slice of SinglrtGrpd 1 cnSinglrt \,\coloneqq\, Grpd_1^{cn} :

GOrbt((Grpd 1 cn) /BG) 0. G Orbt \;\simeq\; \left( (Grpd_1^{cn})_{/ B G} \right)_0 \,.

Under this identification, the adjunction is the canonical left base change on slices composed with the 0-truncation reflection (here):

That the right adjoint RR is fully faithful is most readily seen by direct inspection: A map between G^\widehat{G}-sets on both of which G^\widehat{G} acts only through GG is evidently G^\widehat{G}-equivariant if and only if it is GG-equivariant.


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 in relation to global equivariant homotopy theory see

The notion of fundamental category of a GG-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.