Paths and cylinders
geometric representation theory
representation, 2-representation, ∞-representation
group algebra, algebraic group, Lie algebra
vector space, n-vector space
affine space, symplectic vector space
module, equivariant object
bimodule, Morita equivalence
induced representation, Frobenius reciprocity
Hilbert space, Banach space, Fourier transform, functional analysis
orbit, coadjoint orbit, Killing form
geometric quantization, coherent state
module algebra, comodule algebra, Hopf action, measuring
Geometric representation theory
D-module, perverse sheaf,
Grothendieck group, lambda-ring, symmetric function, formal group
principal bundle, torsor, vector bundle, Atiyah Lie algebroid
geometric function theory, groupoidification
Eilenberg-Moore category, algebra over an operad, actegory, crossed module
Special and general types
The orbit category of a group is the category of “all kinds” of orbits of , namely of all suitable coset spaces regarded as G-spaces.
Given a topological group the orbit category (denoted also ) is the category whose
objects are the homogeneous spaces (-orbit types) , where is a closed subgroup of ,
and whose morphisms are -equivariant maps.
More generally, given a family of subgroups of which is closed under conjugation and taking subgroups one looks at the full subcategory whose objects are those for which .
Sometimes a family, , of subgroups is specified, and then a subcategory of consisting of the where will be considered. If the trivial subgroup is in then many of the considerations of results such as Elmendorf's theorem will go across to the restricted setting.
Relation to -spaces and Elmendorf’s theorem
Elmendorf's theorem (see there for details) states that the (∞,1)-category of (∞,1)-presheaves on the orbit category are equivalent to the localization of topological spaces with -action at the “fixed point weak equivalences”.
Relation to equivariant homotopy theory
The -orbit category is the slice (∞,1)-category of the global orbit category over the delooping :
This means that in the general context of global equivariant homotopy theory, the orbit category appears as follows.
Rezk-global equivariant homotopy theory:
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 -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.
- 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