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.
It is a small topologically enriched category (though of course if is a discrete group, the enrichment of 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 and such that .
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 .
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.
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).
This should not be confused with the situation where a group acts on a groupoid so that one obtains the orbit groupoid.