Morphisms and Isomorphisms Between G-sets
If X and Y are two G-sets, we define a morphism from X to Y to be a function f : X → Y such that f(g.x) = g.f(x) for all g in G and all x in X. Morphisms of G-sets are also called equivariant maps or G-maps.
The composition of two morphisms is again a morphism.
If a morphism f is bijective, then its inverse is also a morphism, and we call f an isomorphism and the two G-sets X and Y are called isomorphic; for all practical purposes, they are indistinguishable in this case.
Some example isomorphisms:
- Every regular G action is isomorphic to the action of G on G given by left multiplication.
- Every free G action is isomorphic to G×S, where S is some set and G acts on G×S by left multiplication on the first coordinate. (S can be taken to be the set of orbits X/G.)
- Every transitive G action is isomorphic to left multiplication by G on the set of left cosets of some subgroup H of G. (H can be taken to be the stabilizer group of any element of the original G-set.the original action.)
With this notion of morphism, the collection of all G-sets forms a category; this category is a Grothendieck topos (in fact, assuming a classical metalogic, this topos will even be Boolean).
Read more about this topic: Group Action