Sets with a Category Action

Let C be a small category and Set the category of sets. We define a C-set to be a functor Ω : C → Set. Thus Ω is simply a diagram of sets, the diagram having the same shape as C: for each object x of C there is specified a set Ω(x) and for each morphism α : x → y there is a mapping of sets Ω(α) : Ω(x) → Ω(y). If C happens to be a group (a category with one object and morphism set G) then a C-set is the same thing as a G-set, since the C-set singles out a set and sends each morphism of C to a permutation of the set. We see that C-sets form a category, the morphisms being natural transformations between the functors. Thus we have a notion of isomorphism of C-sets. Given two C-sets Ω1 and Ω2 we define their disjoint union Ω1 t Ω2 to be the C-set defined at each object x of C by (Ω1 t Ω2)(x) := Ω1(x) t Ω2(x) with the expected definition of Ω1 t Ω2 on morphisms. Let us call a C-set Ω a single orbit C-set or transitive if it cannot be expressed properly as a disjoint union. A C-set Ω may happen to be the disjoint union of two C-sets, or not; if it can be broken up as a disjoint union we can ask if either of the factors is a disjoint union, and by repeating this we end up with a disjoint union of C-sets each of which is transitive.