Given a category , it is often the case that each object in is itself a set with additional structure. Exploding such a category results in a new category whose objects consist of all the elements of all the objects of . The morphisms of are the obvious ones inherited from the morphisms of , i.e. given a morphism
in , we have the corresponding morphism
in .
Given a category whose objects are sets with structure, let
denote the forgetful functor that forgets the additional structure and let
denote the forgetful functor from pointed sets that forgets the point.
The exploded category is the pullback along these two forgetful functors
If is a concrete category, meaning that is faithful and representable by some object , then is simply the coslice category .
Given a vector space , a group , and a representation , let
denote the category with one object and one morphism
for each element in . The action groupoid? is given by
QUESTION: Is a concrete category so that the action groupoid is a coslice category? -Eric