Eric Forgy Exploding and Imploding a Category

Idea

Given a category $C$, it is often the case that each object in $C$ is itself a set with additional structure. Exploding such a category results in a new category $Explode(C)$ whose objects consist of all the elements of all the objects of $C$. The morphisms of $Explode(C)$ are the obvious ones inherited from the morphisms of $C$, i.e. given a morphism

in $C$, we have the corresponding morphism

in $Explode(C)$.

Definition

Given a category $C$ 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 $Explode(C)$ is the pullback along these two forgetful functors

If $C$ is a concrete category, meaning that $F_C$ is faithful and representable by some object $I$, then $\Explode{C}$ is simply the coslice category $(I \to C)$.

Example: Action Groupoid

Given a vector space $V$, a group $G$, and a representation $\rho: G\to End(V)$, let

denote the category with one object $V$ and one morphism

for each element in $G$. The action groupoid? $V//G$ is given by

QUESTION: Is $V\nearrow G$ a concrete category so that the action groupoid is a coslice category? -Eric

Discussion

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