Eric Forgy Exploding and Imploding a Category

Idea

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

XfXX\stackrel{f}{\to}X'

in CC, we have the corresponding morphism

(xX)f x(f(x)X)(x\in X)\stackrel{f_x}{\to}(f(x)\in X')

in Explode(C)Explode(C).

Definition

Given a category CC whose objects are sets with structure, let

F C:CSetF_C: C\to Set

denote the forgetful functor that forgets the additional structure and let

F *:Set *SetF_*: Set_*\to Set

denote the forgetful functor from pointed sets that forgets the point.

The exploded category Explode(C)Explode(C) is the pullback along these two forgetful functors

Explode(C) C Set * F C F * Set \array{Explode(C) \\ \array{ {} & {} & {} & \bullet & {} & {} \\ {} & {} & \swarr & {} & \searr & {} \\ C & \bullet & {} & {} & {} & Set_* \\ {} & {} & {}_{F_C}\searr & {} & \swarr_{F_*} & {} \\ {} & {} & {} & \bullet & {} & {} \\ {} & {} & {} & Set & {} & {} } }

If CC is a concrete category, meaning that F CF_C is faithful and representable by some object II, then ExplodeC\Explode{C} is simply the coslice category (IC)(I \to C).

Example: Action Groupoid

Given a vector space VV, a group GG, and a representation ρ:GEnd(V)\rho: G\to End(V), let

VG.V\nearrow G.

denote the category with one object VV and one morphism

ρ(g):VV\rho(g):V\to V

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

V//G=Explode(VG).V//G=Explode(V\nearrow G).

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

Discussion

category: drafts
Revised on September 5, 2009 at 19:59:33 by Toby Bartels