Re: subgroupoids of V-categories

[Zhen Lin Low <zll22@cam.ac.uk>]

Dear Emily,

I'm sure someone else will have pointed this out by now, but it seems
quite unlikely that the maximal subgroupoid has a V-enrichment in
general! For instance, suppose V is the category of pointed sets: then
a V-category is simply a category with a system of distinguished zero
morphisms, but a zero morphism is hardly ever an isomorphism. (Of
course, one might argue that the maximal V-subgroupoid _does_ in fact
exist and is the full subcategory of zero objects, but that seems
quite uninteresting.)

Perhaps things are better if we focus on those categories V which are
sufficiently Set-like. Suppose V has pullbacks and the "forgetful"
functor U : V -> Set is an isofibration (= transportable functor) and
has a fully faithful right adjoint. Then, for any injective map f :  X
-> U B in Set, there exists a monomorphism g : A -> B in V such that f
= U g and for any morphism h : C -> B in V such that U h = f k in Set,
there is a unique morphism m : C -> A with k = U m and h = g m. For
example, if V = Top, this is just the construction of the subspace
topology on a subset, and if V = sSet, this constructs the maximal
simplicial subset containing a fixed subset of vertices. It is clear
how to use this to make the set of isomorphisms between two objects in
a V-category into a V-object. Composition is inherited if U is
strongly monoidal: the universal property of these "initial lifts"
gives the required factorisations. And there is nothing special about
the maximal subgroupoid: this construction works for any subcategory
whatsoever.

I suppose one could also work "internally" in the cartesian monoidal
case and try to define the V-object of isomorphisms by pulling back
the "name" of the identity morphism along composition and then
projecting down to one of the factors. But I don't see how to
generalise this to the non-cartesian case.

Best wishes,
--
Zhen Lin


[For admin and other information see: http://www.mta.ca/~cat-dist/ ]