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/ ]