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/ ]

Hi Emily, I'm a bit worried about the notion of "local monomorphism". If V is, say, a poset, then every arrow in it is (tautologously) monic, and therefore every V-functor is a local monomorphism. Let us consider, for instance, generalised metric spaces. Suppose C has two vertices, p and q, with C(p,q)=0 and C(q,p)=oo (infinity). Then, for any e>0, we can choose G_e with G_e(p,q)=e and G(q,p)=oo, and the identity-on-objects functor G_e --> C has the property you desire. [I'm sure you realise that the definition of "isomorphism" you gave is equivalent to that of "isomorphism in the underlying category"; the underlying category of C is "the arrow", whereas that of G_e is discrete.] So "the maximal subgroupoid" of C is not defined uniquely, except up to "equivalence of underlying category". Moreover, whenever e>d>0, the i-o-o functor G_e --> C factors through G_d, so there is not even an "optimal" choice of e. More precisely, if one were to define a V-groupoid to be a V-category whose underlying category is a groupoid (which is, implicitly, what you seem to be doing), then I've just shown that the forgetful functor from [0,oo]-groupoids to [0,oo]-categories does not have a right adjoint. Of course, I think it would be better to define a V-groupoid in a more Hopf-y kind of way---e.g., in terms of what one might call a co-Maltsev operation---but that might not suit your purposes, and would seem to make things much more difficult, at least in general. Cheers, Jeff. -------------------------------------------- On Tue, 11/19/13, Emily Riehl <eriehl@math.harvard.edu> wrote: Subject: categories: subgroupoids of V-categories To: "Categories" <categories@mta.ca> Received: Tuesday, November 19, 2013, 1:51 PM Hi all, For general V (closed symmetric monoidal, bicomplete), is there a general way to construct the maximal subgroupoid of a V-category C? I think I know how to *detect* the maximal subgroupoid. A map in C is an isomorphism iff it is representably so: Writing 1 for the monoidal unit, we say f : 1 -> C(x,y) is an iso iff the induced map f_* : C(z,x) -> C(z,y) is an iso in V for all z. So we might say that a V-category G with the same objects and C and an identity-on-objects local monomorphism G -> C is the maximal subgroupoid provided that a morphism f factors through G(x,y) -> C(x,y) just when f is an isomorphism. In examples, this is probably good enough, but I still would feel better if I had a general construction of the maximal subgroupoid. I feel like this should be some sort of weighted limit, perhaps with some additional structure on V? Thanks, Emily [For admin and other information see: http://www.mta.ca/~cat-dist/ ]