From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/7927 Path: news.gmane.org!not-for-mail From: Zhen Lin Low Newsgroups: gmane.science.mathematics.categories Subject: Re: subgroupoids of V-categories Date: Tue, 19 Nov 2013 21:24:47 +0000 Message-ID: References: Reply-To: Zhen Lin Low NNTP-Posting-Host: plane.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 X-Trace: ger.gmane.org 1384912201 18913 80.91.229.3 (20 Nov 2013 01:50:01 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 20 Nov 2013 01:50:01 +0000 (UTC) Cc: Categories To: Emily Riehl Original-X-From: majordomo@mlist.mta.ca Wed Nov 20 02:50:05 2013 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtp3.mta.ca ([138.73.1.186]) by plane.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1Viwvc-00089S-Hi for gsmc-categories@m.gmane.org; Wed, 20 Nov 2013 02:50:04 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:42138) by smtp3.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1ViwuI-0003mq-6n; Tue, 19 Nov 2013 21:48:42 -0400 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1ViwuI-0007eF-Fa for categories-list@mlist.mta.ca; Tue, 19 Nov 2013 21:48:42 -0400 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:7927 Archived-At: 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/ ]