From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/7926 Path: news.gmane.org!not-for-mail From: Emily Riehl Newsgroups: gmane.science.mathematics.categories Subject: subgroupoids of V-categories Date: Tue, 19 Nov 2013 13:51:41 -0500 (EST) Message-ID: Reply-To: Emily Riehl NNTP-Posting-Host: plane.gmane.org Mime-Version: 1.0 Content-Type: TEXT/PLAIN; format=flowed; charset=US-ASCII X-Trace: ger.gmane.org 1384889752 28536 80.91.229.3 (19 Nov 2013 19:35:52 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Tue, 19 Nov 2013 19:35:52 +0000 (UTC) To: Categories Original-X-From: majordomo@mlist.mta.ca Tue Nov 19 20:35:57 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 1Vir5T-0006Jj-8o for gsmc-categories@m.gmane.org; Tue, 19 Nov 2013 20:35:51 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:41049) by smtp3.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1Vir4j-0001O5-Ph; Tue, 19 Nov 2013 15:35:05 -0400 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1Vir4i-0005Cv-JA for categories-list@mlist.mta.ca; Tue, 19 Nov 2013 15:35:04 -0400 Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:7926 Archived-At: 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/ ]