From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/7930 Path: news.gmane.org!not-for-mail From: Ronnie Brown Newsgroups: gmane.science.mathematics.categories Subject: Re: subgroupoids of V-categories Date: Wed, 20 Nov 2013 11:18:18 +0000 Message-ID: References: Reply-To: Ronnie Brown NNTP-Posting-Host: plane.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1385044786 31099 80.91.229.3 (21 Nov 2013 14:39:46 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Thu, 21 Nov 2013 14:39:46 +0000 (UTC) To: Emily Riehl , "categories@mta.ca" Original-X-From: majordomo@mlist.mta.ca Thu Nov 21 15:39:52 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 1VjVQ2-0006oU-FJ for gsmc-categories@m.gmane.org; Thu, 21 Nov 2013 15:39:46 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:43982) by smtp3.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1VjVOM-0004i1-UE; Thu, 21 Nov 2013 10:38:02 -0400 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1VjVOK-00074V-PQ for categories-list@mlist.mta.ca; Thu, 21 Nov 2013 10:38:00 -0400 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:7930 Archived-At: Dear Emily, The following seems relevant to the question and Zhen's answer. A related question is how to obtain a maximal subgroup object of a monoid object wrt tensor in a monoidal category. A particular case concerned is in 59. (with N.D. GILBERT), ``Algebraic models of 3-types and automorphism structures for crossed modules'', {\em Proc. London Math. Soc.} (3) 59 (1989) 51-73. where the monoidal closed category is XMod, that of crossed modules over groupoids, with internal hom written XMOD. For an object C one can define END(C)= XMOD(C,C), but what should be AUT(C)? In this example there is a functor say U: XMod \to Set which takes the zero dimensional part. We then define AUT(C) to be the part of END(C) over the maximal subgroup of END(C)_0. This is a "group-like object". One argument for this is that the nerve N(C) of an object C is a simplicial set, so N(END(C)) is a simplicial monoid; it turns out that N(AUT(C)) is a simplicial group, which gives a kind of justification. The above ideas extend to crossed complexes, using Andy Tonks extension of the Eilenberg-Zilber Theorem from chain complexes to crossed complexes. All this was part of the following scheme for "higher order symmetry" (with applications to .....?!), which clearly should be part of "higher order group theory": sets give models of homotopy 0-types; the automorphisms of a set form a group which is a model of (pointed, connected) homotopy 1-types (ancient history); the automorphisms of a group form (part of the structure of) a crossed module which is a model of (pointed, connected) homotopy 2-types (Mac Lane-Whitehead); and in the above paper: the automorphisms of a crossed module form (part of the structure of) a crossed square, (see also a paper of Norrie), which is a model of (pointed, connected) 3-types (Loday); the automorphisms of a crossed square form ......... (???) Best wishes Ronnie On 19/11/2013 18:51, Emily Riehl wrote: > 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/ ]