From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6064 Path: news.gmane.org!not-for-mail From: Michael Shulman Newsgroups: gmane.science.mathematics.categories Subject: Re: Generalizing Comma Categories Date: Sun, 22 Aug 2010 14:24:55 -0700 Message-ID: References: Reply-To: Michael Shulman NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 X-Trace: dough.gmane.org 1282572377 16307 80.91.229.12 (23 Aug 2010 14:06:17 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Mon, 23 Aug 2010 14:06:17 +0000 (UTC) Cc: categories@mta.ca To: David Leduc Original-X-From: majordomo@mlist.mta.ca Mon Aug 23 16:06:15 2010 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtpy.mta.ca ([138.73.1.139]) by lo.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1OnXfC-0004GJ-Kt for gsmc-categories@m.gmane.org; Mon, 23 Aug 2010 16:06:14 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:34346) by smtpy.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1OnXdS-0000I3-4b; Mon, 23 Aug 2010 11:04:26 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1OnXdM-0006Sx-CP for categories-list@mlist.mta.ca; Mon, 23 Aug 2010 11:04:20 -0300 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6064 Archived-At: This is an interesting and slightly tricky question. There is a notion of "comma 2-category" which has this property; I believe it was first written down by Gray. Given (strict or weak) 2-functors F:A-->C and G:B-->C betwen 2-categories (or bicategories), their comma 2-category (F/G) is defined as follows: * its objects are triples (a, b, s:F(a)-->G(b)) of an object in A, an object in B, and a morphism in C. * its morphisms from (a,b,s) to (a',b',s') are triples (u:a-->a', v:b-->b', z: s;G(v) --> F(u);s') of a morphism in A, a morphism in B, and a 2-cell in C. * its 2-cells from (u,v,z) to (u',v',z') are pairs (x:u-->u', y:v-->v') of a 2-cell in A and a 2-cell in B, such that (F(x);s').z = z'.(s;G(y)). You can check that if A and B are terminal so that F and G pick out objects c, c' of C, then (F/G) is the hom-category C(c,c') regarded as a locally discrete 2-category. There is a "dual" comma 2-category which would give you C(c,c')^op instead (assuming I didn't screw up and write down the dual version above). The tricky part is that while an ordinary comma category is expressible as a certain kind of weighted limit in the 2-category Cat, and can therefore be generalized to comma objects in other 2-categories, the comma 2-category as defined above is *not*, as far as I know, a weighted limit in the (strict or weak) 3-category 2-Cat. It does have a universal property, though: just as an ordinary comma category is equipped with a natural transformation filling a square in a universal way, the comma 2-category is equipped with a *lax* natural transformation. But there is no 3-category whose 2-cells are lax natural transformations. I believe that this universal property can be expressed as a weighted limit in Gray_lax enriched categories, where Gray_lax denotes 2-Cat with the lax version of the Gray tensor product, but I don't know any references for such a point of view. I'd be very interested to hear of other work on this question; it seems potentially important in developing a notion of exactness for 3-categories. Mike On Fri, Aug 20, 2010 at 6:46 PM, David Leduc wrote: > Hi, > > Thank you for all the replies to my previous questions. > It is very helpful. > Now, I have another (trivial) question. > > If I have two constant functors Delta_X and Delta_Y that respectively > returns X and Y in C, then their comma category is the discrete category > C(X,Y). > > How do comma categories generalize to bicategories? > > When C is a 2-category, I would like the "comma category" of the > "functors"(?) Delta_X and Delta_Y be the (not necessarily discrete) category > C(X,Y). > [For admin and other information see: http://www.mta.ca/~cat-dist/ ]