From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/7022 Path: news.gmane.org!not-for-mail From: Steve Lack Newsgroups: gmane.science.mathematics.categories Subject: Re: Dualizing comma categories Date: Mon, 31 Oct 2011 09:29:18 +1100 Message-ID: References: Reply-To: Steve Lack NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 (Apple Message framework v1084) Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: quoted-printable X-Trace: dough.gmane.org 1320065443 28275 80.91.229.12 (31 Oct 2011 12:50:43 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Mon, 31 Oct 2011 12:50:43 +0000 (UTC) Cc: categories To: David Leduc Original-X-From: majordomo@mlist.mta.ca Mon Oct 31 13:50:39 2011 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtpx.mta.ca ([138.73.1.4]) by lo.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1RKrK2-00063g-HQ for gsmc-categories@m.gmane.org; Mon, 31 Oct 2011 13:50:38 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:35908) by smtpx.mta.ca with esmtp (Exim 4.76) (envelope-from ) id 1RKrIz-0000cI-Qb; Mon, 31 Oct 2011 09:49:33 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1RKrIy-0002OE-7C for categories-list@mlist.mta.ca; Mon, 31 Oct 2011 09:49:32 -0300 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:7022 Archived-At: Dear David, As usual, constructing colimits in Cat (and other concrete categories = or 2-categories) is more difficult than constructing limits.=20 As you suspected, a cocomma object over the initial object is just a = coproduct. This is completely general, not just true in Cat. Seen as a general = 2-categorical fact, it becomes the same fact that you mentioned: a comma object over a terminal object = is just a product.=20 I won't try to describe the general cocomma object, but another special = case of a cocomma=20 object is the *collage* of an arrow f:A-->B. This is the universal = diagram containing arrows=20 i:A->C and j:B->C and a 2-cell jf->i. It can be seen as a cocomma object = of f and the identity 1_A. This special case is easy to describe in Cat. The object-set of C = is the disjoint=20 union of the object-sets of A and of B. A morphism in C between objects = of A is a=20 morphism in A; a morphism in C between objects of B is a morphism of B. = There is a morphism fa->a for each a in A, and these are the only morphisms from = objects of B to objects of A; there are no morphisms from objects of A to = objects of B.=20 (Thus the morphisms can be described by the 2x2 matrix with entries A, = f, 0, B; this can=20 be made precise if you think of the underlying span of a category as a = matrix.) Steve Lack. On 30/10/2011, at 11:34 PM, David Leduc wrote: > Hi, >=20 > A comma category is a comma object in the 2-category Cat of categories > and functors. And a comma object is defined by a universal property. > Now, one can dualize the notion of comma object by turning around the > 1-cells and/or 2-cells in its definition. My question is: when we > instantiate those dualized definitions to Cat, what do we obtain? In > other words, what is a "co-comma category"? >=20 > For example, since the product of two categories is a special case of > comma category, I would expect that the coproduct of two categories is > a special case of "co-comma category". >=20 > Thanks! >=20 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]