From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/7091 Path: news.gmane.org!not-for-mail From: Ross Street Newsgroups: gmane.science.mathematics.categories Subject: Re: The category of categories as a 3-limit Date: Fri, 2 Dec 2011 13:17:19 +1100 Message-ID: References: Reply-To: Ross Street NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 (Apple Message framework v936) Content-Type: text/plain; charset=WINDOWS-1252; format=flowed; delsp=yes Content-Transfer-Encoding: quoted-printable X-Trace: dough.gmane.org 1322833205 15012 80.91.229.12 (2 Dec 2011 13:40:05 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Fri, 2 Dec 2011 13:40:05 +0000 (UTC) Cc: categories To: David Leduc Original-X-From: majordomo@mlist.mta.ca Fri Dec 02 14:40:00 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 1RWTLK-0002XC-Ck for gsmc-categories@m.gmane.org; Fri, 02 Dec 2011 14:39:58 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:33005) by smtpx.mta.ca with esmtp (Exim 4.76) (envelope-from ) id 1RWTKQ-0004jf-1V; Fri, 02 Dec 2011 09:39:02 -0400 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1RWTKO-0003B8-Bf for categories-list@mlist.mta.ca; Fri, 02 Dec 2011 09:39:00 -0400 Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:7091 Archived-At: On 02/12/2011, at 12:17 PM, David Leduc wrote: >> http://www.maths.mq.edu.au/~street/Sketch.pdf > > Thank you very much but I am afraid I am already stuck at the first =20= > sentence. > Why such a diagram in Cat would be a theory? A (limit) sketch in the sense of Ehresmann is a family of cones on a =20 category C. A cone is a natural transformation pointing right in a triangle whose =20= top horizontal side is X ------> 1 and whose bottom vertex is C. A cone is the special case of what I =20 called a theory with A =3D X and B =3D D =3D 1 (the terminal category). Now let A be the coproduct (disjoint union) of all the categories X in =20= the sketch and let B =3D D be the discrete category obtained by adding up all the 1s, one for each =20 index of the family. Take u : A --> B to be the induced map on the coproducts. Take t to be =20= the identity. The cones give a single natural transformation tau using the 2-universal =20 property of coproduct. So a sketch on C is the same as one of my theories with B discrete and =20= t an identity. The extra flexibility of having B not necessarily discrete and tagging =20= on the t was aiming at theories in the sense of John Isbell [General functorial semantics. I. Amer. J. Math. 94 =20 (1972), 535=96596; MR0396718 (53 #580)]. Ross [For admin and other information see: http://www.mta.ca/~cat-dist/ ]