From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5051 Path: news.gmane.org!not-for-mail From: "Pieter Hofstra" Newsgroups: gmane.science.mathematics.categories Subject: RE: Non-free cocompletions Date: Fri, 10 Jul 2009 11:09:45 -0400 Message-ID: Reply-To: "Pieter Hofstra" NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset="iso-8859-1" Content-Transfer-Encoding: quoted-printable X-Trace: ger.gmane.org 1247323829 27418 80.91.229.12 (11 Jul 2009 14:50:29 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Sat, 11 Jul 2009 14:50:29 +0000 (UTC) To: Original-X-From: categories@mta.ca Sat Jul 11 16:50:22 2009 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from mailserv.mta.ca ([138.73.1.1]) by lo.gmane.org with esmtp (Exim 4.50) id 1MPdu9-0000v0-Tz for gsmc-categories@m.gmane.org; Sat, 11 Jul 2009 16:50:22 +0200 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1MPd9B-0001SQ-GT for categories-list@mta.ca; Sat, 11 Jul 2009 11:01:49 -0300 Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5051 Archived-At: There are situations where standard completion constructions are = insufficient to accurately describe the relationship between two = categories. The motivating example in the paper "Relative completions" = (JPAA 192, 2004) was the presentation of the Effective topos as an exact = completion of the category of partitioned assemblies. This presentation = relies on the axiom of choice in Sets, and therefore does not work when = we work over an arbitrary base topos. The solution is to define a = relative version of the exact completion which preserves quotients of = equivalence relations coming from the base topos. More precisely, it is = defined by first freely adding all quotients, but then formally = inverting the canonical comparison morphisms between the new quotients = and the old ones from the base. Best regards, Pieter -----Original Message----- From: categories@mta.ca on behalf of Andree Ehresmann Sent: Mon 7/6/2009 4:08 AM To: Categories Subject: categories: Non-free cocompletions =20 Vaughan Pratt writes > Incidentally, of what use are non-free cocompletions? Is there any reason not to define "cocompletion" to make it free? I can indicate two important uses of non-free cocompletions, and more precisely cocompletions for particular classes of diagrams preserving some given colimits: 1. The construction of what, with Charles, we called the "prototype" and the "type" associated to a sketch (in "Categories of sketchd structures", Cahiers Top. et Geom. Diff. III-2, 1972) 2. The "complexification process" which, with Jean-Paul Vanbremeersch, we use extensively in our model for hierarchical evolutionary systems ("Memory Evolutive Systems: Hierarchy, Emergence, Cognition", Elsevier 2007) Kindly Andree [For admin and other information see: http://www.mta.ca/~cat-dist/ ]