From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6925 Path: news.gmane.org!not-for-mail From: selinger@mathstat.dal.ca (Peter Selinger) Newsgroups: gmane.science.mathematics.categories Subject: Re: partial categories Date: Wed, 28 Sep 2011 23:18:59 -0300 (ADT) Message-ID: References: Reply-To: selinger@mathstat.dal.ca (Peter Selinger) NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit X-Trace: dough.gmane.org 1317299829 18533 80.91.229.12 (29 Sep 2011 12:37:09 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Thu, 29 Sep 2011 12:37:09 +0000 (UTC) Cc: categories@mta.ca To: eriehl@math.harvard.edu Original-X-From: majordomo@mlist.mta.ca Thu Sep 29 14:37:03 2011 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtpx.mta.ca ([138.73.1.30]) by lo.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1R9FrG-00066s-Ve for gsmc-categories@m.gmane.org; Thu, 29 Sep 2011 14:36:59 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:34906) by smtpx.mta.ca with esmtp (Exim 4.76) (envelope-from ) id 1R9Fq1-0002cc-8x; Thu, 29 Sep 2011 09:35:41 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1R9Fpz-0004NR-Gb for categories-list@mlist.mta.ca; Thu, 29 Sep 2011 09:35:39 -0300 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6925 Archived-At: Dear Emily, one notion of "partial category" that has been studied is Freyd's notion of "paracategory". It differs from what you wrote below as follows: instead of taking compositions of two arrows as the primitive operation, one takes compositions of n arrows as primitive operations, for all n. So if f1, ..., fn are n arrows (so that the codomain of each is the domain of the next), then [f1, ..., fn] is their composition, which may be defined or undefined. In the case of (total) categories, adding n-ary compositions makes no difference, since they are already definable in terms of identities and binary composition. But in the partial case, it does make a difference, as it is possible, for example, that [f,g,h] is defined, but [f,g] and [g,h] are undefined. The axioms are: (a) [] : A -> A is defined (the composition of the empty path from A to A) (b) [f] is defined and equal to f, for all arrows f, (c) if ff, gg, hh are are (composable) paths, and if [gg] is defined, then [ff,[gg],hh] is defined if and only if [ff,gg,hh] is defined, and in this case, they are both equal. The main representation theorem is: Every reflexive subgraph of a category is a paracategory; conversely, every paracategory can be faithfully completed to a category. It is not the only possible notion of partial category, and may not always be the notion that one wants. -- Peter Emily Riehl wrote: > > A colleague of mine is wondering if anyone has studied "partial > categories," by which she means directed graphs with identities but with > only some compositions (including all identity compositions) defined. > > A partial category can be thought of as a category enriched in pointed > sets (with smash product as tensor and S^0 as unit). The slogan is that > the basepoint in each hom-set stands in for "does not exist". But enriched > functors don't give the right notion of maps; these should preserve > identities and all specified compositions. Enriched functors behave > appropriately with regards to the identites but may "forget" extant > arrows and in particular need not preserve composites. So perhaps this > perspective is not useful. > > I'll happily pass along any suggestions. > > Thanks, > Emily Riehl > [For admin and other information see: http://www.mta.ca/~cat-dist/ ]