From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6922 Path: news.gmane.org!not-for-mail From: Claudio Hermida Newsgroups: gmane.science.mathematics.categories Subject: Re: partial categories Date: Wed, 28 Sep 2011 21:11:24 -0400 Message-ID: References: Reply-To: Claudio Hermida 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: dough.gmane.org 1317299669 17419 80.91.229.12 (29 Sep 2011 12:34:29 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Thu, 29 Sep 2011 12:34:29 +0000 (UTC) Cc: categories@mta.ca To: Emily Riehl Original-X-From: majordomo@mlist.mta.ca Thu Sep 29 14:34:24 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 1R9Fom-0004xf-Ev for gsmc-categories@m.gmane.org; Thu, 29 Sep 2011 14:34:24 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:45737) by smtpx.mta.ca with esmtp (Exim 4.76) (envelope-from ) id 1R9Fll-0002Bx-Km; Thu, 29 Sep 2011 09:31:17 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1R9Flk-0004Ia-3N for categories-list@mlist.mta.ca; Thu, 29 Sep 2011 09:31:16 -0300 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6922 Archived-At: > > On Wed, Sep 28, 2011 at 4:34 PM, Emily Riehl wro= te: > >> 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 enrich= ed >> 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 >> >> >> > Dear Emily Riehl, > > The notion you seem to refer to is that which has appeared in the > literature as 'precategory' as used in > > - Universal Aspects of Probabilistic Automata > L. Schr=F6der and P. Mateus Mathematical Structures in Computer Science, Volume 12 Issue 4, August 2002 - Precategories for Combining Probabilistic Automata P. Mateus, A. Sernadas, C. Sernadas *Electronic Notes in Theoretical Computer Science*, 1999, Pages 169-186 CTCS '99, Conference on Category Theory and Computer Science Enriching over pointed sets you will end up with functors which must preserve the undefined composites, something a little awkward to require. The above treatment does not do that, but the concepts are developed ad hoc from scratch. A more sophisticated notion of 'partial category' is that of *paracategory*= , introduced by P. Freyd. Here, the partial composites must satisfy a certain 'saturation' condition. The general context for such a theory is that of *partial algebras*, which admit an abstract notion of *saturation*. In this context, saturation is equivalent to a representation result which embeds any partia= l algebra in a total one (using the notion of Kleene morphism, which reflect definedness of composites, rather than preserving them) . This theory is given in * - *Paracategories I: Internal Paracategories and Saturated Partial Algebras C. Hermida, P. Mateus *Theoretical Computer Science* Volume 309, Issues 1-3, 2 December 2003, Pages 125-156 Such a general theory can then be instantiated in various contexts: categories, multicategories and indexed categories are treated in - Paracategories II: adjunctions, fibrations and examples from probabilisti= c automata theory C. Hermida, P. Mateus *Theoretical Computer Science* Volume 311, Issues 1-3, 23 January 2004, Pages 71-103 which develops the basic 2-category theory of paracategories, inlcuding adjunctions, limits, etc. The most relevant example (for which paracategories were introduced) is that of bivariant functors and dinatural transformations, which constitute a cartesian closed paracategory. Finally, I'd like to mention that paracategories have been used to study certain models of quatum compuation in the thesis of O. Malherbe: Categorical models of computation: Partially traced categories and presheaf models of quantum computation by *Malherbe, Octavio, * Ph.D., *UNIVERSITY OF OTTAWA , 2010, 215 pages; NR73903 * Sincerely, Claudio Hermida [For admin and other information see: http://www.mta.ca/~cat-dist/ ]