From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/8554 Path: news.gmane.org!not-for-mail From: Uwe Egbert Wolter Newsgroups: gmane.science.mathematics.categories Subject: Re: Re: Partial functor Date: Mon, 16 Mar 2015 14:42:35 +0100 Message-ID: References: Reply-To: Uwe Egbert Wolter NNTP-Posting-Host: plane.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=windows-1252; format=flowed Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1426516402 9987 80.91.229.3 (16 Mar 2015 14:33:22 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Mon, 16 Mar 2015 14:33:22 +0000 (UTC) To: Christopher King , categories@mta.ca Original-X-From: majordomo@mlist.mta.ca Mon Mar 16 15:33:12 2015 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtp3.mta.ca ([138.73.1.127]) by plane.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1YXW4W-0007gf-HU for gsmc-categories@m.gmane.org; Mon, 16 Mar 2015 15:32:48 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:40983) by smtp3.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1YXW3x-0004KB-5O; Mon, 16 Mar 2015 11:32:13 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1YXW3v-0006xX-RQ for categories-list@mlist.mta.ca; Mon, 16 Mar 2015 11:32:11 -0300 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:8554 Archived-At: On 2015-03-15 18:01, Christopher King wrote: > David Leduc googlemail.com> writes: > >> Hi, >> >> A partial functor from C to D is given by a subcategory S of C and a >> functor from S to D. What is the appropriate notion of natural >> transformation between partial functors that would allow to turn small >> categories, partial functors and those "natural transformations" into >> a bicategory? The difficulty is that two partial functors from C to D >> might not have the same definition domain. >> >> [For admin and other information see: http://www.mta.ca/~cat-dist/ ] >> >> > I know this is late, but I find a quite obvious notion for it. Why not turn > your partial functor into a regular functor from C->D+1 (1 and + are the > terminal object and coproduct in the category of categories.) Now you can just > use regular natural transformations. > I think this construction will not work since the set-theoretic difference C\S is not a subcategory of C while 1 is a subcategory of D+1. The collection of all partial functors from C to D is a partial ordering due to the inclusion of definition domains. For each subcategory S of C you have the functor category [S->C] and each inclusion functor In_S,S':S->S' gives you a functor from [S'->D] into [S->D]. Combining both structures (via an appropriate variant of the Grothendieck construction] you should get a category with objects all partial functors (S,F:S->D) and morphisms (In_S,S', \alpha:F => In_S,S';F'):(S,F)->(S',F'). Composition of partial functors is given by pullback (inverse image) construction. I don't know if this gives a bicategory put maybe it helps to have a look in the paper of Barry Jay "Partial Functions, Ordered Categories, Limits and Cartesian Closure (1993) " http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.48.6433 Uwe [For admin and other information see: http://www.mta.ca/~cat-dist/ ]