From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/8569 Path: news.gmane.org!not-for-mail From: henry@phare.normalesup.org Newsgroups: gmane.science.mathematics.categories Subject: Re: Partial functors .. Date: Wed, 18 Mar 2015 09:44:45 +0100 Message-ID: References: Reply-To: henry@phare.normalesup.org NNTP-Posting-Host: plane.gmane.org Mime-Version: 1.0 Content-Type: text/plain;charset=iso-8859-1 Content-Transfer-Encoding: 8bit X-Trace: ger.gmane.org 1426681842 24087 80.91.229.3 (18 Mar 2015 12:30:42 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 18 Mar 2015 12:30:42 +0000 (UTC) Cc: "Categories list" To: "David Yetter" Original-X-From: majordomo@mlist.mta.ca Wed Mar 18 13:30:33 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 1YYD7I-0000GD-59 for gsmc-categories@m.gmane.org; Wed, 18 Mar 2015 13:30:32 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:44034) by smtp3.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1YYD6h-0004EO-VQ; Wed, 18 Mar 2015 09:29:56 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1YYD6g-0006YP-Gy for categories-list@mlist.mta.ca; Wed, 18 Mar 2015 09:29:54 -0300 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:8569 Archived-At: I don't this this works: if (g .g') is in the domain of the partial functor and g is not, then g will be sent to 0 by the extension of the partial functor, hence g.g' will also be sent to 0 which shouldn't be the case because it is in the domain of the functor. Also it is not entirely clear to me if the original question wanted functor defined on a full subcategory or on an arbitrary sub-category. But in both case, my opinion would be that the only reasonable notion is to talk about "partial natural transformations" which are defined on a subcategory included in the domain of definition of both functor (and hence are natural transformation between ordinary functor). Best wishes, Simon Henry. > The previous suggestion of considering functors to D + 1 was a false start > for reasons Fred and Uwe pointed out, but it suggests a better approach: > consider functors to the category D~ formed from D by freely adjoining a > zero object. Arrows not in S now have somewhere to go (the zero arrow > with the appropriate source and target). > > I think at the one-categorical level, taking Hom(C,D) to be the > zero-preserving functors from C~ to D~, and letting C and D range over all > small categories gives a category isomorphic to that of small categories > with partial functors as arrows. > > Natural transformations between (zero-preserving) functors from C~ to D~ > would > then give a reasonable notion of partial natural transformations. It > certainly captures some, at least, of the natural transformations "more > partial" than their source functor, since there will be a zero natural > transformation between any two partial functors, corresponding to a > "defined nowhere" partial natural transformation when zero-ness is > interpreted as undefined as it was in the correspondence between > zero-preserving functors from C~ to D~ and partial functors from C to D. > > I'm not sure how this fits with the restrictions Robin points out. It > seems to allow more partial natural transformations than Robin's > observation, since zero arrows can fill in whenever the image object under > either the source or target functor is undefined, a partial natural > transformation to be a natural transformation between the restrictions of > the two partial functors to the intersections of their domain of > definition (or a subcategory thereof). > > Best Thoughts, > David Yetter [For admin and other information see: http://www.mta.ca/~cat-dist/ ]