From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/8562 Path: news.gmane.org!not-for-mail From: David Yetter Newsgroups: gmane.science.mathematics.categories Subject: Re: Partial functors .. Date: Tue, 17 Mar 2015 15:04:54 +0000 Message-ID: References: Reply-To: David Yetter NNTP-Posting-Host: plane.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset="iso-8859-1" Content-Transfer-Encoding: quoted-printable X-Trace: ger.gmane.org 1426610887 9174 80.91.229.3 (17 Mar 2015 16:48:07 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Tue, 17 Mar 2015 16:48:07 +0000 (UTC) To: Categories list Original-X-From: majordomo@mlist.mta.ca Tue Mar 17 17:47:56 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 1YXuel-0005pA-N5 for gsmc-categories@m.gmane.org; Tue, 17 Mar 2015 17:47:51 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:42608) by smtp3.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1YXue8-0000mK-Mo; Tue, 17 Mar 2015 13:47:12 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1YXue5-0006Gx-3h for categories-list@mlist.mta.ca; Tue, 17 Mar 2015 13:47:09 -0300 In-Reply-To: Accept-Language: en-US Content-Language: en-US Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:8562 Archived-At: 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: c= onsider functors to the category D~ formed from D by freely adjoining a zer= o object. Arrows not in S now have somewhere to go (the zero arrow with th= e appropriate source and target).=0A= =0A= I think at the one-categorical level, taking Hom(C,D) to be the zero-preser= ving functors from C~ to D~, and letting C and D range over all small categ= ories gives a category isomorphic to that of small categories with partial= functors as arrows.=0A= =0A= Natural transformations between (zero-preserving) functors from C~ to D~ wo= uld=0A= then give a reasonable notion of partial natural transformations. It certa= inly captures some, at least, of the natural transformations "more partial"= than their source functor, since there will be a zero natural transformati= on between any two partial functors, corresponding to a "defined nowhere" p= artial 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.=0A= =0A= I'm not sure how this fits with the restrictions Robin points out. It seem= s to allow more partial natural transformations than Robin's observation, s= ince zero arrows can fill in whenever the image object under either the sou= rce or target functor is undefined, a partial natural transformation to be = a natural transformation between the restrictions of the two partial functo= rs to the intersections of their domain of definition (or a subcategory the= reof).=0A= =0A= Best Thoughts,=0A= David Yetter=0A= ________________________________________=0A= From: Robin Cockett =0A= Sent: Monday, March 16, 2015 6:12 PM=0A= To: Categories list=0A= Subject: categories: Partial functors ..=0A= =0A= David Leduc googlemail.com> writes:=0A= =0A= > A partial functor from C to D is given by a subcategory S of C and a=0A= > functor from S to D. What is the appropriate notion of natural=0A= > transformation between partial functors that would allow to turn small=0A= > categories, partial functors and those "natural transformations" into=0A= > a bicategory? The difficulty is that two partial functors from C to D=0A= > might not have the same definition domain.=0A= =0A= =0A= Here is a basic and quite natural interpretation (if someone has not=0A= already pointed this out):=0A= =0A= One can have a n.t F =3D> G iff F is less defined than G and on their comm= on=0A= domain (which is just the domain of F) there is a natural transformation=0A= from F =3D> \rst{F} G. Partial functors, of course, form a restriction=0A= category so they are naturally partial order enriched (by restriction).=0A= This 2-cell structure must simply respect this partial order ...=0A= =0A= This is certainly not the only possibility, unfortunately ... for example= =0A= why not also allow partial natural transformations ... which are less=0A= defined than the functor. Here one does have to be a bit careful: a=0A= natural transformation must "know" the subcategory it is working with ...= =0A= thus defining the natural transformation as a function on arrows (rather=0A= than just objects) is worthwhile adjustment (see MacLane page 19, Excercise= =0A= 5). This then works too ....=0A= =0A= I hope this helps.=0A= =0A= -robin=0A= =0A= [For admin and other information see: http://www.mta.ca/~cat-dist/ ]