From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/8565 Path: news.gmane.org!not-for-mail From: "Sergei Soloviev" Newsgroups: gmane.science.mathematics.categories Subject: Re: Partial functors .. Date: Tue, 17 Mar 2015 22:05:16 +0100 Message-ID: References: Reply-To: "Sergei Soloviev" NNTP-Posting-Host: plane.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset="utf-8" Content-Transfer-Encoding: quoted-printable X-Trace: ger.gmane.org 1426681620 20104 80.91.229.3 (18 Mar 2015 12:27:00 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 18 Mar 2015 12:27:00 +0000 (UTC) Cc: "Categories list" To: "David Yetter" Original-X-From: majordomo@mlist.mta.ca Wed Mar 18 13:26:50 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 1YYD3h-00071Y-TB for gsmc-categories@m.gmane.org; Wed, 18 Mar 2015 13:26:50 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:44003) by smtp3.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1YYD2w-0003L2-Ll; Wed, 18 Mar 2015 09:26:02 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1YYD2v-0006Lg-97 for categories-list@mlist.mta.ca; Wed, 18 Mar 2015 09:26:01 -0300 in-reply-to: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:8565 Archived-At: Dear David, of recent papers, there are for exemple K. Dosen, Z. Petric Coherence in substructural categories, Studia Logica, vol. > 70 (2002), pp. 271-296 (available at: http://arXiv.org) > > where the diagonal case (Relevant categories) is treated separately. = The > cartesian case is treated in > > K. Dosen, Z. Petric, The maximality of cartesian categories, Mathemat= ical > logic Quarterly, vol. 47 (2001), pp. 137-144 (available at: > http://arXiv.org) > > and also in our book Proof-Theoretical Coherence, Chapter 9. (availab= le at > http://www.mi.sanu.ac.rs/~kosta/coh.pdf). > There is also a recent paper > K. Dosen, On Sets of Premises (available at: http://arXiv.org) As to the historical aspect, there was a theorem in an old paper by Gri= gori Mints, published in Russian in 1980 (reprinted in "Selected papers in Proof Theory", Bibliopolis, 1992), bu= t I have to find the paper and exact formulation of his theorem. regards, Sergei Soloviev Le Mardi 17 Mars 2015 16:04 CET, David Yetter a =C3=A9= crit: > 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 ap= proach: consider functors to the category D~ formed from D by freely a= djoining 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 s= mall categories gives a category isomorphic to that of small categorie= s 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 "mor= e partial" than their source functor, since there will be a zero natur= al transformation between any two partial functors, corresponding to a = "defined nowhere" partial natural transformation when zero-ness is inte= rpreted as undefined as it was in the correspondence between zero-pres= erving 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. I= t seems to allow more partial natural transformations than Robin's obse= rvation, since zero arrows can fill in whenever the image object under = either the source or target functor is undefined, a partial natural tra= nsformation to be a natural transformation between the restrictions of = the two partial functors to the intersections of their domain of defini= tion (or a subcategory thereof). > > Best Thoughts, > David Yetter > =5F=5F=5F=5F=5F=5F=5F=5F=5F=5F=5F=5F=5F=5F=5F=5F=5F=5F=5F=5F=5F=5F=5F= =5F=5F=5F=5F=5F=5F=5F=5F=5F=5F=5F=5F=5F=5F=5F=5F=5F > From: Robin Cockett > Sent: Monday, March 16, 2015 6:12 PM > To: Categories list > Subject: categories: Partial functors .. > > David Leduc googlemail.com> writes: > >> 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 sm= all >> categories, partial functors and those "natural transformations" in= to >> a bicategory? The difficulty is that two partial functors from C to= D >> might not have the same definition domain. > > > Here is a basic and quite natural interpretation (if someone has not = > already pointed this out): > > One can have a n.t F =3D> G iff F is less defined than G and on thei= r common > domain (which is just the domain of F) there is a natural transformat= ion > from F =3D> \rst{F} G. Partial functors, of course, form a restrict= ion > category so they are naturally partial order enriched (by restriction= ). > This 2-cell structure must simply respect this partial order ... > > This is certainly not the only possibility, unfortunately ... for exa= mple > why not also allow partial natural transformations ... which are less= > defined than the functor. Here one does have to be a bit careful: a= > natural transformation must "know" the subcategory it is working with= ... > thus defining the natural transformation as a function on arrows (rat= her > than just objects) is worthwhile adjustment (see MacLane page 19, Exc= ercise > 5). This then works too .... > > I hope this helps. > > -robin > > > [For admin and other information see: http://www.mta.ca/~cat-dist/ ] = [For admin and other information see: http://www.mta.ca/~cat-dist/ ]