From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/2740 Path: news.gmane.org!not-for-mail From: Vaughan Pratt Newsgroups: gmane.science.mathematics.categories Subject: Re: Questions on dinatural transformations. Date: Thu, 01 Jul 2004 10:01:28 -0700 Message-ID: <200407011701.i61H1SRH005609@coraki.Stanford.EDU> NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=us-ascii X-Trace: ger.gmane.org 1241018865 5570 80.91.229.2 (29 Apr 2009 15:27:45 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:27:45 +0000 (UTC) To: Original-X-From: rrosebru@mta.ca Fri Jul 2 07:48:09 2004 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 02 Jul 2004 07:48:09 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 1BgLaL-0005ma-00 for categories-list@mta.ca; Fri, 02 Jul 2004 07:48:01 -0300 X-Mailer: exmh version 2.6.3 04/04/2003 with nmh-1.0.4 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 3 Original-Lines: 51 Xref: news.gmane.org gmane.science.mathematics.categories:2740 Archived-At: >From: noson@sci.brooklyn.cuny.edu >I was hoping that the category of small categories, functors and >dinat transformations... There's a category problem already at this point. Dinats don't go between functors F,G:C->D, they go between sesquifunctors F:C^op x C->D and differ from n.t.'s of that type by only being defined on the diagonal of C^op x C. The off-diagonal and non-identity-morphism entries in F,G only participate in the dinaturality condition, not in the transformation itself. >a) It is well known that there is no vertical >composition of dinatural transformations. >How about horizontal composition? Before you can compose dinats horizontally you have to be able to compose the sesquifunctors they bridge. I don't know how others do this, but if I had to compose G:D^op x D -> E with F:C^op x C -> D, my inclination would be to restrict the evident composite G(F(a,b),F(c,d)) to a=d, b=c (i.e. where the variances match up). That is, GoF:C^op x C -> E is defined by G(F(c,c'),F(c',c)) on object pairs (c',c) of C^op x C, with the expected extension to morphism pairs (f',f) where f':c'->d' in C^op (i.e. f':d'->c' in C) and f:c->d in C, namely G(F(f,f'),F(f',f)): G(F(c,c'),F(c',c)) -> G(F(d,d'),F(d',d)). With that (or some) choice of sesquifunctor composition one can then ask about horizontal composition tos where s:F->F', t:G->G'. How would you whisker a dinatural on the left, i.e. apply the whisker G:D^op x D->E on the left to the dinat s:F->F' on the right where F,F':C^op x C->D? For natural transformations, G is just a functor G:D->E, so this is just a matter of applying G pointwise to each s_c. For dinaturals however, G is a sesquifunctor. What do you want a sesquifunctor to do to a morphism s_c? Maybe there's some span-like thing one can do here but I don't see it. For dinaturals, vertical composition may turn out to be easier than horizontal, in that it at least makes sense provided one solves the shape-matching problem somehow. In doing so one also solves another problem, that dinaturality is too weak a condition, typically admitting transformations on the internal hom that aren't Church numerals (Pare & Roman, JPAA 128 33-92 for Set, Pratt, TCS 294:3, bottom of p461, for Chu(Set,K) and chu(Set,K) which awkwardly seem to need different treatments). Mike Barr has a notion of strong dinatural (unpublished?), and the notion of binary (more generally n-ary) logical transformation also works well here when definable on the category of interest. Vaughan Pratt