From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/2743 Path: news.gmane.org!not-for-mail From: Claudio Hermida Newsgroups: gmane.science.mathematics.categories Subject: Re: Questions on dinatural transformations. Date: Fri, 02 Jul 2004 20:20:11 -0400 Message-ID: <40E5FBBB.4020701@cs.queensu.ca> References: <200407011701.i61H1SRH005609@coraki.Stanford.EDU> NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1241018867 5587 80.91.229.2 (29 Apr 2009 15:27:47 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:27:47 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Sat Jul 3 12:24:20 2004 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sat, 03 Jul 2004 12:24:20 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 1BgmNA-0002Tg-00 for categories-list@mta.ca; Sat, 03 Jul 2004 12:24:12 -0300 In-Reply-To: <200407011701.i61H1SRH005609@coraki.Stanford.EDU> Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 6 Original-Lines: 109 Xref: news.gmane.org gmane.science.mathematics.categories:2743 Archived-At: Vaughan Pratt wrote: >>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)). > There is a `canonical' choice of composition for such `sesquifunctors' (what follows is presumably folklore and written up somewhere). Consider the category SDCat of *self-dual* categories: objects are categories C, equipped with a duality c: C -> C^op (with c^op c = id), and morphisms F: (C,c) -> (D,d) are functors F:C -> D such that F^op c = d F. The forgetful SDCat -> Cat admits both adjoints (and SDCat is actually both monadic and comonadic over Cat): the right adjoint takes a category A to (A^op x A, s) where s is the switch isomorphism (the second projection \pi' : A^op x A -> A is the counit of the adjunction). We thus get a comonad G on Cat, and sesquifunctors are the morphisms of the resulting Kleisli category Cat_G, which tells us how to compose G:D^op x D -> E with F:C^op x C -> D. The composite is G(F^op s, F)\delta, which agrees indeed with the formula above. (This is of course the composite in SDCat via the adjunction) > >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 > > The counterexamples mentioned in P. Scott's posting concern the lack of a well-defined *vertical* composition of dinaturals. If one persists on endowing them with such a composite, one possible approach is to accept the partiality of this composition and work with *paracategories*. Pushing this simple idea to its logical conclusion leads to a decent enough basic theory, which allows to make sense of the fact that `dinats into a ccc form a cartesian-closed paracat'. See Hermida, C; Mateus, P. Paracategories. II. Adjunctions, fibrations and examples from probabilistic automata theory. Theoret. Comput. Sci. 311 (2004), no. 1-3, 71--103. (also available at my homepage http://maggie.cs.queensu.ca/chermida) Making a 2-dimensional structure with dinats, using their partial vertical composition, leads to consider enrichment over ParCat (the cartesian closed category of paracategories). But whiskering (and therefore *horizontal* composition) is bound to be a partial operation as well, so one has to broaden/weaken ParCat to accommodate this fact. Ultimately, the kind of composite required in Yanosfky's posting: S,S':C^op x C---->B T,T':C^op x C ----> B^op U,U':B^op x B --->A \alpha: S--->S' dinat \alpha': T--->T' dinat \beta: U--->U' dinat ------------------------------------------------------------------- \beta \circ (\alpha',\alpha) suggests a *partial multicategory* structure (as introduced in the article above), with homs in ParCat. Claudio Hermida