From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/3290 Path: news.gmane.org!not-for-mail From: Vaughan Pratt Newsgroups: gmane.science.mathematics.categories Subject: Re: dualities Date: Wed, 03 May 2006 09:40:43 -0700 Message-ID: <25729.005438234$1241019207@news.gmane.org> 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 1241019207 8086 80.91.229.2 (29 Apr 2009 15:33:27 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:33:27 +0000 (UTC) To: categories Original-X-From: rrosebru@mta.ca Wed May 3 17:43:28 2006 -0300 X-Keywords: X-UID: 167 Original-Lines: 91 Xref: news.gmane.org gmane.science.mathematics.categories:3290 Archived-At: John Baez wrote: > [...] > So, can we find equally nice examples [of representable dualities] where C and D are instead > 2-categories? In particular, can we find examples where C and D > are 2-categorical generalizations of the 1-categorical examples > we already know? > > In particular, he suggested taking the example where C is the > category of finite distributive lattices and finding an analogous > example where C is the 2-category of (maybe finite, in some sense?) > distributive categories. Enrico Vitale just sent me the answer for that one: C = the 2-category of idempotent-closed categories, D = the 2-category of presheaf categories. This categorifies C = Pos, D = StoneDLat by passing from 2 to Set as the enriching autonomous category (so in that sense one could say we were in 3-CAT all along, though presumably only trivially so by virtue of only having identity modifications when V = 2, I think). Although I'd heard the phrase "Morita equivalence" many times over the years, it meant nothing to me until recently when Bill Lawvere was talking about graphs as presheaves on the monoid consisting of the three monotone functions on the ordinal 2 and I finally woke up to the connection between splitting the two idempotents and ME (the equivalence, not the condition). The idempotent closure of that monoid, meaning the result of splitting the idempotents, is just the initial segment of Delta of length 2, aka the ordinals 1 and 2 and their monotone functions. The impact on the models, here graphs, is that splitting the idempotents results in giving the self-loops that were playing the role of vertices their own datatype V, as coded by the ordinal 1. This new category of graphs is not the old one as its objects now have vertices in their own right, but it is equivalent to the old one. {2} and {1,2}, each made a category with respectively 3 and 7 monotone functions, are Morita equivalent: they have equivalent idempotent closures, and homming into Set maps them to equivalent categories, the iff that makes Morita equivalence important. ME is the kernel of idempotent closure, which is a categorification, with Set in place of 2, of the functor Ord --> Pos (Ord the category of preordered sets, Pos of posets) that collapses the cliques. The reason there is no representable duality between Ord and a suitable cousin of StoneDLat (FinOrd and FinDLat for the Stonaphobes) is that preorders are equivalent to posets and the Yoneda embedding taking elements of P to primes in 2^P, while fully faithful, is only good up to equivalence. (The homfunctor being transposed here is the order <= : P\op x P --> 2.) The categorification of this, meaning in this case not the passage from 2-CAT to 3-CAT but from enrichment in 2 to enrichment in Set, still has to deal with equivalence in the same way (though here it goes with the territory and so is less noticeable than back down at Ord vs. Pos where we tend to think isomorphism rather than equivalence). But Hom: C\op x C --> Set is not itself an equivalence but only a "retract that retracts retracts", the essence of Morita equivalence (a dual of Freyd's "trivial for a trivial reason"?). In order to take the "log to the base Set" we can't really "retract all the retracts" because we may need to keep some of them around but then which ones (like picking a dense subset of a continuum: which subset?). We can however put them all in, which is to say, split all the idempotents, so we do that in order to get a normal form. The rest of this duality is then the triviality that the internal hom of CAT is contravariant in its first argument. Morita equivalence is the only thing to be worried about. Proposition 5.28 of Kelly's "Basic Concepts of Enriched Category Theory", namely that Cauchy completion (Kelly's name for the enriched counterpart of idempotent closure) permits taking logs to any autonomous base V, then produces a proper class of dualities, one for every autonomous V. In particular we can recover Pos\op ~ StoneDLat by taking V = 2. (Pos and Ord, preordered sets, while not equivalent any more than CAT and its subcategory of idempotent-closed categories are equivalent, have equivalent objects which is all we need ask of a duality.) There are two "good" 3-object V's, the non-Heyting one of which enriches the "prossets" that Haim Gaifman and I wrote about in LICS'87, so these have their dual objects in the same way, by homming into 3, a construct I talked about incomprehensibly at the Newton Institute meeting on geometry in computation some years ago, not recognizing that it was a duality. Metric spaces, another duality there. And so on. But then every such duality has its subdualities, for example Set\op ~ CABA as a subduality of Pos\op ~ StoneDLat, so a great many more dualities there. Enrico also mentioned the Gabriel-Ulmer duality for locally finitely presentable categories, and the Adamek-Lawvere-Rosicky duality for varieties. Are these in addition to the above or can they be recovered from them? Likewise for the duality Peter Johnstone mentioned? Vaughan Pratt