From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/7048 Path: news.gmane.org!not-for-mail From: Vaughan Pratt Newsgroups: gmane.science.mathematics.categories Subject: Re: The boringness of the dual of exponential Date: Thu, 10 Nov 2011 23:47:12 -0800 Organization: Stanford University Message-ID: References: Reply-To: Vaughan Pratt NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Trace: dough.gmane.org 1321020269 19481 80.91.229.12 (11 Nov 2011 14:04:29 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Fri, 11 Nov 2011 14:04:29 +0000 (UTC) To: categories@mta.ca Original-X-From: majordomo@mlist.mta.ca Fri Nov 11 15:04:25 2011 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtpx.mta.ca ([138.73.1.4]) by lo.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1ROriQ-0000HB-4J for gsmc-categories@m.gmane.org; Fri, 11 Nov 2011 15:04:22 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:56448) by smtpx.mta.ca with esmtp (Exim 4.76) (envelope-from ) id 1ROrhE-0006Qv-60; Fri, 11 Nov 2011 10:03:08 -0400 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1ROrhC-0002OX-Nq for categories-list@mlist.mta.ca; Fri, 11 Nov 2011 10:03:06 -0400 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:7048 Archived-At: On 11/9/2011 7:58 AM, RJ Wood wrote: > Your observation about lack of symmetry in \set is underscored by the > fact that the yoneda functor for \set has a left adjoint which has > a left adjoint which has a left adjoint which has a left adjoint but > but the co-yoneda functor for \set has a right adjoint that fails to > preserve even finite sums. > R_j Richard modestly omitted that he and our esteemed moderator showed (necessarily by classical reasoning) that what Richard just said characterized \set up to equivalence. On 11/10/2011 1:29 AM, Prof. Peter Johnstone wrote: > One point that no-one has mentioned yet is that you can't have > exponentiation and its dual in the same category, unless it is a > preorder. Peter modestly omitted that he is the go-to category theorist when it comes to toposes. He also omitted that he was confining himself to cartesian closed categories when he mentioned his point, understandable given that every topos is cartesian closed. To expand a little on my (1965) classmate Ross Street's counterexample of Set^op, Set x Set^op is yet another counterexample. Here I've one-upped Ross (I must be getting competitive in my dotage) by contradicting Peter and giving a counterexample in which both exponentiation *and* dual exponentiation are present simultaneously. How did I know that? Well, Set x Set^op is equivalent (in fact isomorphic) to Chu(Set, 1). For *any* set K, both exponentiation and dual exponentiation are admissible in Chu(Set,K), product being of the tensor kind in this case. How did I know *that*? Well, every Chu category is a *-autonomous category in the sense of Barr 1979. If you don't know why every *-autonomous category contains both exponentiation and dual exponentiation, then like Ebert and Siskel I'm not going to give away the plot and you'll just have to fork out to see the movie, or steal it if you're a nerd, or watch this space (someone is bound to be a spoiler). Open question. At this year's CT, conveniently held 3 km from my sister's house so I could bike in, I talked about TAC's, or topoalgebraic categories. These are defined by picking two sets of objects from an arbitrary category (TAC's for dummies), some details of which may be found at http://boole.stanford.edu/pub/sortprop.pdf . (At question time PTJ insightfully observed that TACs would cause immense confusion if I submitted my write-up to TAC.) A Chu category is precisely a dense complete TAC for which J and L are singleton monoids. That is, one sort and one property, both rigid. The open question: Characterize those dense complete TAC's admitting both exponentiation and dual exponentiation. Chu categories do so, but what about others? Many thanks to Ross, Richard, and Jeff Eggers for their respective roles in the representation of TAC objects (A,r,X) over a profunctor K as a profunctor morphism r: AX --> K. (Ross and I would call them bimodules, much as Mike Barr calls monads triples.) But toposes are fun too. *-autonomous categories are to Democrats as toposes are to Republicans. Vaughan (donkey) Pratt [For admin and other information see: http://www.mta.ca/~cat-dist/ ]