From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/4886 Path: news.gmane.org!not-for-mail From: Steve Lack Newsgroups: gmane.science.mathematics.categories Subject: Re: sketch theory Date: Tue, 26 May 2009 08:09:03 +1000 Message-ID: Reply-To: Steve Lack NNTP-Posting-Host: lo.gmane.org Content-Type: text/plain; charset="US-ASCII" Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1243294563 27583 80.91.229.12 (25 May 2009 23:36:03 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Mon, 25 May 2009 23:36:03 +0000 (UTC) To: John Baez , categories Original-X-From: categories@mta.ca Tue May 26 01:35:56 2009 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from mailserv.mta.ca ([138.73.1.1]) by lo.gmane.org with esmtp (Exim 4.50) id 1M8jhz-0005Xn-Rq for gsmc-categories@m.gmane.org; Tue, 26 May 2009 01:35:56 +0200 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1M8jFi-0005fn-4O for categories-list@mta.ca; Mon, 25 May 2009 20:06:42 -0300 Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:4886 Archived-At: On 25/05/09 3:03 PM, "John Baez" wrote: > Steve Lack writes: > > You ask about a sketch for cartesian closed categories. Have a look at >> at the paper "A presentation of topoi as algebraic relative to categories >> or >> graphs (Dubuc-Kelly, J. Alg. 81: 420-433, 1983). This describes something >> even tighter: the category of cartesian closed categories is monadic over >> the category of graphs. >> > > Thanks! > > And thanks to everyone else for their helpful comments. I'm behind on > answering my emails. > > In this approach, does each pair of objects in a ccc come with a chosen > product and exponential? Are the morphisms of ccc's are required to preserve > these on the nose? Yes, that's right on both counts, but see below. > > At first I was a bit shocked to hear of a sketch for ccc's, because the > internal hom is contravariant in one variable. But I guess that as long as > we treat ccc's purely 1-categorically that's no problem. But then I guess > we pay the price of "undue strictness". Right? As you say, if you work 1-categorically, you are stuck with undue strictness. And as you imply, there is an impediment to a fully 2-categorical approach because of the contravariance of the internal hom. But there is a way around this. You work 2-categorically, but not over the 2-category Cat, but over the 2-category of categories, functors, and natural _isomorphisms_. (Kelly & co call this 2-category Cat_g, with g presumably standing for groupoidal, since this is not just enriched in Cat but in groupoids.) Then the internal hom does indeed become a 2-functor Cat^2_g-->Cat_g. Having these invertible 2-cells now allows you to consider pseudomorphisms of algebras, which preserve structure up to isomorphism, thus alleviating the problem of undue strictness. It doesn't completely solve it - since we only have invertible 2-cells, we don't have a notion of lax morphism; or, more precisely, the notion of lax morphism we get is just that of pseudomorphism. Similarly, some constructions might not give what we hoped for. For example, the cotensor C^2 in Cat_g is not the category of arrows in C, it's the category of invertible arrows in C. All the best, Steve. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]