From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/3282 Path: news.gmane.org!not-for-mail From: Michael Barr Newsgroups: gmane.science.mathematics.categories Subject: Re: dualities Date: Mon, 1 May 2006 15:06:07 -0400 (EDT) Message-ID: <40799.1670807407$1241019203@news.gmane.org> NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII X-Trace: ger.gmane.org 1241019202 8055 80.91.229.2 (29 Apr 2009 15:33:22 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:33:22 +0000 (UTC) To: Categories list Original-X-From: rrosebru@mta.ca Mon May 1 20:56:56 2006 -0300 X-Keywords: X-UID: 112 Original-Lines: 28 Xref: news.gmane.org gmane.science.mathematics.categories:3282 Archived-At: First, let me say I have avoided contributing to this thread because I don't understand what Vaughan is asking. He knows, as well as anyone, since he put them on the map, about Chu categories. He knows about *-autonomous categories as well. So what is the question, really? The simplest answer to Mamuka's question is the duality between vector spaces (over any field, including the 2 element field) and linearly compact vector spaces. In the case of a finite field, linear compactness is the same as the ordinary topological kind. One proof of this fact is that the category of finite dimensional spaces is self-dual and if two categories are dual, the inductive completion of one is dual to the projective completion of the other. For finite dimensional vector spaces, the inductive completion is vector spaces and the projective completion is linearly compact ones. Another example is the obvious duality between finite sets and finite boolean algebras that gives Stone duality on one hand and the duality between Set and CABA on the other, depending which one you complete which way. Most examples I am aware of of self-dualities are Chu categories (or chu categories). And if V_k is the category of k-vector spaces, then Chu(V_k,k) (an object is a pair of spaces and a bilinear pairing into k) is *-autonomous, as is chu(V_k,k) of separated extensional pairs. Peter's example is one of the very few *-autonomous categories I cannot relate to Chu. Complete (say inf) semi-lattices is another. Michael