From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/4123 Path: news.gmane.org!not-for-mail From: Michael Barr Newsgroups: gmane.science.mathematics.categories Subject: On defining *-autonomous categories Date: Sun, 16 Dec 2007 21:45:53 -0500 (EST) Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII; format=flowed X-Trace: ger.gmane.org 1241019732 11808 80.91.229.2 (29 Apr 2009 15:42:12 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:42:12 +0000 (UTC) To: Categories list Original-X-From: rrosebru@mta.ca Mon Dec 17 12:16:15 2007 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 17 Dec 2007 12:16:15 -0400 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1J4IS7-00041o-21 for categories-list@mta.ca; Mon, 17 Dec 2007 12:04:23 -0400 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 17 Original-Lines: 28 Xref: news.gmane.org gmane.science.mathematics.categories:4123 Archived-At: I got a note from a student named Benjamin Jackson saying that someone had told him that to define a *-autonomous category, you need only a symmetric monoidal category and a contravariant involution * such that Hom(A @ B,C*) = Hom(A,(B @ C)*). Here "=" means natural equivalence and @ is the tensor product. Not only is this apparently true but that equivalence is needed only for A = I, the tensor unit. Of course, what is going on is that all the coherence is built-in to the structure of a symmetric monoidal category. First define A --o B = (A @ B*)*. Then the isomorphism above implies that Hom(A,B) = Hom(I @ A,B) = Hom(I,(A @ B*)*) = Hom(I,A --o B) Next we see that (A @ B) --o C = (A @ B @ C*)* = A --o (B @ C*)* = A --o (B --o C) and, applying Hom(I,-), that Hom(A @ B,C) = Hom(A,B --o C) Also we have that A --o I* = (A @ I)* = A* and A --o B = (A @ B*)* = (B* @ A)* = B* --o A* which gives the structure of a *-autonomous category with dualizing object I*. The trouble with this is that generally speaking it is the --o which is obvious and the tensor is derived from it. The coherences involving internal hom alone are much less well-known, although they are included in the original Eilenberg-Kelly paper in the La Jolla Proceedings. Still I am surprised that I never noticed this before.