From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/960 Path: news.gmane.org!not-for-mail From: Tom Leinster Newsgroups: gmane.science.mathematics.categories Subject: Re: one-object closed categories Date: Thu, 10 Dec 1998 19:20:03 +0000 (GMT) Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=US-ASCII Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1241017388 28565 80.91.229.2 (29 Apr 2009 15:03:08 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:03:08 +0000 (UTC) To: categories@mta.ca Original-X-From: cat-dist Fri Dec 11 01:06:22 1998 Original-Received: (from Majordom@localhost) by mailserv.mta.ca (8.8.8/8.8.8) id WAA26514 for categories-list; Thu, 10 Dec 1998 22:45:04 -0400 (AST) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f X-Mailer: ELM [version 2.4 PL25] Original-Sender: cat-dist@mta.ca Precedence: bulk Original-Lines: 40 Xref: news.gmane.org gmane.science.mathematics.categories:960 Archived-At: > From: maxk@maths.usyd.edu.au (Max Kelly) > > Tom observed that an abelian monoid is a symmetric monoidal closed > category with one object, and asked whether anyone had studied categories > enriched in such a closed category. > [...] > > Anyway, I had a brief look at V-categories for such a V tonight, but with > too few details so far to say much about them before bedtime. Queer little > creatures, aren't they? A V-category A has objects a, b. c. and so on, but > each A(a,b) is the unique object * of V. All the action takes place at the > level of j: I --> A(a,a) and M: A(b,c) o A(a,b) --> A(a,c). [...] Since I asked the question I've found a few examples; they've all got the same flavour about them, so I'll just do my favourite. If V is the commutative monoid, then a V-enriched category is a set A plus two functions [-,-,-]: A x A x A ---> V [-]: A ---> V satisfying [a,c,d] + [a,b,c] = [a,b,d] + [b,c,d] [a,a,b] + [a] = 0 = [a,b,b] + [b] for all a, b, c, d. The example: let A be a subset of the plane. Choose a smooth path P(a,b) from a to b for each (a,b) in A x A, and define [a,b,c] to be the signed area bounded by the loop P(a,b) then P(b,c) then (P(a,c) run backwards); also define [a] to be -(area bounded by P(a,a)). (There's meant to be an orientation on the plane, so that areas can be negative.) Then the equations say obvious things about area - don't think I'm up to that kind of ASCII art, though. Tom