From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/962 Path: news.gmane.org!not-for-mail From: Peter Selinger Newsgroups: gmane.science.mathematics.categories Subject: Re: Re: one-object closed categories Date: Fri, 11 Dec 1998 22:15:22 -0500 (EST) Message-ID: <199812120315.WAA01623@dirichlet.math.lsa.umich.edu> References: 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 1241017391 28575 80.91.229.2 (29 Apr 2009 15:03:11 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:03:11 +0000 (UTC) Cc: categories@mta.ca To: T.Leinster@dpmms.cam.ac.uk (Tom Leinster) Original-X-From: cat-dist Sun Dec 13 16:10:50 1998 Original-Received: (from Majordom@localhost) by mailserv.mta.ca (8.8.8/8.8.8) id PAA19965 for categories-list; Sun, 13 Dec 1998 15:09:52 -0400 (AST) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f In-Reply-To: from "Tom Leinster" at Dec 10, 98 07:20:03 pm X-Mailer: ELM [version 2.4 PL25] Original-Sender: cat-dist@mta.ca Precedence: bulk Original-Lines: 53 Xref: news.gmane.org gmane.science.mathematics.categories:962 Archived-At: > From Tom Leinster: > > 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. A few remarks: In the case where V is an abelian group, the first axiom already implies the other two if we define [a] = -[a,a,a]. Namely, by letting a=b in the first axiom, it follows that [a,a,c] is independent of c. If V is an abelian group, then one can get an example of the above structure from an arbitrary map {-,-} : A x A ---> V by letting [a,b,c] = {a,b}+{b,c}-{a,c} and [a] = -{a,a}. Tom's "area" example is of this form. In fact, if V is an abelian group, then *any* example of a V-enriched category is (non-uniquely) of the form described in the previous paragraph: Fix some x in A (if any), and define {a,b} = [a,b,x]. What about the non-group case? In general, [a,b,c] need not always be invertible in V. In fact, [a,b,a] need not be invertible. For a simple example of this, let V be the natural numbers and define [a] = 0, [a,b,c] = 0, if a=b or b=c, 1, if a,b,c pairwise distinct, 2, otherwise (i.e., if a=c but a,b distinct). This indeed works. Best wishes, -- Peter Selinger > 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.