From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/3019 Path: news.gmane.org!not-for-mail From: Marco Grandis Newsgroups: gmane.science.mathematics.categories Subject: Re: horizontal composition Date: Thu, 2 Feb 2006 08:15:45 +0100 Message-ID: NNTP-Posting-Host: main.gmane.org Content-Type: text/plain; charset=US-ASCII; delsp=yes; format=flowed Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1241019048 6914 80.91.229.2 (29 Apr 2009 15:30:48 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:30:48 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Thu Feb 2 05:31:09 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 02 Feb 2006 05:31:09 -0400 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1F4alP-0003P5-OY for categories-list@mta.ca; Thu, 02 Feb 2006 05:28:27 -0400 X-Mailer: Apple Mail (2.746.2) Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 2 Original-Lines: 34 Xref: news.gmane.org gmane.science.mathematics.categories:3019 Archived-At: Dear Jean (and dear colleagues), Everyone knows you defined bicategories (and Charles Ehresmann 2- categories). The point of those two messages is that one can equivalently define a 2-category or a bicategory using the following primitive operations (and suitable axioms): - the vertical composition of cells, - the whisker composition of cells with maps (instead of the horizontal composition of cells). (Which is precisely what we concretely do in Cat, when we define horizontal composition of natural transformations: we use vertical composition and whiskering, showing that the two possible ways of defining horizontal composition give the same result, by the relevant part of the middle-four interchange axiom - which I was calling "reduced interchange".) All this has some importance in homotopy, which is why I was interested in it. For instance, take chain complexes with their homotopies: then the vertical composition of homotopies is (strictly) associative, whiskering is also associative (in the appropriate sense), but reduced interchange fails and you do not have a horizontal composition of homotopies. Such a structure is a sesqui- category in Ross Street's sense - actually one might say "sesqui- groupoid". With best regards Marco