From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/1078 Path: news.gmane.org!not-for-mail From: Michael Barr Newsgroups: gmane.science.mathematics.categories Subject: Re: Monoidal structure, take II Date: Thu, 18 Mar 1999 12:46:52 -0500 (EST) Message-ID: References: <36F0E6ED.7EF7BB45@loria.fr> NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII X-Trace: ger.gmane.org 1241017552 29412 80.91.229.2 (29 Apr 2009 15:05:52 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:05:52 +0000 (UTC) To: categories@mta.ca Original-X-From: cat-dist Thu Mar 18 16:36:24 1999 Original-Received: (from Majordom@localhost) by mailserv.mta.ca (8.9.3/8.9.3) id OAA26035 for categories-list; Thu, 18 Mar 1999 14:07:46 -0400 (AST) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f In-Reply-To: <36F0E6ED.7EF7BB45@loria.fr> Original-Sender: cat-dist@mta.ca Precedence: bulk Original-Lines: 81 Xref: news.gmane.org gmane.science.mathematics.categories:1078 Archived-At: I did not quite understand Francois' construction. However, my first reaction to a question like that is that it ought to be a homotopy. So I will say what a homotopy reduces to in this case and leave it to Francois to decide if this is what he has. I suspect that rather few people know what a simplicial homotopy is and, of those, rather few have ever actually verified one. I am in that minority^2, so perhaps I tend to see them where they are not the most natural, but I think it quite remarkable that they can arise where no real topology (but some combinatorics) is present. I have to begin by saying how a graph becomes a simplicial set. Actually, that is a lie, since unless you are dealing with reflexive graphs--that are equipped with a selected loop at each vertex--you will only get a face complex. But homotopies are still definable. A category is a simplicial set by taking for n-simplexes composable n-tuples of arrows. This doesn't work for graphs, since the "interior faces" (all except the lowest and highest numbered) all depend on composition. But there is a face complex in which an n-simplex is simply an n-simplex in the graph. So a 2-simplex is a triangle--obviously non-commutative and a 3-simplex is a tetrahedron and so on. You can describe a composable n-tuple in a category as commutative n-simplex, so this isn't so different. Now given this, if f,g: X --> Y are graph morphisms, what is a homotopy? Well, write X as d^0,d^1: X_1 --> X_0 and similarly for Y. Then f consists of f_0: X_0 --> Y_0 and f_1: X_1 --> Y_1 giving a serially commutative square. Just a homomtopy between functors turns out to be simply a natural transformation, a homotopy in this case turns out to consist of a function p_0: X_0 --> Y_1 and a function p_1: X_1 --> Y_1 such that there is a diagram (not, of course commutative; what a diagram does is specify source and target) as follows. In this diagram I assume x: x^0 --> x^1 in X, and f(x): y^0 --> y^1 and g(x): z^0 --> z^1 in Y. f(x) y^0 -----------> y^1 | \ | | \ | | \ | | \ | | \ | | \ | | \ | p_0(x^0)| p_1(x)\ |p_0(x^1) | \ | | \ | | \ | | \ | | \ | | \ | v g(x) vv z^0 -----------> z^1 So if this is what Francois was saying, then the answer is it a homotopy of face complexes. Of course, if you replaced X_1 by X_1 + X_0, you would have a reflexive graph and I assume (I have not checked this) you would then get a simplicial homotopy. BTW, homotopies do not generally compose--and the ones described here do not appear to either. Categories are special because of their internal composition. It makes me wonder if the well-known failure of dinats to compose could be related to this in some way. Having seen Francois' clarification, I think this is exactly what he had. Michael ------------------------------------------------------------------- History shows that the human mind, fed by constant accessions of knowledge, periodically grows too large for its theoretical coverings, and bursts them asunder to appear in new habiliments, as the feeding and growing grub, at intervals, casts its too narrow skin and assumes another... Truly the imago state of Man seems to be terribly distant, but every moult is a step gained. - Charles Darwin, from "The Origin of Species"