From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/3062 Path: news.gmane.org!not-for-mail From: Marco Grandis Newsgroups: gmane.science.mathematics.categories Subject: Re: Undirected graph citation Date: Fri, 3 Mar 2006 10:04:57 +0100 Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 (Apple Message framework v746.2) Content-Type: text/plain; charset=US-ASCII; delsp=yes; format=flowed Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1241019073 7091 80.91.229.2 (29 Apr 2009 15:31:13 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:31:13 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Fri Mar 3 14:53:33 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 03 Mar 2006 14:53:33 -0400 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FFFHE-0003uD-4w for categories-list@mta.ca; Fri, 03 Mar 2006 14:45:20 -0400 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 8 Original-Lines: 67 Xref: news.gmane.org gmane.science.mathematics.categories:3062 Archived-At: Undirected versus directed Going along with the last messages of C Berger and FW Lawvere, I would like to list the following parallel notions, undirected versus directed. Of course, it is not a question of saying which is better, but only of separating them to make things clearer. --- Undirected: - symmetric simplicial sets (sss) - simplicial complexes (classical) = sets with distinguished subsets = sss where each simplex is determined by its vertices - undirected graphs - groupoids (fundamental groupoids) - abelian groups (homology groups) - spaces - classical metric spaces - undirected algebraic topology --- Directed: - simplicial sets - "directed simplicial complexes" (not classical) = sets with distinguished words = simplicial sets where each simplex is determined by (the family of) its vertices - directed graphs - categories (fundamental categories) - preordered abelian groups ("directed homology groups") - "directed spaces" (preordered, locally preordered, etc.) - generalised metric spaces (Lawvere) - "directed algebraic topology" --- Spaces are plainly an undirected structure. Note that their singular simplicial set already has a natural symmetric structure (by "permuting vertices" on tetrahedra); there is no need of symmetrising it and loosing information. Classical algebraic topology is mostly undirected (since spaces, groupoids, abelian groups are so), but it has also used directed structures, like simplicial sets, for undirected purposes: simulating spaces and computing undirected algebraic structures, like groupoids and homology groups. The study of "directed algebraic topology" is quite recent. (There are some papers on that in my web page, from which one can see the literature; present applications are concerned with concurrency and rewriting. But the general aim should be modeling non-reversible phenomena.) Finally, I would like to point out - once more - that the term "simplicial complex" is highly confusing: this notion (as Bill recalls) is a simplified version of a symmetric simplicial set, while the corresponding simplified version of a simplicial set is a "set with distinguished words" (the reflexive cartesian closed subcategory of "objects determined by their vertices", in the presheaf topos of simplicial sets). But I have noticed that people can get nervous about terminology, and it might be better to forget about this last point. Marco Grandis