From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/2929 Path: news.gmane.org!not-for-mail From: Philippe Gaucher Newsgroups: gmane.science.mathematics.categories Subject: Re: semi-categories Date: Fri, 2 Dec 2005 13:25:46 +0100 Message-ID: <200512021325.46133.gaucher@pps.jussieu.fr> References: <1133389475.438e26a388e62@mail2.buffalo.edu> Reply-To: gaucher@pps.jussieu.fr NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset="windows-1252" Content-Transfer-Encoding: quoted-printable X-Trace: ger.gmane.org 1241018989 6398 80.91.229.2 (29 Apr 2009 15:29:49 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:29:49 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Fri Dec 2 14:02:54 2005 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 02 Dec 2005 14:02:54 -0400 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1EiFCS-0006IF-W4 for categories-list@mta.ca; Fri, 02 Dec 2005 14:00:01 -0400 X-ME-UUID: 20051202122454739.B4669280019F@mwinf1008.wanadoo.fr User-Agent: KMail/1.7.1 In-Reply-To: <1133389475.438e26a388e62@mail2.buffalo.edu> Content-Disposition: inline Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 4 Original-Lines: 96 Xref: news.gmane.org gmane.science.mathematics.categories:2929 Archived-At: Le mercredi 30 Novembre 2005 23:24, vous avez =E9crit : > Perhaps it has not been sufficiently emphasized that semi-categories an= d > the like are not really "generalizations" of categories (though formall= y > they may appear so).=20 Indeed, at least in my case, a flow must not be viewed as a generalizatio= n of=20 the notion of small categories. Let me explain a little bit what I am doi= ng=20 with these objects. I was not very explicit in my previous post. And so t= he=20 terminology I use is not in "competition". I want to model HDA, at least those coming from precubical sets. I use a = set=20 of states X^0 and between each state A and B of the HDA, there is a=20 topological space P_{A,B}X whose elements represent the non-constant=20 execution paths from A to B. The topology of this space models the=20 concurrency of the situation between A and B. And execution paths can be=20 composed with a strictly asssociative law. There does not necessarily exi= st a=20 loop from a given state A to itself : so P_{A,A}X can be empty. This fact= is=20 one reason among several other ones why I remove the identity maps. Inside this model, I am able to define what is a dihomotopy equivalence. = The=20 main problem to define dihomotopy is that some contractible parts of "the= =20 directed spaces of execution paths" must not be contracted. Otherwise in = the=20 categorical localization, the relevant geometric information is lost. In=20 particular, initial and final states must be unchanged by a dihomotopy=20 equivalence. A very simple example : take two execution paths going from = one=20 initial state 0 to one final state 1. If contractions in the direction of= time=20 are allowed, one finds in the same equivalence class a loop. Some example= s of=20 unwanted final states are deadlocks of concurrent systems : a deadlock is= =20 nothing else but a final state from a geometric viewpoint. Flows allow to= =20 propose a solution of this problem : in fact I introduced this notion of=20 flows on purpose, to make the following solution work.=20 The first kind of dihomotopy equivalence is a morphism f:X->Y such that=20 f^0:X^0->Y^0 is a bijection and such that Pf:PX->PY is a weak homotopy=20 equivalence. It turns out that there is a model structure on Flows whose = weak=20 equivalences are exactly the preceding kind of morphisms. By imposing the= =20 condition f^0:X^0->Y^0 bijective, we do not take any risk : nothing is=20 contracted in the direction of time. So no geometric information is lost.= But=20 this kind of identification is too rigide ! The second kind of dihomotopy= =20 equivalence is generated by taking a representative set of inclusions of=20 posets P1\subset P2, where P1 and P2 are finite bounded posets and where = the=20 inclusions preserve the bottom element and the top element (which are=20 different by hypothesis in a bounded poset). For example, the inclusion o= f=20 posets {0<1}\subset{0