From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/2196 Path: news.gmane.org!not-for-mail From: Vaughan Pratt Newsgroups: gmane.science.mathematics.categories Subject: Re: More Topos questions ala "Conceptual Mathematics" Date: Thu, 20 Feb 2003 16:57:26 -0800 Message-ID: References: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=us-ascii X-Trace: ger.gmane.org 1241018481 2956 80.91.229.2 (29 Apr 2009 15:21:21 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:21:21 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Thu Feb 20 21:33:29 2003 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 20 Feb 2003 21:33:29 -0400 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 18m1z8-0001FB-00 for categories-list@mta.ca; Thu, 20 Feb 2003 21:28:18 -0400 X-Mailer: exmh version 2.1.1 10/15/1999 In-Reply-To: Message from "Stephen Schanuel" of "Thu, 20 Feb 2003 13:48:16 EST." <000701c2d910$a0faa480$39a14244@grassmann> Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 52 Original-Lines: 57 Xref: news.gmane.org gmane.science.mathematics.categories:2196 Archived-At: >From: Stephen Schanuel >you'll learn why Boolean algebra, so familiar in sets, needs >to be replaced by Heyting algebra in more general toposes. I would expand this beyond Heyting algebras to quantales, residuated monoids, etc. See http://boole.stanford.edu/pub/seqconc.pdf for an example of a situation, namely event structures as a model catering simultaneously to concerns of branching time and "true" concurrency, that has traditionally been handled in a Boolean way. That paper extends event structures to three- and four-valued logics of behavior. This particular extension (expansion, augmentation) doesn't generalize the two-valued Boolean logic of event structures to Heyting algebras. There are exactly two three-element idempotent commutative quantales. Obviously the three-element Heyting algebra is one of them, and this HA does find application in drawing a distinction between accidental and causal temporal precedence, a topic Haim Gaifman looked into around 1988. The other, which isn't a Heyting algebra, is at the core of the notion of transition as the intermediate state between "ready" and "done," more on this in the above-cited paper. This is not to say that there is no topos-theoretic approach to this extension. In particular the above paper briefly mentions the presheaf category Set^FinBip where FinBip is the category of finite bipointed sets, as a notion of cubical set. Cubical sets certainly provide one algebraically attractive model of true concurrency that works roughly the same way as the one based on this 3-element quantale---both of them entail cubical structure---but I've been finding the latter a more elementary and natural tool for working with cubes, at least for my purposes---homologists may find limitations that I don't seem to run into. An advantage of Set^FinBip is that it accommodates cyclic structures (iterative concurrent automata), whereas the one based on 3' as I've been calling this 3-element quantale works with acyclic cubical sets, calling for iteration to be unfolded, much as formal languages "are" unfolded grammars. (Come to think of it, I don't know anything about the subobject classifier of Set^FinBip. If someone has a succinct description of it I'd be very grateful.) The main point here is that there *is* a logic of behavior that is close to but not quite intuitionistic, at least not in the strict Heyting algebra sense. Furthermore it is not a question of just finding the smallest Heyting algebra in which the above quantale embeds, since there isn't one that preserves the ordered monoid structure: a Heyting algebra must have its monoid unit at the top, which 3' as "the other three-element quantale" doesn't. So whatever relationship obtains between the subobject classifier of Set^FinBip and 3', it's not an ordered-monoid embedding of the latter in the former. See http://boole.stanford.edu/pub/seqconc.pdf for more details. Vaughan