From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/10336 Path: news.gmane.io!.POSTED.blaine.gmane.org!not-for-mail From: Neil Barton Newsgroups: gmane.science.mathematics.categories Subject: How does the logic of Set^P vary with the properties of P? Date: Sat, 5 Dec 2020 22:18:32 +0100 Message-ID: Reply-To: Neil Barton Mime-Version: 1.0 Content-Type: text/plain; charset="UTF-8" Injection-Info: ciao.gmane.io; posting-host="blaine.gmane.org:116.202.254.214"; logging-data="2075"; mail-complaints-to="usenet@ciao.gmane.io" To: categories@mta.ca Original-X-From: majordomo@rr.mta.ca Wed Dec 09 17:10:58 2020 Return-path: Envelope-to: gsmc-categories@m.gmane-mx.org Original-Received: from smtp2.mta.ca ([198.164.44.74]) by ciao.gmane.io with esmtps (TLS1.2:ECDHE_RSA_AES_256_GCM_SHA384:256) (Exim 4.92) (envelope-from ) id 1kn23M-0000Px-VS for gsmc-categories@m.gmane-mx.org; Wed, 09 Dec 2020 17:10:57 +0100 Original-Received: from rr.mta.ca ([198.164.44.159]:45364) by smtp2.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1kn1wU-0003C8-Ow; Wed, 09 Dec 2020 12:03:50 -0400 Original-Received: from majordomo by rr.mta.ca with local (Exim 4.92.1) (envelope-from ) id 1kn201-0003Uy-RC for categories-list@rr.mta.ca; Wed, 09 Dec 2020 12:07:29 -0400 Precedence: bulk Xref: news.gmane.io gmane.science.mathematics.categories:10336 Archived-At: Dear All, I am very suspicious the answer to this (family of) question(s) is well-known, but I couldn't find anything after a bit of searching so I'll ask anyway. (I've also tried asking on MathOverflow, if anyone is interested: https://mathoverflow.net/questions/378167/how-do-properties-of-a-partial-order-mathbbp-affect-the-logic-of-the-functo) I am interested in how the logic associated with the algebra of subobjects in the functor category Set^P (for a partial order P) varies with different properties of P. Thus far, all I've been able to find is: Fact 1. P is (weakly) linearly-ordered iff the logic of the topos is intuitionistic logic with the classical tautology (phi rightarrow psi) vee (psi rightarrow phi) added (otherwise known as Dummett's Logic). Fact 2. If P has a least element then the topos is disjunctive (i.e. if y:1 to Omega and z:1 to Omega are truth-values, then y cup z = true iff y = true or z = true). I *think* this implication can be reversed, but I'm not sure. I was wondering if anything more is known about how the logic of the topos varies according to the properties of P (and vice versa)? I'd be interested in any information here, but to make things more concrete, is it known: Q1. If the logic is affected when P is directed or has incompatible elements? Q2. If P has incompatible elements, does the size of the largest antichain matter? Q3. What if P doesn't have a least element? (In particular can Fact 2's implication be reversed?) Q4. P has (or doesn't have) a maximal element? (An aside: In the presentation I'm most familiar with (namely Goldblatt's book) there is a restriction that P be a small category. I don't know whether this is essential for the results, or just made for metamathematical ease/queasiness of dealing with a functor category that can't be represented as anything small.) Thanks for any pointers. Best Wishes, Neil [For admin and other information see: http://www.mta.ca/~cat-dist/ ]