From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/10339 Path: news.gmane.io!.POSTED.blaine.gmane.org!not-for-mail From: Steve Vickers Newsgroups: gmane.science.mathematics.categories Subject: Re: How does the logic of Set^P vary with the properties of P? Date: Wed, 9 Dec 2020 23:06:44 +0000 Message-ID: References: Reply-To: Steve Vickers Mime-Version: 1.0 (1.0) Content-Type: text/plain; charset="utf-8" Content-Transfer-Encoding: quoted-printable Injection-Info: ciao.gmane.io; posting-host="blaine.gmane.org:116.202.254.214"; logging-data="14611"; mail-complaints-to="usenet@ciao.gmane.io" Cc: To: Neil Barton Original-X-From: majordomo@rr.mta.ca Thu Dec 10 03:11:56 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 1knBQy-0003i4-KX for gsmc-categories@m.gmane-mx.org; Thu, 10 Dec 2020 03:11:56 +0100 Original-Received: from rr.mta.ca ([198.164.44.159]:45464) by smtp2.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1knBKx-0006hD-Pe; Wed, 09 Dec 2020 22:05:43 -0400 Original-Received: from majordomo by rr.mta.ca with local (Exim 4.92.1) (envelope-from ) id 1knBOa-0007Av-LF for categories-list@rr.mta.ca; Wed, 09 Dec 2020 22:09:28 -0400 In-Reply-To: Precedence: bulk Xref: news.gmane.io gmane.science.mathematics.categories:10339 Archived-At: Set^P is the category of sheaves over the ideal completion of P, so its glob= al elements of =CE=A9 are in bijection with the opens - that is to say the S= cott opens - of Idl(P). But they are in bijection with the Alexandrov opens o= f P, that is to say the up-closed subsets. Looking at Fact 1, if P has a bottom, and U and V are up-closed with union t= he whole of P, then one of them contains bottom and hence is the whole of P.= The converse is not true. Consider P the natural numbers with reverse numeri= cal order - and hence an infinite downward chain. It has the disjunction pro= perty, but no bottom. Steve. > On 9 Dec 2020, at 16:09, barton.neil.alexander@gmail.com wrote: >=20 > =EF=BB=BFDear All, >=20 > 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. >=20 > (I've also tried asking on MathOverflow, if anyone is interested: > https://mathoverflow.net/questions/378167/how-do-properties-of-a-partial-o= rder-mathbbp-affect-the-logic-of-the-functo) >=20 > 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: >=20 > 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). >=20 > 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 =3D true > iff y =3D true or z =3D true). I *think* this implication can be reversed,= > but I'm not sure. >=20 > 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: >=20 > Q1. If the logic is affected when P is directed or has incompatible elemen= ts? >=20 > Q2. If P has incompatible elements, does the size of the largest > antichain matter? >=20 > Q3. What if P doesn't have a least element? (In particular can Fact > 2's implication be reversed?) >=20 > Q4. P has (or doesn't have) a maximal element? >=20 > (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.) >=20 > Thanks for any pointers. >=20 > Best Wishes, >=20 > Neil >=20 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]