From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/10338 Path: news.gmane.io!.POSTED.blaine.gmane.org!not-for-mail From: ptj@maths.cam.ac.uk Newsgroups: gmane.science.mathematics.categories Subject: Re: How does the logic of Set^P vary with the properties of P? Date: 09 Dec 2020 19:12:04 +0000 Message-ID: References: Reply-To: ptj@maths.cam.ac.uk Mime-Version: 1.0 Content-Type: text/plain; format=flowed; charset=ISO-8859-1 Injection-Info: ciao.gmane.io; posting-host="blaine.gmane.org:116.202.254.214"; logging-data="9896"; mail-complaints-to="usenet@ciao.gmane.io" Cc: categories@mta.ca To: Neil Barton Original-X-From: majordomo@rr.mta.ca Thu Dec 10 03:10:46 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 1knBPp-0002QC-QV for gsmc-categories@m.gmane-mx.org; Thu, 10 Dec 2020 03:10:45 +0100 Original-Received: from rr.mta.ca ([198.164.44.159]:45446) by smtp2.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1knBJo-0006VI-Nf; Wed, 09 Dec 2020 22:04:32 -0400 Original-Received: from majordomo by rr.mta.ca with local (Exim 4.92.1) (envelope-from ) id 1knBNX-00079J-FT for categories-list@rr.mta.ca; Wed, 09 Dec 2020 22:08:23 -0400 In-Reply-To: Precedence: bulk Xref: news.gmane.io gmane.science.mathematics.categories:10338 Archived-At: There is quite a lot in the literature about how properties of a poset P (or more generally a small category C) are reflected in logical properties of the topos [C,Set]. In particular, `Fact 1' is in my paper `Conditions related to De Morgan's Law' in Springer LNM 753 (1979). Regarding `Fact 2', the existence of a least element of P is not necessary for [P,Set] to satisfy the disjunction property; the necessary and sufficient condition is that P^op should be directed. (I'm afraid I don't know a reference for this.) On the other hand, if you strengthen to the infintary disjunction property (if \bigvee \phi_i is provable, then some \phi_i is provable), you do get a condition equivalent to P having a least element. The reason why one restricts to small categories is that smallness of C is used in the proof that [C^op,Set] is a topos -- though actually, as Hans Engenes pointed out in Math. Scand. 34 (1974), it's sufficient (and necessary) to require that each slice category C/A is equivalent to a small category. (Thus, for example, if Ord is the ordered class of ordinals then [Ord^op,Set] is a topos.) Peter Johnstone On Dec 9 2020, Neil Barton wrote: >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/ ]