CAUTION: The Sender of this email is not from within Dalhousie. Dear Neil, You might be interested in the systematic treatment of this kind of questions given in my paper "Topologies for intermediate logics" (Mathematical Logic Quarterly 60 (4-5), 335-347 (2014)), which describes how to characterize the classes of Grothendieck toposes whose internal logic satisfies a given assertion in the theory of Heyting algebras and also introduces natural analogues of the double negation and De Morgan topologies on a topos for a wide class of intermediate logics. Best, Olivia -----Messaggio originale----- Da: ptj@maths.cam.ac.uk <ptj@maths.cam.ac.uk> Inviato: mercoledì 9 dicembre 2020 20:12 A: Neil Barton <barton.neil.alexander@gmail.com> Cc: categories@mta.ca Oggetto: categories: Re: How does the logic of Set^P vary with the properties of P? 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-parti al-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/ ]