CAUTION: The Sender of this email is not from within Dalhousie. 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-ord...)

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/ ]

CAUTION: The Sender of this email is not from within Dalhousie. Set^P is the category of sheaves over the ideal completion of P, so its global elements of Ω are in bijection with the opens - that is to say the Scott opens - of Idl(P). But they are in bijection with the Alexandrov opens of 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 the 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 numerical order - and hence an infinite downward chain. It has the disjunction property, but no bottom. Steve.

On 9 Dec 2020, at 16:09, barton.neil.alexander@gmail.com 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-ord...)

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/ ]