Re: How does the logic of Set^P vary with the properties of P?

[Steve Vickers <s.j.vickers@cs.bham.ac.uk>]

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-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
> 


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