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

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

[Neil Barton]

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/ ]

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

[ptj]

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/ ]

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

[Steve Vickers]

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
> 


[For admin and other information see: http://www.mta.ca/~cat-dist/ ]