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

[ptj@maths.cam.ac.uk]

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
>


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