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$\DeclareMathOperator\true{\mathsf{true}}$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 am interested in how the logic associated with the algebra of subobjects in the functor category $\mathsf{Set}^\mathbb{P}$ (for a partial order $\mathbb{P}$) varies with different properties of $\mathbb{P}$. Thus far, all I've been able to find is:

Fact 1. $\mathbb{P}$ 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 (so called Dummett's logic).

Fact 2. If $\mathbb{P}$ 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 $\mathbb{P}$ (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 $\mathbb{P}$ is directed or has incompatible elements?

Q2. If $\mathbb{P}$ has incompatible elements, does the size of the largest antichain matter?

Q3. What if $\mathbb{P}$ doesn't have a least element? (In particular can Fact 2's implication be reversed?)

Q4. $\mathbb{P}$ 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 $\mathbb{P}$ be a small category. I don't know whether this is essential for the results, or just made for metamathematical ease/queasiness.)

Thanks for any pointers!

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    $\begingroup$ If $P$ isn't small then functor categories out of it generally fail to be locally small (an easy example is if $P$ is a discrete proper class). The category $[P^{op}, \text{Set}]$ of presheaves on $P$ also generally fails to be the free cocompletion of $P$ (which is one of its more useful properties when $P$ is small); for that you want to restrict to the "small" presheaves (the ones which are a small colimit of representables). $\endgroup$
    – Qiaochu Yuan
    Commented Dec 5, 2020 at 2:13
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    $\begingroup$ Even with the restriction to small presheaves, the lack of a small generating set means some things no longer exist (= are representable as smal presheaves) “for free”. For example, subobject classifiers and exponential. $\endgroup$
    – Zhen Lin
    Commented Dec 5, 2020 at 4:37
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    $\begingroup$ @AsafKaragila en.wikipedia.org/wiki/Functor_category $\endgroup$
    – Gabe Goldberg
    Commented Dec 5, 2020 at 17:28
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    $\begingroup$ @AsafKaragila I guess the objects of the category are commuting systems $\langle X_p,f_{pq} : p\in \mathbb P\rangle$ of sets $X_p$ and functions $f_{pq} : X_p\to X_q$ and an arrow from $\langle X_p,f_{pq} : p\in \mathbb P\rangle$ to $\langle Y_p,g_{pq} : p\in \mathbb P\rangle$ is a sequence $\langle h_p : p\in \mathbb P\rangle$ of functions $h_p : X_p\to Y_p$ such that all the diagrams commute, i.e., $h_q\circ f_{pq} = g_{pq}\circ h_p$. $\endgroup$
    – Gabe Goldberg
    Commented Dec 5, 2020 at 17:51
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    $\begingroup$ @SimonHenry is right about the internal logic. But externally when you mean logic of the truth values, which are subobjects of 1, or equivalently global elements of $\Omega$, as you do in stating Fact 2, then the answer to Q3 is yes, and worth verifying. $\endgroup$
    – Colin McLarty
    Commented Dec 6, 2020 at 0:51

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