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When explaining how Heyting categories can model first order logic it would be nice to be able to give some small example and contrast it with Set-semantics. I realized however that I don't know of any Heyting category which is not also a topos. It would be nice to have more concrete examples to give.

A search netted me a master thesis that discussed the Heyting category structure on some categories of graphs. Do you have other examples of Heyting categories that are not toposes?

EDIT: Syntactic categories are Heyting categories which are not necessarily toposes as pointed out by godelian. I am however looking for examples not arising out of logic (:

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    $\begingroup$ Any first-order (intuitionistic) theory gives rise to a small Heyting category: its syntactic category. See e.g. Johnstone $\endgroup$
    – godelian
    Commented May 24, 2020 at 16:31
  • $\begingroup$ Yes, that's true, I had something more concrete in mind though, that I could show someone who is less familiar with categorical logic. $\endgroup$
    – rosensymmetri
    Commented May 24, 2020 at 16:32
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    $\begingroup$ In the category of sets, take the full subcategory whose objects are the countable sets. This looks to me like a Heyting category, also satisfying classical logic. $\endgroup$
    – Andreas Blass
    Commented May 24, 2020 at 22:33
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    $\begingroup$ A similar sort of example is the category of classes (relative to some model of set theory). $\endgroup$
    – Mike Shulman
    Commented Nov 8, 2020 at 7:36

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In case someone else stumbles upon this question: there are several examples of "natural" Heyting categories arising in algebraic logic. Ghilardi and Zawadowski (2002) have some results concerning this, which imply that the dual category to finitely presented Heyting algebras is a Heyting category. With duality theory this can be given a more concrete description, for instance with Esakia duality. There are other examples of this kind, intimately related to the question of when a variety of a model completion, which have been explored in that literature. Moreover, such examples are not toposes, as also shown by Ghilardi and Zawadowski.

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