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I have briefly recorded the equivalence of FinSet${}^{op}$ with finite Booplean algebras at FinSet – Properties – Opposite category. Then I linked to this from various related entries, such as finite set, power set, Stone duality, opposite category.
(I thought we long had that information on the $n$Lab, but it seems we didn’t)
Added to FinSet a remark on the opposite category $FinSet^{op}$ from a constructive perspective:
“In constructive mathematics, for any flavor of finite, $\mathcal{P}$ defines an equivalence of $FinSet$ with the opposite category of that of those complete atomic Heyting algebras whose set of atomic elements is finite (in the same sense as in the definition of $FinSet$).”
I don’t know whether for some values of finite, this characterization can be made more interesting, i.e. whether we can give a condition which does not explicitly mention the set of atomic elements.
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