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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeJan 23rd 2013
• (edited Jan 23rd 2013)

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)

1. 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.

• CommentRowNumber3.
• CommentAuthorJohn Baez
• CommentTimeOct 22nd 2020

• CommentRowNumber4.
• CommentAuthorHurkyl
• CommentTimeOct 22nd 2020

Typo fix: $FinSet^op$ is freely generated by finite limits (not finite colimits).

2. Does FinSet have a global choice operator?

• CommentRowNumber6.
• CommentAuthorDavidRoberts
• CommentTimeOct 24th 2022

I doubt it without some Choice, because surely that would imply that the ff, surjective-on-objects functor from the category of pointed finite sets and arbitrary functions to FinSet had a section. Even if one took a skeleton of FinSet and restricted to that, it would imply that the function $u\colon \mathcal{N}\to \mathbb{N}$ has a section, where $|u^{-1}(n)|= n$.

• CommentRowNumber7.
• CommentAuthorHurkyl
• CommentTimeOct 24th 2022
• (edited Oct 24th 2022)

(Nitpick: the image of $FinSet_* \to FinSet$ only surjects onto the full subcategory of nonempty sets)

I don’t think we require functions between sets preserve the choices, do we? I.e. we’re not asking for $FinSet_* \to FinSet_{\ncong \varnothing}$ to have a section or even $Core(FinSet_*) \to Core(FinSet_{\ncong \varnothing})$, but instead it’s $Ob(FinSet_*) \to Ob(FinSet_{\ncong \varnothing})$ we want to have a section.

• CommentRowNumber8.
• CommentAuthorDavidRoberts
• CommentTimeOct 25th 2022

@Hurkyl,

regarding non-empty: my mistake (but, constructively, it would even be inhabited sets). If you have a surjection at the level of objects, then you get a section of the functor (modulo the correction) that is an adjoint inverse, and in fact I think these are isomorphic (supposing we had chosen a small skeleton): the set of section of the surjection on objects, thought of as a discrete category, and the category of sections of the functor that are also adjoint inverses. This follows the construction in CWM of a unique adjoint inverse from a certain section at the object level.

• CommentRowNumber9.
• CommentAuthorHurkyl
• CommentTimeOct 25th 2022

Ah, ignore most of #7 then. I misread and thought you were referring to $FinSet_*$ rather than the category you were actually defining.