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

    I have briefly recorded the equivalence of FinSet op{}^{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 nnLab, but it seems we didn’t)

  1. Added to FinSet a remark on the opposite category FinSet opFinSet^{op} from a constructive perspective:

    “In constructive mathematics, for any flavor of finite, 𝒫\mathcal{P} defines an equivalence of FinSetFinSet 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 FinSetFinSet).”

    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

    Added facts about the universal properties of FinSet and its opposite.

    diff, v17, current

    • CommentRowNumber4.
    • CommentAuthorHurkyl
    • CommentTimeOct 22nd 2020

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

    diff, v18, current

  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:𝒩u\colon \mathcal{N}\to \mathbb{N} has a section, where |u 1(n)|=n|u^{-1}(n)|= n.

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

    (Nitpick: the image of FinSet *FinSetFinSet_* \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 *FinSet FinSet_* \to FinSet_{\ncong \varnothing} to have a section or even Core(FinSet *)Core(FinSet )Core(FinSet_*) \to Core(FinSet_{\ncong \varnothing}), but instead it’s Ob(FinSet *)Ob(FinSet )Ob(FinSet_*) \to Ob(FinSet_{\ncong \varnothing}) we want to have a section.

    • CommentRowNumber8.
    • CommentAuthorDavidRoberts
    • CommentTimeOct 25th 2022


    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 *FinSet_* rather than the category you were actually defining.

    • CommentRowNumber10.
    • CommentAuthorGuest
    • CommentTimeMay 25th 2023

    adding section about the equivalence between FinSet and the category of finite T 1T_1-topological spaces

    diff, v30, current

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeMay 25th 2023
    • (edited May 25th 2023)

    have hyperlinked more of the technical terms (eg. Top)

    diff, v31, current

    • CommentRowNumber12.
    • CommentAuthorTodd_Trimble
    • CommentTimeMay 25th 2023

    Re #10: this seems awfully elaborate. If you asked ordinary mathematicians how to prove it, they might say this: in a T 1T_1-space, all points are closed; since finite unions of closed sets are closed, this implies that every subset of a finite T 1T_1-space is closed, i.e., a finite T 1T_1-space is discrete. But the category of finite discrete spaces is clearly equivalent to the category of finite sets.

    Is there a reason for doing it in this much more complicated way?

    • CommentRowNumber13.
    • CommentAuthorDavidRoberts
    • CommentTimeMay 26th 2023

    A nicer version might look at how finite T 1T_1 spaces correspond to the discrete preorders inside finite preorders, using the equivalence between finite preorders and finite Alexandroff spaces, and if one is especially motivated, how the R 0R_0 and T 0T_0 spaces sit between these.