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    • CommentRowNumber1.
    • CommentAuthorMike Shulman
    • CommentTimeAug 18th 2013

    I’m not entirely happy with the introduction (“Statement”) to the page axiom of choice. On the one hand, it implies that the axiom of choice is something to be considered relative to a given category CC (which is reasonable), but it then proceeds to give the external formulation of AC for such a CC, which I think is usually not the best meaning of “AC relative to CC”. I would prefer to give the Statement as “every surjection in the category of sets splits” and then discuss later that analogous statements for other categories (including both internal and external ones) can also be called “axioms of choice” — but with emphasis on the internal ones, since they are what correspond to the original axiom of choice (for sets) in the internal logic.

    (I would also prefer to change “epimorphism” for “surjection” or “regular/effective epimorphism”, especially when generalizing away from sets.)

    • CommentRowNumber2.
    • CommentAuthorTobyBartels
    • CommentTimeAug 19th 2013

    I agree.

    • CommentRowNumber3.
    • CommentAuthorMike Shulman
    • CommentTimeAug 24th 2013

    Since no one objected, I went ahead and made this change.

    • CommentRowNumber4.
    • CommentAuthorThomas Holder
    • CommentTimeOct 16th 2018

    I added the characterization of IAC toposes as Boolean étendues.

    diff, v66, current

    • CommentRowNumber5.
    • CommentAuthorMike Shulman
    • CommentTimeMay 2nd 2020

    Added the fact (thanks to Alizter for finding it) that AC is equivalent to the statement “if two free groups have equal cardinality, then so do their generating sets”.

    diff, v70, current

    • CommentRowNumber6.
    • CommentAuthorMike Shulman
    • CommentTimeMay 2nd 2020

    Oops, of course the sets have to be infinite.

    diff, v70, current

    • CommentRowNumber7.
    • CommentAuthorTodd_Trimble
    • CommentTimeMay 2nd 2020

    Seems like a somewhat roundabout way of putting it: can’t we just say that for infinite XX, that XX and F(X)F(X) have the same cardinality? Am I missing something?

    • CommentRowNumber8.
    • CommentAuthorTodd_Trimble
    • CommentTimeMay 3rd 2020

    Okay, I guess never mind my question. The direction that the indicated statement plus ZF implies AC doesn’t look easy.

    • CommentRowNumber9.
    • CommentAuthorjesuslop
    • CommentTimeDec 7th 2021

    Fixed image broken link in References-General, image uploaded.

    diff, v73, current

    • CommentRowNumber10.
    • CommentAuthorzskoda
    • CommentTimeOct 20th 2022
    • (edited Oct 20th 2022)

    Some of the equivalent statements to AC listed like

    • That every essentially surjective functor is split essentially surjective.
    • That every fully faithful and essentially surjective functor between strict categories is a strong equivalence of categories.

    are suspicious as they talk about classes rather than sets. Isn’t it that AC for classes is a stronger statement ? Should one just put modifier small ?

  1. Added equivalent statement

    That any cartesian product of any family of inhabited sets is inhabited.

    And it’s type theoretic analogue in the section on type theory

    That any dependent product of any family of pointed sets is pointed.


    diff, v76, current

  2. Surjections are families of inhabited sets, not families of sets.


    diff, v76, current

    • CommentRowNumber13.
    • CommentAuthorUrs
    • CommentTimeOct 20th 2022
    • (edited Oct 20th 2022)

    Re #10:

    These two clauses (here and here) were added in rev 75 by Anonymous and in rev 48 by Mike, respectively.

    It seems clear that they are meant to be applied to small categories. For the clause mentioning strict categories this is almost explicit, since the entry strict category speaks as if smallness is the default assumption.

    I have now added the smallness qualifier to both items, and also the strictness qualifier to the former.

    diff, v77, current

  3. adding a paragraph explaining that the traditional axiom of choice using bracket types implies function extensionality in type theory.


    diff, v78, current

  4. adding reference


    diff, v78, current

  5. updated reference


    diff, v80, current

    • CommentRowNumber17.
    • CommentAuthorMike Shulman
    • CommentTimeFeb 24th 2023

    In May, an Anonymous editor added the following as an equivalent of AC:

    That every subset ABA \subseteq B in a universe 𝒰\mathcal{U} comes with a choice of injection i:ABi:A \hookrightarrow B. Constructive mathematicians usually use subsets equipped with the structure of an injection, as those are usually more useful than general subsets with the mere property of being a subset.

    I don’t know what this means. How can a subset fail to come with a choice of injection defined by i(a)=ai(a)=a?

    • CommentRowNumber18.
    • CommentAuthorDavidRoberts
    • CommentTimeFeb 26th 2023

    Perhaps by “subset ABA\subseteq B” it is meant a pair AA, BB such that |A||B||A|\le |B|, or I guess that there merely exists an injection, and this is quantified over all pairs satisfying this cardinality condition. This feels like a Choice principle, but perhaps too strong?

    • CommentRowNumber19.
    • CommentAuthorMike Shulman
    • CommentTimeMar 1st 2023

    I suppose that’s possible. That should follow from (global) choice, since you are choosing a set of specific injections from a family of nonempty sets of injections, and it certainly implies choice taking A=1A=1.

    • CommentRowNumber20.
    • CommentAuthorDavidRoberts
    • CommentTimeMar 2nd 2023

    Yes, it felt like Global Choice to me, which is stronger than the passage claims (plain AC). So, regardless, it will need editing a bit more.

    • CommentRowNumber21.
    • CommentAuthorMike Shulman
    • CommentTimeMar 2nd 2023

    Perhaps the relativization to “a universe” was intended to weaken it from global choice to ordinary choice?