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It is well known that even if one works with say ZFC as a base theory, Grothendieck topoi do not in general satisfy even fairly weak axioms like countable choice or small violations of choice and one can find explicit counterexamples. However, they do satisfy a weak form of choice called WISC, and do so even when only ZF + WISC is chosen as the base theory.

The question is: are any other choice principles inherited by Grothendieck topoi like this? The question may also include axioms that are mutually incompatible with the axiom of choice but still imply taking a position on the matter while being fairly natural, such as for example the axiom of determinacy which implies countable choice AND the negation of full choice.

Edit: By "inherited by grothendieck topoi", I mean that for any topos T which satisfies said choice axiom A, the following two properties must hold:

  1. For any category C, the category of functors from C to T (which may be viewed as presheaves in T on the dual category of C) must satisfy A.
  2. Any topos t which has a geometric embedding in T must also satisfy A (which up to equivalence is the same as restricting T to sheaves with respect to some Lawvere-Tierny topology on T).

Doing 1 and then 2 is basically a generalization of defining Grothendieck topoi over sets but to arbitrary topoi. WISC as I understand it has the property of being preserved by those two operations regardless of which topos you start with.

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    $\begingroup$ The special adjoint functor theorem - and its consequence like the existence of free model for infinitary algebraic theories - hold in all Grothendieck topos (assuming choice in the base - though assuming the special adjoint functor theorem in the base is probably enough). This result fails in ZF, but I don't know if it is a consequence of WISC or some other choice principle that holds in Grothendieck topos. $\endgroup$
    – Simon Henry
    Commented Sep 18, 2022 at 16:21
  • $\begingroup$ Don't you need to be a little careful how your phrase this question? Specifically, different formulations of classically equivalent choice principles may fail to be intuitionistically equivalent. The nLab article on Grothendieck topoi says that some constructive version of the axiom of multiple choice holds, for instance. $\endgroup$
    – James E Hanson
    Commented Sep 18, 2022 at 16:55
  • $\begingroup$ Does $\neg \neg \mathsf{AC}$ hold in every Grothendieck topos? $\endgroup$
    – James E Hanson
    Commented Sep 18, 2022 at 16:55
  • $\begingroup$ @JamesHanson No it doesn't holds in any topos - in most sheaves topos (ex Sh([0,1]) ) the axiom of choice is nowhere true, that is $\neg AC$ holds. There are also boolean toposes that don't satisfies AC. Note that there are some technical difficulties here as AC involves quantification on objects, so it is not technically a proposition in the internal logic and so it doesn't make sense to take its negation - however one can make sense of this for Grothendieck toposes using the stack semantics and the fact that Grothendieck toposes are "autological" (arxiv.org/abs/1004.3802) $\endgroup$
    – Simon Henry
    Commented Sep 18, 2022 at 23:18
  • $\begingroup$ @JamesHanson I believe that is for Grothendieck topoi over set (rather than over an arbitrary topos with the same starting property). I clarified my question to describe what I mean by "inherited by". $\endgroup$
    – saolof
    Commented Sep 20, 2022 at 17:51

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