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It is known that one can formulate certain large cardinal axioms (e.g. Vopenka's principle--see Mike Shulman's answer to Harry Gindi's mathoverflow question "Reasons to believe Vopenka's Principle/huge cardinals are consistent") in terms of Category theory.

Can Reinhardt cardinals (and consequently Kunen's inconsistency result regarding them) be formulated in terms of Category theory as well? As a 'soft question' I am wondering if such a formulation might show forth (so to speak) any possible "deep inconsistency" (to use Peter Koellner's term) regarding the hypothesis that $ZF+\text{ exists a Reinhardt cardinal}$ is consistent?

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    $\begingroup$ I've been thinking lately it would be a good idea to link elementary embeddings and category theoretic constructions as discussed at eg golem.ph.utexas.edu/category/2015/04/five_quickies.html#c048923. One issue to note is that measurables and certain elementary embeddings aren't the same in the absence of Choice, and this makes things more interesting. $\endgroup$
    – David Roberts ♦
    Commented May 10, 2015 at 9:36
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    $\begingroup$ @David: I merely wanted to point out that if we are referring to possible cases that elementary embeddings and ultrapowers are not quite as nicely behaved as in $\sf ZFC$, these two papers should be mentioned. :-) $\endgroup$
    – Asaf Karagila ♦
    Commented May 10, 2015 at 11:52
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    $\begingroup$ It is not really correct to talk about Reinhardt cardinals in mere ZF, since the concept is not first-order expressible in the language of set theory, unless it is self-contradictory (proof: if being Reinhardt were first-order expressible and $\kappa$ is the least Reinhardt cardinal, with elementary embedding $j:V\to V$, then $j(\kappa)$ is also the least Reinhardt cardinal, a contradiction). Rather, formalizing Reinhardt cardinals requires some formal treatment of classes, and so one is working in GB or ZF($j$). $\endgroup$
    – Joel David Hamkins
    Commented May 10, 2015 at 12:36
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    $\begingroup$ @JoelDavidHamkins Thanks for the clarification. If I remember correctly, global choice was in both Kelley's and Morse's axiomatizations, but it was also in Gödel's version of GB (Axiom E, I think); I'm not sure about Bernays, but von Neumann had it in the form "all proper classes are in bijection with V". So it seems the presence of choice in a system named after someone need not agree with its presence in that person's axioms. (That's certainly the case for Zermelo and ZF.) $\endgroup$
    – Andreas Blass
    Commented May 10, 2015 at 23:24
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    $\begingroup$ @Andreas: Now I feel less guilty for not thinking about $\sf KM$ as something with global choice included. :-) $\endgroup$
    – Asaf Karagila ♦
    Commented May 11, 2015 at 5:47

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I suspect that no very satisfying answer to this question is known, so let me just point out that there are a lot of different things one might mean by a "category-theoretic understanding" of a large-cardinal axiom. I am far from an expert, so I hope that others can help flesh this out and correct me, but here are some themes that I happen to have seen in the literature:

  1. Set theorists have responded to the paucity of elementary embeddings $V \to V$ by considering elementary embeddings $V \to M$ where $M$ is not $V$. Whereas category theory suggests studying endofunctors $\mathsf{Set} \to \mathsf{Set}$ which satisfy nice properties weaker than being elementary embeddings (actually the first approach can be viewed as a special case of the second, since $M$ is generally assumed to be a non-elementary substructure of $V$, so we can compose $V \to M \subset V$ to get an endofunctor $V \to V$). This approach is much less developed; the only examples I happen to have come across are at the level of measurable cardinals:

    • As David Roberts alluded to in the comments, Isbell essentially showed that a full subcategory of $\mathsf{Set}$ is codense iff it is not bounded by any measurable cardinal. A subcategory $A \subseteq B$ is codense (when $B$ is suitably complete) iff its codensity monad (a certain endofunctor $B \to B$, namely the right Kan extension of the inclusion $A \to B$ along itself) is isomorphic to the identity. The codensity monad of a full subcategory $A \subseteq \mathsf{Set}$ is the functor $\mathsf{Set} \to \mathsf{Set}$ sending $X$ to the set of $A$-complete ultrafilters on $X$, hence from a category-theoretic perspective, measurable cardinals can be viewed as connected to the study of certain endofunctors $\mathsf{Set} \to \mathsf{Set}$ which are not elementary embeddings.

    • Another relationship between endofunctors of $\mathsf{Set}$ and measurable cardinals is Trnková's result (later rediscovered by Blass) that there is a measurable cardinal if and only if there is a left exact endofunctor of $\mathsf{Set}$ not isomorphic to the identity. Both of these results are related to Borger's result that the endofunctor of $\kappa$-complete ultrafilters is terminal among endofunctors of $\mathsf{Set}$ preserving $\kappa$-small coproducts.

  2. I'm having a harder time getting a grip on where category-theoretic statements of Vopenka's Principle fit, largely out of ignorance. But it seems like roughly, any ZFC-independent statement about accessible categories is equivalent to Vopenka's Principle, making it hard to see how to generalize the approach. The idea that a large cardinal axiom can be equivalent to a structure axiom about essentially arbitrary classes of first-order models is pretty cool -- I guess this is analogous to having combinatorial statements equivalent to smaller large cardinals. It provides a pretty compelling picture of what Vopenka's Principle means.

  3. Type theory has a natural categorical semantics. And it's natural to try to formulate large cardinal axioms in terms of the universes in type theory. There's some work on this by Palmgren, but I think it's dealt mostly with smaller large cardinal axioms.

  4. Let me just throw in another possibility that I haven't seen studied, but which could be. One could look for statements high in consistency strength by asking questions about nice functors between categories that don't model ZFC, such as toposes and geometric morphisms. This might go in the same direction as asking about ZF+Reinhardt cardinals, but much further. I have no idea whether this could be a legitimate source of strong statements. EDIT Andreas Blass points out in the comments that there is an example of this at the level of measurable cardinals studied by Adelman and Blass.

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    $\begingroup$ Regarding your first point, every elementary embedding $j:V\to M$ into a transitive class $M$ (and this includes basically all the usual large cardinal embeddings) is also a $\Delta_0$-elementary embedding $j:V\to V$, simply because every transitive class $M$ is $\Delta_0$-elementary in $V$. The larger large cardinals then insist on greater absoluteness between $M$ and $V$, such as with strongness or supercompactness, and so the usual large cardinal development fits into the paradigm you describe. $\endgroup$
    – Joel David Hamkins
    Commented May 11, 2015 at 12:27
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    $\begingroup$ Thanks for citing my paper, but, to give proper credit, I must point out that the main results were discovered earlier by Trnkova and Reiterman. See: Corrections to: “Exact functors and measurable cardinals” Pacific J. Math. 73 (1977), no. 2, 540. $\endgroup$
    – Andreas Blass
    Commented May 11, 2015 at 14:13
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    $\begingroup$ A special case of your item 4 is the topic of: Adelman, Murray(5-MCQR); Blass, Andreas(1-MI) Exact functors, local connectedness and measurable cardinals. (Italian summary) Rend. Sem. Mat. Fis. Milano 54 (1984), 9–28 (1987). $\endgroup$
    – Andreas Blass
    Commented May 11, 2015 at 14:15
  • $\begingroup$ @AndreasBlass: Thanks! I've updated my answer to reflect your comments. $\endgroup$
    – Tim Campion
    Commented May 11, 2015 at 14:42
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    $\begingroup$ I think that a $\Delta_0$-elementary embedding of the universe is implied by a measurable cardinal (according to Cantor's Attic, a measurable cardinal is equivalent to an elementary embedding from $V$ to some transitive class $M$, and the inclusion $M \to V$ is always a $\Delta_1$-elementary embedding). I would guess that a $\Sigma_n$-elementary embedding would be given by a logical functor that preserves classifying objects for $\Sigma_n$ formulas in the stack semantics. $\endgroup$
    – Tim Campion
    Commented Jun 13, 2015 at 15:42

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