Large cardinals approached through $\infty$-categories

Question

I am an undergraduate student (rising junior) majoring in philosophy and mathematics. For some time, I have been interested in homotopy type theory and so-called "univalent foundations". On the other hand, I am also interested in modern set theory, especially independence proofs and some questions related to large cardinals.

While I have learned some areas which allow one to interpret set theory in a "univalent" light (perhaps most notably through the analogies between forcing and some constructions in the theory of topoi), I find some questions are still relatively esoteric. To my knowledge (where I am very open to any corrections), for example, there does not seem to be an easy way to think about large cardinals from a univalent point of view, and apart from some applications to braid groups, they do not seem to "pop up" in areas of mathematics outside of logic (again, could be wrong here). I am wondering, however, whether we can think about them "homotopically" somehow. The ideas that follow are vague and I will be fleshing them out, so I'd love your feedback.

Let us take, for example, some inaccessible cardinal $κ$. I'm trying to think about how I might construct a functor from $\mathrm{\mathbf{Set}}_κ$ (taken as a subcategory of $\mathrm{\mathbf{Set}}$ or some appropriate quotient structure) to the $\infty$-category $\mathrm{\mathbf{Spaces}}$ that in some sense “preserves” the properties of κ as a cardinal. Let me illustrate more clearly what I mean. By definition an inaccessible cardinal is a cardinal $κ$ that is uncountable and satisfies $κ$-closure properties for smaller cardinals. In the context of the category $\mathrm{\mathbf{Spaces}}$, this could be thought of a sort of category closed under homotopy (co)limits up to $κ$, which translates the cardinal property to a diagrammatic one. This, in turn, could be used to perhaps define the combinatorial properties of cardinal arithmetic in a "homotopy-theoretic" way, in a sense making them more "natural" to deal with from the univalent POV. I know there are also some elegant results I could exploit if I focus on inaccessible cardinals specifically, like the correspondence with Grothendieck universes, so I get the sense I might not be shooting in the dark entirely (especially if I were to approach this from say, Grothendieck-Tarski set theory, where inaccessible cardinals exist, rather than ZFC).

I am still a novice as far as abstract homotopy theory goes (I am currently trying to work through some basic texts on the subject), and I have not yet started reading Kanamori's treatise on large cardinals (I am still working slowly through Kunen's "Set Theory", and trying to learn the material well). Is this an approach that could yield any interesting fruits? If so, is there any existing literature on similar matters I could consult? I have not yet found anything, and would like feedback and directions to take from here.

Answer 1

This is not really an answer to your question, but too long for a comment. It is also answering something in a slightly different direction, along the lines of "what can homotopy theory say about cardinals?".

In particular it's not about large cardinals but rather about quite small ones : the $\aleph_n$'s, or more precisely their corresponding ordinals $\omega_n$.

Namely there is a funny phenomenon that, say in categories of modules over rings, if you look at derived limits indexed by $\omega_n$, they have possibly a $\lim^{n+1}$, but no $\lim^{n+2}$.

The most famous example of this is $n=0$, where there is a very conceptually clear homotopical/$\infty$-categorical explanation : the poset $\omega$, viewed as a simplicial set, is categorically equivalent to a $1$-dimensional simplicial set, in other words the corresponding $\infty$-category is free on a graph.

Now this has other consequences besides vanishing of $\lim^2$ : it allows you to build and understand functors out of $\omega$ very nicely, and really simplifies one's life woth coherences and so on.

I'm not aware of a similar explanation for $\omega_n$ - I thought I had found one but I convinced myself (not super rigorously so this could be wrong) that the corresponding statement was wrong over $\omega_n$, namely that it is not free on an $n+1$-dimensional simplicial set. The proof of the $\lim^{n+2}$ statement from earlier is more indirect and does not seem to tell you much about the "homotopical structure of $\omega_n$".

Now here is the point : I believe this would be interesting to figure out for $\omega_n$, but furthermore it seems like there is some deep thing happening in the interaction between cardinals larger than $\omega$ and homotopy theory. Producing and understanding functors out of them is typically difficult, and unlike $\omega$ is by no means an automatic process (there is some sense in which transfinite induction doesn't work in homotopy theory - of course that's not literally true).

And this is, in some sense, only the tip of the iceberg: presumably there are deeper structures at play that one could try to understand, and there might be legitimately fun ways to understand the homotopical structure of large cardinals, beyond their set theoretic structure. But this is maybe just too optimistic and based on (so far) a single example and $\omega$-many non-understood ones...