Large Cardinal Assumptions in Category Theory
One thing that's funny about being a grad student in the intersection of model theory and category theory is navigating how to feel about large cardinals. On the one hand, most of my time is spent doing category theory, which tends to be quite casual/unaware about the subtitles of large cardinal assumptions; on the other, I'm cursed with some knowledge of set theory---enough to know that there's more to say, but not enough to have a perfectly refined take on the matter. As a result, my feelings about large cardinal assumptions (specifically Grothendieck universes) has had a lot of changes. This blog post is about my past and current feelings on the matter.
Initially, I thought that the problem of large cardinals was a matter of consistency, and so, since the consistency strength of large cardinals was higher (lower? I forget the convention on direction here) then ZFC, it was probably a best avoided practice. However, I later realized that, at least for the assumption of inaccessible cardinals (Grothendieck universes), this is really not much of a concern. In particular, there's nothing really special about the consistency strength of ZFC, and the assumption of inaccessible cardinals is rather mild in the scale of things (if we are concerned about consistency, then replacement and powerset are much better targets then inaccessible cardinals for our concerns). Indeed, the assumption of even many inaccessible cardinals is sometimes not even depicted on the large cardinal charts, and set theorist are often assuming much worse in their day to day.
So, for a period of time, I thought that assuming Grothendieck universes was basically fine, and that, although category theorists could do a better job of noting when they make such assumptions, it didn't really matter that much. But then I did my master's thesis. For some technical reasons, I needed to consider presheaves on a large category but without making large cardinal assumptions. The tool for such a task is to consider the category of small presheaves, which is the free cocompletion of a possible large category. This category is legitimate and has many of the same properties as a presheaf category, but many notable properties (such as the existence of limits!) need not hold in this category. This makes the situation very different from taking a presheaf category by assuming universes. And so the crux of the issue is this: what the category theorist uses a single cardinal assumption to solve, usually encodes several separate assumptions. In particular, the meaning of being small as a category, a small (co)limit or a member of the category of sets could be separate things. Category theorists are not completely unaware of this issue, using terms like small, large, very large, ect, as a way to differentiate between various sizes encoded in their assumptions, but because this isn't very closely accounted for, it's hard to really say what precise assumptions are necessary in some of these constructions.
It's worth making clear again, this is not a cause to expect inconsistency. There should be some set theory in which things work out, but which set theory becomes unclear. This is made worse by the fact that different Grothendieck universes often disagree about properties of smaller sets. So I was once again convinced that we just shouldn't assume universes such assumptions.
But then recently I've been learning about independence relations in model theory, which naturally give rise to the notion of a monster model. That is, a class sized model, often defined to be saturated (for some intuition, a saturated model is basically a major generalization of the notion of an algebraically closed filed; it is a model in which all types are realized). Model theory is often more convenient inside a monster model and the existence of a monster require the existence of inaccessible cardinals; yet, model theoriests rarely claim there theorems to take place outside of set theories equiconsistent with ZFC. The way the pull this off is by computing what sort of large cardinals are needed for their constructions, meaning that the assumptions for types, the monster and automorphisms of the monster are kept separate but are related through cardinal arithmetic. The reason this is useful is that, although ZFC can't prove the existence of large cardinals, it can still talk about what would happen in a large cardinal if it existed so long as its properties are well specified. Thus, if the conclusion of a theorem doesn't itself infer the existence of large cardinals, one can often deduce, implicitly, that there is a proof of a given theorem in ZFC even if the proof that was used uses large constructions.
So now my opinion is that category theory should go about large cardinal assumptions more in this way, though I reserve the right to change my opinion.
