It's all signs but it isn't

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 suppose we have a large but locally small category C for which went to consider its category of presheaves. It is understandable why one would want to do so. After all, the yonda lemma is awesome! And presheaf categories have many nice properties, like, for example, being locally presentable. However, despite what n-lab might tell you, the category of presheaves on a large category does not exist in general (even when it is locally small).

So first we can consider (as was suggested in the last post) the category of small presheaves on C. This is the category of functors C^op → Set that are the left kan extension of a small functor. This category is legitimate (even without universes) and is, equivalently, the free cocompletion of our category C. As mentioned, this might lack certain limits. It is also not presentable but instead class-accessible because it has a class (rather then a set) of presentable objects (representable functors in this case) generating the category under filtered colimits. Limits can be recovered with some mild assumptions about C, but the presentability not so much. We have also added the possibly cumbersome (how cumbersome is debatable) task of checking whether our functors of interest are small.

Okay, so maybe we assume universes instead. But what does this actually do? For if C is defined in terms of the category of sets (as a functor category of its own or as a collection of sets with some structure or satisfying some condition) we will expand, by assuming an inaccessible cardinal, what it means to be an object of C, thus producing the same problem we had in the first place. So our assumption must be so that C is left untouched, making the assumption of a universe already two assumptions rather then one. But this still leaves a question, should our new inaccessible cardinal k be included in the category of sets?

Let us first consider the case where we assume an inaccessible cardinal k but leave the category of sets as the category of sets < k (the category of “small” sets if you like). This is seemingly the most natural assumption considering that we wish to keep C the same. Now Psh(C) is an actual category, but what properties does it have? Is it presentable now? Does it have limits? The answer is complicated by the ambiguity of what we should call a limit or colimit, for now it is meaningful to speak of (co)limits indexed by categories of size λ for λ not in the category of sets. Indeed, this category is no longer the free cocompletion of C and is not even locally small. The limits and colimits that do exist may also fail to be computable pointwise, since the diagram might not produce a (co)limit in Set.

Okay so we add k to the category of sets. Now C is small and so we recover the usual theory of presheaf categories. But now, as mentioned, there is a disconnect between the cardinal assumptions for the category of sets and for our categories of interest. This isn’t a contradiction, but it’s at least awkward. Of course, you might say that you don’t necessarily care about, say, inaccessible cardinal sized groups or topological spaces, and so this awkwardness is not much of a problem. And perhaps it isn’t, but what about categories of say the form Psh(Psh(D))? Even if D is small, Psh(D) should be large useless we restrict to presheaves into the category of sets of size less then k; call it Psh_{<k}(D). In this case, Psh(Psh_{<k}(D)) would be a presheaf category, but Psh_{<k}(D) wouldn’t be, as it would be missing the limits of size k and greater.

1. First and foremost, its barely Maths. We use logic in maths, sure, but a lot of logic isn't itself maths. That would be like saying that maths is a subfield of physics, just because it's its most well-known application.

Perhaps this is naive, but to me maths could be loosely and circularly defined as "the studied structures* of mathematics, and their applications to science". Here a structure is a Thingy With Axioms, and implicitly includes working in ZFC if there are sets. Maybe you're allowed to care about the continuum hypothesis, but if either assuming it or its contradiction gives better results (meaning that it allows us to prove more things with applications to the real world), then we should just do that. That's maths now. You can study whatever perverse axiom system you like, but that isn't maths. If you care about interpretation, it's Philosophy; if you don't, it's Logic. Maths cares about both intrinsic properties and interpretation of a particular set of "realistic" axioms, and the useful structures we define from them. I also don't think proofs are true if and only if they _are_ fully rigorous, we just the community to agree that they could be. That's good enough.

This is definitely circular, because what about set theory? But yeah, what about set theory? What is up with that?

2. The "interesting/surprising" results aren't that interesting anymore. Gödel is just the previous generation's Milnor/Freedman. I grew up with incompleteness, it doesn't surprise me. It's been a key part of maths since forever, as far as I'm concerned. I've been aware of exotic and non-smoothable manifolds for as long as I've been aware of manifolds, pretty much, so it isn't surprising that they exist. It's just a technical construction exercise. Same as Banach–Tarski isn't a paradox, it's just the statement that some sets aren't measurable, which isn't weird actually! Maybe there are some good results in there, but are they worth it? Not from the logic/model theory courses I watched my friends take.

On this note, I am excited to see what this is for future generations. Maybe they'll get more and more specialised, but I hope something else fun, well-known, and surprising is proven in our lifetimes, that our children will think is natural and boring.

3. Don't you people like fun?? I get that there's some fun in proving a surprising result, but the actual process?? Do you not like intuition? A physical vibe?? Maybe this is the topologist in me, but even the algebraists and analysists are always drawing shapes and diagrams of some kind, and talking about symmetries and actions. Maybe forcing is powerful, but has it any whimsy? I think not.

The reason this can often get confused is that early mathematical logic was motivated in this more philosophical aim. It was thought that mathematics could be see as the elementary consequences of first order logic, and when this turned out not to be the case, it was thought that one could at least see mathematics as the elementary consequences of a enumerable set of first order axioms, which, thanks to Godel, also turned out to be wrong. However, these early results are often presented as examples of the study of mathematical logic when, in reality, they are the justification for the study of mathematical logic. After all, it wouldn't be very intresting to study, say, models of various theories if mathmatics was the study of a single unifying theory with a single model.

This is a stand-alone version of a post that was split in two reblog chains under a different mostly unrelated post.

There are non-concretizable categories, aka categories which can not be thought of as sets and functions.

A concrete category is a category with a faithful (= injective on each hom-set, but not necessarily on objects) functor to the category of sets.

An classical example of a non-concretizable category is Ho(Top), the homptopy category of Topological spaces.

But why is this relevant?

Like with any size issue you can trick your way into ignoring it when you have large enough cardinals with respect to the category you are working with and are willing to step up in the size hierarchy. Similarily there are trivial counterexamples (if you allow categories with proper hom-classes).

Whenever we have a category K and a universe of set theory V in which this category is set-sized we have an easy concreatization F: K→V which maps objects X to ∪{K(A,X) | A∈K} and morphisms f∈K(X,Y) to -○f.

And if a category has a proper class sized hom-class there cannot be a concretization.

However there is the relevant case of proper class sized categories with set-sized hom-sets. For example the category of topologigal spaces. If we care about "all" topological spaces the trick of taking an universe of set theory with larger cardinals is no longer viable, since this would mean we can also have more topological spaces. Morally similar to how circumventions of various other size issues don't fundamentally invalidate those size issues.

But topological spaces are still concretizable, by simply forgetting the topological structure. The interesting insight is that this can not be done for all categories. Not all categories with set sized hom-sets can be thought of as sets with additional structure and morphisms which are particular functions, hence my original claim.

Ho(Top) has set sized hom-sets, since those are surjected on from the hom-sets of Top. Yet it can not be thought of as sets and functions.

In some way you can reinterpret the fact that not all locally small categories are concretizable as "the hom-sets being small does not imply that the objects can be represented via small sets".

How to prove this?

Theorem (Isbell, Freyd). A category which has all finite limits is concretizable if and only if it is well-powered when restricted to regular subobjects.

In fact this theorem holds for a weakening of the finite limits condition, but that basically just adds notational overhead.

Here an object being well-powered means that it has a partially ordered set of subobjects (= monomorphisms to the object). (This corresponds to the intuition "the powerset of the object exists. Ofc monomorphisms to an object are always partially ordered up to isomorphisms by monomorphisms between their domains, the important part here is that it is only set-sized. Also you can only take the domains of those monomorphisms as the subobjects, which also works.)

A monomorphism (or subobject) is regular if it is the equalizer of two morphisms in the category. (So intuetively if it can be defined as "the part of the domain of two morphisms on which they agree". This has little to do with the notion of regularity in set theory.)

The "⇒" of the theorem is due to Isbell and is quite easy to prove by

just applying the definitions to get that faithful functors are injective on equivalence classes of regular subobjects,

and the fact that sets have power-sets.

Try it yourself! (The "⇐" direction is due to Freyd and more complicated.)

where Freyd's result about the homotopy category of topological spaces is a more general phenomenon about homotopy categories of model categories.