# Foundations and contradictions of Scholze's work: the category of presentable infinity categories contains itself

Preface: I am not an expert in the work of Scholze, or anything for that matter.

## Question

Has Scholze stated what axioms he is using to develop his theory of motives and analytic geometry. In the following talk he seems to produce a contradiction to both ZFC and NBG; https://www.youtube.com/watch?v=I1lChD8fuGc

He seems to draw on the work of Lurie for his foundations, however Lurie doesnt make clear what his categorical foundations are in "Higher Topos Theory" or "Higher Algebra" so I'm not sure that you can import foundations from his work. Indeed, maybe this question is more aptly named, "Has Lurie specified his axioms?". Lurie references Mac Lane who seems to use universes, however Scholze says something along the lines of avoiding universes, but also talks about categories containing the category of all sets, which sounds like NBG.

I will provide a brief explanation of the contradiction as I see it, which occurs at the 30 min and 30 sec mark.

## The statement

First we let $$\kappa$$ be a regular cardinal. Scholze states that for the purposes of his theory we may as well take $$\kappa = \aleph_1$$. He then defines the following category $$\mathrm{Pr}^\kappa = \{\kappa-\text{compactly generated presentable }(\infty, 1)-\text{cats}, \text{colimit and }\kappa-\text{compact object preserving functors}\}$$

He then states that $$\mathrm{Pr}^\kappa \in \mathrm{Pr}^\kappa$$ which should properly mean that it is an object in the category $$\mathrm{Pr}^\kappa \in \mathrm{Ob}(\mathrm{Pr}^\kappa)$$

## Why this shouldn't be allowed

In NBG to be an element of something is to be a set, and therefore we reduce to the case of ZFC. In ZFC no set can contain itself by the axiom of regularity / well-foundedness.

It seems that there must be something tricky going on here using the fact that $$\mathrm{Pr}^\kappa$$ is a category and so as a set it is more than just its collection of objects. But this seems fraught with danger. It really seems to depend on how you set up category theory, not just ZFC+I vs NBG, but which particular sets you define categories to be etc.

A relevant thread that upon an amateurs perusal seems to support that this conclusion cannot be is; Is the $\infty$-category of presentable $\infty$-categories presentable?

Another relevant thread (that I havent had time to look through carefully yet) is Scholze's post; Reflection principle vs universes

Thanks.

• The statement is indeed mysterious, and I would also like an answer to this question as to how the apparent contradiction is resolved. Commented Aug 1 at 11:46
• In another context, a similar apparent contradiction occurs on the top of page 28 of arxiv.org/pdf/1312.2204. In that paper, it is stated that $\mathfrak{Top}^{\text{ét}}_\infty/\mathcal E\cong \mathcal E$ which, given that $\mathcal E\in\mathfrak{Top}^{\text{ét}}_\infty/\mathcal E$, seems to imply $\mathcal E\in \mathcal E$. Commented Aug 1 at 12:18
• One way to make sense of this is to recall that $\mathrm{Pr}^\kappa$ can either be presented as the (very large) category of $\kappa$-presentable (large) categories and cocontinuous functors preserving $\kappa$-compact objects or as the large category of $\kappa$-cocomplete small categories and $\kappa$-cocontinuous functors. If we take the first occurrence of $\mathrm{Pr}^\kappa$ to be the latter and the second to be the former, this is easier to interpret. Technically, it amounts to the claim that $\mathrm{Pr}^\kappa$ is $\kappa$-compactly generated which is (remarkably) true. Commented Aug 1 at 12:18
• $\mathrm{Pr}^\kappa\in \mathrm{Pr}^\kappa$ is a true and completely harmless statement, even if it cannot directly be expressed in classical set-theoretic foundations. The essential issue is not one of size (there are standard ways to work with "large" objects in set-theoretic foundations), but it is that $\infty$-groupoids/types/anima are more general than sets, and the axiom of regularity simply does not hold for them. A "set-theoretic translation" of the statement would look like $A\in\mathrm{Pr}^\kappa$ and $A\simeq \mathrm{Pr}^\kappa$, which does not violate regularity. Commented Aug 1 at 12:23
• @Rilem cf ncatlab.org/nlab/show/structural+set+theory and references there Commented Aug 1 at 12:26

A toy example of the same phenomenon: say a category is contractible if it’s equivalent to the terminal category (i.e. codiscrete and inhabited). Let $$\newcommand{\E}{\mathcal{E}}\E$$ be the category of contractible categories, and functors between them up to natural isomorphism. Then $$\E$$ is a contractible category; so $$\E \in \E$$?

Reading over everything carefully, to form $$\E$$ at all, we must have meant it was the category of small contractible categories (or some other standard way to handle size issues). Then $$\E$$ is not small, so it’s not literally an object of itself, but it is essentially small, so it is an object of itself up to equivalence. So $$\E \in \E$$ is true up to equivalence: there is some $$\E' \in \E$$ and equivalence $$\E' \simeq \E$$.

I understand the situation with Scholze’s $$\mathrm{Pr}^\kappa$$ (and the Lurie example mentioned by André Henriques in comments) as exactly analogous. $$\mathrm{Pr}^\kappa$$ itself must be formally defined with some size restriction on its objects; that will prevent it from being literally an object of itself. However, it’s compactly generated, so up to equivalence, it is an element of itself.

• There is a subtle difference between this example and the $\mathrm{Pr}^\kappa$ situation: $\mathcal E$ is really a 2-category of 1-categories, and to contemplate the statement $\mathcal E\in\mathcal E$ you have to allow yourself to view $\mathcal E$ as a 3-category of 2-categories (and then as a 4-category, a 5-category, etc.). So the actual failure of regularity only occurs if you go all the way to $\infty$-categories. As I expressed in my comment to the OP, the point is not one of size but is really about the nature of $\infty$-things. Commented Aug 1 at 13:09
• @MarcHoyois: I don’t entirely agree, unless I’m misunderstanding you. I was careful to make $\mathcal{E}$ into a 1-category by modding out its morphisms (leaving its class of objects unchanged) — so it really is a 1-category of 1-categories. I think the apparent conflict with regularity doesn’t come from the $\infty$-nature, but from working with objects structurally rather than in coding-dependent ways. Of course, these aren’t unrelated, since working fully structurally inevitably pushes one towards infinity-categorical language, and possibly infinity-categorical foundations… Commented Aug 1 at 13:18
• @PeterLeFanuLumsdaine I think you did misunderstand my comment. My point is that statements like $\mathcal E\in\mathcal E$ and $\mathrm{Pr}^\kappa\in\mathrm{Pr}^\kappa$ can be be regarded as literally true in the $\infty$-world, but without the concept of anima they can only be true up to something (as in your post/comments). Commented Aug 1 at 14:38
• @MarcHoyois: I agree with the first half of that point, that we can and should see statements as $\mathcal{E} \in \mathcal{E}$ to be literally true — what I’m suggesting (in comments on the question, I guess I didn’t say it in my answer!) is that this doesn’t necessarily require “move to the $\infty$-world”, but just the more general “take $\in$ to be something other than material-set-theoretic $\in$”, and that this is very natural in several other settings beside the $\infty$-world. Commented Aug 1 at 15:52
• @PeterLeFanuLumsdaine I agree that even classically there is no harm in saying $\mathcal E\in\mathcal E$, but ultimately this is because there exists a univalent setting where it is actually true (univalence is what I meant above by "the nature of $\infty$-things"). Classical category theory emulates univalence, but only in the $\infty$-world does it actually hold. Anyway, I don't want to keep trying to make this point, because it seems that everyone else here is discussing things completely orthogonal to it. Commented Aug 2 at 13:52

I think Marc’s point is worth emphasizing. While it is true that universe restrictions are what save us from paradox, (so that technically what we have is $$Pr_\kappa \in \Pr_\kappa’$$ for $$Pr_\kappa \neq Pr_\kappa’$$) this phenomenon is still much stranger than the fact that the category of small categories is an object of the category of large categories, because the universe restrictions we end up doing do not affect the fact that $$Pr_\kappa \simeq Pr_\kappa’$$!

To be more precise, we have to be a bit gauche and actually specify our set-theoretic foundations. Let’s work in ZFC with a single universe $$V_\lambda$$ (where $$\kappa \in V_\lambda$$ is a regular cardinal). Define $$Pr_\kappa$$ to comprise those categories whose object sets are subsets of $$V_\lambda$$, morphism sets live in $$V_\lambda$$, and which satisfy the usual definition of $$\kappa$$-presentability in $$V_\lambda$$; a morphism in $$Pr_\kappa$$ is a left adjoint functor preserving $$\kappa$$-compact objects. [1] Then we get some dumb thing like $$Pr_\kappa \in V_{\lambda+2} \setminus V_{\lambda+1}$$, depending on exactly how categories are coded up in terms of ordered pairs and on exactly how ordered pairs are defined. Hence we don’t have $$Pr_\kappa \in Ob(Pr_\kappa)$$. Nevertheless, it turns out that the object set of $$Pr_\kappa$$ is only of size $$\lambda$$, and not of size $$2^\lambda$$. And the morphism sets of $$Pr_\kappa$$ are of size $$<\lambda$$. And indeed, $$Pr_\kappa$$ is equivalent to a category which is an element of $$Pr_\kappa$$.

I think the reason this seems bizarre to me is that when we make universe restrictions in category theory, we’re used to thinking of them as being restrictions on size. Most of the time when I say “small set”, I only care about the cardinality of the set, and not about whether it actually is an element of $$V_\lambda$$. But in order to avoid the paradox here, you have to remember that “small set” is not really just a restriction on the size of the set — it’s a restriction on the universe the set belongs to.

But really, for category-theoretic purposes, as Marc emphasizes, the universe-evasion maneuver was completely invisible. This is quite different from the paradox-evasion we do when we say “well, sure, I don’t actually want to work with a category with all products — just small ones”, where the set-theoretical size issues had a legitimate category-theoretic interpretation. On the contrary, I think the correct way to think about this category-theoretically is that $$Pr_\kappa \in Pr_\kappa$$ is literally true, and there’s something about not-containing-yourself which is not forbidden category-theoretically. I don’t pretend to have a fleshed out formal apparatus to back this up, but I’m pretty convinced that the fact that $$Pr_\kappa \in Pr_\kappa$$ is hinting at something new and interesting.

[1] I’m going to blithely pretend everything is a 1-category and not an $$(\infty,1)$$-category here, it’s not that important.

• I think it’s also worth emphasizing that this is a fact which surprises many category theorists when they first learn it. My first reaction was “surely $Pr_\kappa \in Pr_{\kappa^+}$, but $Pr_\kappa \not \in Pr_\kappa$, right?”, and I suspect that’s most category theorists’ intuition, but nope! $\kappa$ stays the same. My second reaction was “wait, did I miss some unexpected universe-juggling which is necessary?” but nope, there’s no change in the notion of “small” here. Commented Aug 1 at 19:14
• One thing that’s really fascinating is that the concluion “$Pr_\kappa \in Ob(Pr_\kappa’) \wedge Pr_\kappa \simeq Pr_\kappa’$” is not really surprising-looking category-theoretically. After all, we could be defining the objects of $Pr_\kappa’$ in some stupid,cheating, completely arbitrary way. What makes the whole thing interesting (in a hard-to-precisify way) is that the passage from $Pr_\kappa’$ back to $Pr_\kappa$ is the most standard, reflexive, invisible move possible. Commented Aug 1 at 19:18
• Also — although $Pr_\kappa$ is an object of $Pr_\kappa$, it is not a $\kappa$-compact object. So the left adjoint functor $F : Set \to Pr_\kappa$ sending a point to $Pr_\kappa$ is a morphism in $Pr$ but not a morphism in $Pr_\kappa$. You might interpret this as saying that $Pr_\kappa$ is a “point” of $Pr_\kappa$, but not an “internal point” or something like that. (Here I’m thinking of left-adjoint functors from $Set$ as being like “points” because $Set$ is the unit of the Lurie tensor product on $Pr$ (which restricts to a tensor product on $Pr_\kappa$).) Commented Aug 2 at 3:38
• This is a more interesting and much more subtle point than my answer. I think mine is the main answer of the question as written, but I guess this is behind the underlying discomfort that’s why OP asked it. Like Mike in the Zulip, I think the main subtlety is working out what (if anything) should really be surprising here. Superficially, it looks analogous to Russellian paradoxical territory, but when one makes it precise (e.g. “$\newcommand{\C}{\mathcal{C}}\C$ is in the (essential) image of a certain functor $\C \to \mathrm{Cat}$”) it looks much less analogous to that, and not so surprising, Commented Aug 2 at 10:02