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.

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