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In this beautiful talk by Colin McLarty, McLarty quotes Grothendieck:

It would be nice to have a context where one doesn't add any real axioms to set theory, and yet one can work with categories without too much afterthought and trembling. Take each functor category $C^D$ as another category etc.

and then he remarks:

An advertisement for my work: I can now provide this context. I can provide a context radically weaker than even Zermelo–Fraenkel set theory in which you can do all of SGA4 with no afterthought and trembling and all the functor categories are categories.

Question: What is this context McLarty is referring to? How can such a context be possible without using universes?

user333306
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    See here – Wojowu Aug 02 '21 at 14:16
  • @Wojowu This is a related discussion with lots of comments and answers, but does it really answer the question? Please explain a bit more. – Martin Brandenburg Aug 02 '21 at 15:14
  • @MartinBrandenburg Sorry, I did mean it just as a "related discussion". I'm not sure to what extent (if any) OP would find it satisfactory. – Wojowu Aug 02 '21 at 15:32
  • I eat universes for breakfast. – Monroe Eskew Aug 02 '21 at 16:53
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    https://arxiv.org/abs/1102.1773 – David Roberts Aug 02 '21 at 20:52
  • @DavidRoberts Thank you very much! It would be nice if someone could summarize the main ideas in this article as an anwer to my question. I skimmed through the article and as far as I understand it, McLarty works in a formal system with the types "set", "class", and "collection", which is conservative over the "set"-fragment of itself (which implies that concrete consequences of results in SGA, which where originally proved using "large" structures such as toposes, are provable in fact in finite order arithmetic). But I'm wondering, how can he talk in 3.2 about presheaves without ... – user333306 Aug 05 '21 at 16:34
  • having introduced classes at this point? Also, it seems to me that in this context not all functor categories are categories, as McLarty's system uses the types "class" and "collection" to avoid exactly these kinds of size issues. – user333306 Aug 05 '21 at 16:35
  • @MonroeEskew What is the point of your comment? – user333306 Aug 05 '21 at 16:39

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