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fosco (Mar 22 2025 at 09:26):I'm reading Krömer's Tool and object, and at some point, page 255, I found this
Why is that so?! I tried looking for the original text of Lacombe, which is here https://archives-bourbaki.ahp-numerique.fr/files/original/561b398381c8d0ea41f74d38ef10705a.pdf but very difficult to parse for me. Even more because it doesn't feel to be explaining the problem much more than it is stating it...
There is also this other note by Krömer https://eudml.org/doc/252066 stating the problem.
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I would like to understand what is going on here, and whether this is "true" or a matter of philosophical stance.
Morgan Rogers (he/him) (Mar 25 2025 at 10:03):You're asking why it's "not expressible in a foundation of sets and proper classes"? I expect it's a matter of perceived limitations on what can be constructed in set/proper class foundations. To make formal the domain of the functor, you need to construct the product of functor categories between large categories. In simple foundations, the collection of functors can be 'too large' to form a proper class that one can take to be the objects of the functor category... A more nuanced resolution of size issues is typically enough to address this, I think.