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Patrick Nicodemus (Jan 04 2024 at 04:54):This paper is pretty interesting and hard to find a copy of so I'm posting it here to share it as I think others might also find it interesting. Rather than Grothendieck universes it proposes working in the category of sets in V_\lambda, parametrically in lambda for all sufficiently nice lambda (regular uncountable, say). It introduces a notion of "slow-growing categories" which i found pretty intriguing.
The paper is by V. K. Rao at Ohio State.
vidhyanath-rao-foundations.pdf
David Michael Roberts (Jan 04 2024 at 06:31):Thanks for this! I saw you add it to the nLab ( :partying_face: ) but couldn't find more than a partial Google Books preview.
Matteo Capucci (he/him) (Jan 04 2024 at 20:08):"A category C is called slowly growing if the underlying set of C is contained in V and every
small set of morphisms of C is contained in a small subcategory of C"
How could the second clause fail? If I have a small set of morphisms, then they span at most twice as many objects, and twice a small set is still small, innit?
Mike Shulman (Jan 04 2024 at 20:16):"small" here doesn't refer just to size, but to membership in a specific ZFC-style set. In particular, something small can be isomorphic to something large. So the condition is a non-structural one about what the objects and morphisms of the category "actually are" in the ZFC sense.
Matteo Capucci (he/him) (Jan 04 2024 at 20:18):ooh I see
Matteo Capucci (he/him) (Jan 04 2024 at 20:18):thanks!