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Kevin Carlson (Aug 27 2024 at 03:36):How do you actually prove that the 2-category of elementary toposes and logical functors is 2-monadic over categories? I supposedly know this is true (and the nLab agrees) but the BPS 2-monads paper doesn't have it; they just have monadicity of the 1-category and 2-monadicity of the 2-category over groupoids, which is a common cheat for structures that involve some contravariant operations. I'm interested both in a reference and in some general lore if, say, it's actually easy now to upgrade the 1-monadicity result to 2-monadicity.
Nathanael Arkor (Aug 27 2024 at 05:29):Elementary toposes are algebras for a 2-monad on the 2-category of categories, functors, and natural isomorphisms, which is sometimes denoted (which is not the same as a 2-category of groupoids). This is consistent with both what the nLab states and what Blackwell–Kelly–Power state. They're not algebras for a 2-monad on the 2-category of categories, functors, and natural transformations, for the contravariance problems that you mention.
Kevin Carlson (Aug 27 2024 at 06:27):Ah, thanks, I was looking for groupoids in the objects and not the homs and so didn’t read the whole definition. That’s comforting that there’s not some sudden leap back to monadicity over the full 2-category when you go from Cartesian closed to a topos.