Several things part 4

-If you need a ten minute break from whatever important thing you’re doing, and you read German, you could do worse than looking at this short essay I wrote for the 2021 Jahrbuch of the Max-Planck-Gesellschaft. Big thanks to Christian Kaiser for doing the translation!

-In an earlier post, I mentioned that I didn’t appreciate monads. Thanks to some very lucid explanations from CJ, this is no longer the case. In retrospect, two things were confusing me:
i. If A\to B is a ring map, then describing B-modules in terms of A-modules with extra structure (namely as modules for the monad given by -\otimes_A B) is the canonical example of monadic descent, and it works for any ring map, while the more familiar faithfully flat descent is an example of comonadic descent.
ii. The perverted and misleading terminology of “algebras” for a monad, used e.g. on wikipedia, for the concept which should obviously be called “modules”. At least Lurie uses the word module in his books.

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