-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 is a ring map, then describing -modules in terms of -modules with extra structure (namely as modules for the monad given by ) 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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