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@Urs: I do not quite agree with the sentence “This is unrelated to other notions of monads” in Beilinson monad.
One can indeed view the Beilinson monad as the monad of an adjoint equivalence between $\mathfrak{Coh} \mathbb{P}^n$ (interpreted as the heart of $D^b \mathbb{P}^n$ and some category of linear complexes over an exterior algebra (the Koszul dual of the Cox ring of $\mathbb{P}^n$).
All right, if there is a useful way to make a relation, would be great if you could add explanation in the entry!
But it still seems that Beilinson chose the name “monad” in this case entirely uninfluenced by the category theoretic concept of monad, no?
(There are few things which may not be expressed in terms of some monad in some way…)
(Sorry, I should’ve just added my comment to http://nforum.mathforge.org/discussion/1563/monad/)
I was not referring to the historical origin of the naming in my comment.
Beilinson generalized the $3$-term Horrocks monad (a complex with $3$ split vector bundles) introduced in
G. Horrocks, Vector bundles on the punctured spectrum of a ring, 1964, Proc. London Math. Soc. (3) 14, 689-713
to a complex with $2n+1$ vector bundles on $\mathbb{P}^n$. So the question is whether Horrocks was influenced by the category theory concept back then (btw Horrocks’ papers are full of categorical concepts). I don’t know when the word monad was used for the very first time instead of Godement’s “standard construction” (Benabou in 1967 is 3 years after Horrocks).
What I was referring to is probably what you expressed by saying: (There are few things which may not be expressed in terms of some monad in some way…)
Okay. Whatever useful information, such as above, you have to add to the entry, please do. Whether it relates to monads in the sense of category theory or not!
OK, I will, probably next week. I will then need to create at least one additional page about the Tate resolutions/complexes in this context.
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