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following Zoran’s suggestion I added to the beginning of the Idea-section at monad a few sentences on the general idea, leading then over to the Idea with respect to algebraic theories that used to be the only idea given there.
Also added a brief stub-subsection on monads in arbitrary 2-categories. This entry deserves a bit more atention.
I’ve reorganized monad a bit, trying to remove some of the cruft that had gathered there. Also I’ve added a new section on Street’s bicategory of monads in a given bicategory, and moved the section on the successor monad to its own page.
there is also a notion of a monad in the context of holomorphic vector bundles (or coherent sheaves) over projective spaces (or more generally, (compact) complex manifolds): A monad in this context is a complex of holomorphic vector bundles $0 \to A \to B \to C \to 0$ which is exact at A and C. If I’m not mistaken, these first appeared in
G. Horrocks, Vector bundles on the punctured spectrum of a ring, 1964, Proc. London Math. Soc. (3) 14, 689-713
and often appear in the same context as the Beilinson spectral sequence.
Is there any relation to the monoids in the category of endofunctors? Is it an example? If so, how? Is it worth mentioning this in the article on monads?
ad 3: The name is not motivated by any kind of similarity to monoids, as far as I know. Once we create the $n$Lab page for it, it should be Beilinson’s monad, what is the standard full name for it.
i was unable to find the page you mentioned, but thanks anyway.
I just checked up on Geof Horrocks in Wikipedia and that construction is known as the ADHM construction. (NB. the H is not Horroocks here but Hitchin.) This is relevance to instantons (or so it say on Wikipedia). Wikipedia also calls it the monad construction. (Perhaps disambiguation is called for. I know of another use of the term monad in non-standard analysis, and lots of others, see the wikipedia page on monad and the nLab entry).
No, Tim. ADHM construction is quite specific and later. Though it does involve Beilinson monads in some formalism. ADHM construction is from mid 1970s, see the paper under Yang-Mills instanton:
and for a noncommutative version the paper of Kapustin, Kuznetsov and Orlov which does explicitly talk about (noncommutative) Beilinson monads.
I have to say that I disagree with Wikipedia (although I’m not sure what the standard terminology is): From my point of view, the monad construction is just a part of the ADHM construction (another part being the Twistor construction which identifies instantons with certain real algebraic bundles over $\mathbb{P}^3$ and makes it possible to use monads at all)
also note that the ADHM construction was reformulated by W. Nahm in a way that does not use any monads at all (at least not explicitly).
I have started Beilinson monad.
Also created monad (disambiguation).
here is the reference to Nahm’s paper:
As David Corfield noticed in another thread, the amount of discussion of monadicity on the $n$Lab had been rather lacking.
As a first-aid means, I have
added a brief paragraph with the basic statements to monad – Properties – Relation to adjunctions
copied the same to adjoint functor – Properties – Relation to monads
cross-linked the two entries monadic adjunction and monadic functor (which didn’t know of each other and which should maybe be merged alltogether).
More deserves to be done here, eventually.
Is there something ’modal’ going on here, an adjunction suspended between two moments - the Eilenberg-Moore adjunction and the Kleisli adjunction?
In Section 1 we read:
the concept of a monad is the horizontal categorification of that of a monoid.
however horizontal categorification suggests that the h.c. of a monoid is a category, so I’m wondering how useful the above remark is.
Well, I think the preceding sentence explains what was meant by that. Also, “categorification” to my mind isn’t quite well-defined (“decategorification” fares better).
Also, categories are a special case of monads. (They are monads in Span.) But perhaps it would be better for the sentence to say “a horizontal categorification” rather than “the horizontal categorification” (and now it does).
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