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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).
Absolutely. If you do have the energy to improve this entry, please do.
Added:
An elementary proof of the equivalence between infinitary Lawvere theories and monads on the category of sets is given in Appendix A of
While statement is (probably) true for Cartesian closed categories, the example of the double-dual monad for vector spaces is not an example of double-dual monads for Cartesian closed categories; indeed, the Vect_k double dual monad makes use of the other monoidal product on Vect_k, namely the tensor product, which is explicitly, and very importantly here, the product which induces the appropriate tensor-hom adjunction to apply.
In any case, (assuming that the Cartesian product adjunction is the internal hom functor) all Cartesian closed categories are automatically closed symmetric monoidal categories (so automatically bi-closed monoidal categories), so the remark falls under this more expansive and contextually accurate case one way or the other.
Lillian Ryan Uhl
The comment #27 is probably referring to the last paragraph in this example.
While looking over it, I have added some more hyperlinking and adjusted some wording in this example.
Thanks for the alert. I’ll forward this to the technical team.
Thanks for highlighting. The reference to Voutas’ article was deleted in revision 93 and actually announced so in comment #26 above, but nobody reacted.
In general, though, if there is anything cite-worthy in a pdf sitting otherwise unpublished somewhere on the web, then we shoould upload it to the nLab server to preempt the almost inevitable link rot.
I have done so now for Voutas’ file (the second pdf link here)
back to #30:
Christian Sattler has kindly deleted the spurious entry “monad+”, along with a slew of similar ghost entries.
So the issue at hand should be solved. But apparently it arises due to a bug deep in the database and may reoccur over time.
The first commutative diagram is a bit confusing I think, since those diagonal arrows are unlabeled. From what I can tell they are identity 2-cells?
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