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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeJul 8th 2010
    • (edited Jan 14th 2014)

    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.

    • CommentRowNumber2.
    • CommentAuthorFinnLawler
    • CommentTimeSep 16th 2010
    • (edited Sep 16th 2010)

    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.

    • CommentRowNumber3.
    • CommentAuthorphx
    • CommentTimeJan 9th 2012

    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 0ABC00 \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?

    • CommentRowNumber4.
    • CommentAuthorzskoda
    • CommentTimeJan 9th 2012
    • (edited Jan 9th 2012)

    ad 3: The name is not motivated by any kind of similarity to monoids, as far as I know. Once we create the nnLab page for it, it should be Beilinson’s monad, what is the standard full name for it.

    • CommentRowNumber5.
    • CommentAuthorphx
    • CommentTimeJan 9th 2012

    i was unable to find the page you mentioned, but thanks anyway.

    • CommentRowNumber6.
    • CommentAuthorTim_Porter
    • CommentTimeJan 10th 2012
    • (edited Jan 10th 2012)

    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).

    • CommentRowNumber7.
    • CommentAuthorzskoda
    • CommentTimeJan 10th 2012
    • (edited Jan 10th 2012)

    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:

    • M.F. Atiyah, N.J. Hitchin, V.G. Drinfeld, Yu.I. Manin, Construction of instantons, Physics Letters 65 A, 3, 185–187 (1978) pdf

    and for a noncommutative version the paper of Kapustin, Kuznetsov and Orlov which does explicitly talk about (noncommutative) Beilinson monads.

    • CommentRowNumber8.
    • CommentAuthorphx
    • CommentTimeJan 10th 2012
    • (edited Jan 10th 2012)

    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 3\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).

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeJan 10th 2012

    I have started Beilinson monad.

    Also created monad (disambiguation).

    • CommentRowNumber10.
    • CommentAuthorphx
    • CommentTimeJan 10th 2012

    here is the reference to Nahm’s paper:

    • W. Nahm. Self-dual monopoles and calorons. In Group theoretical methods in physics (Trieste, 1983), pages 189-200. Springer, Berlin, 1984. pdf
    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeJan 14th 2014
    • (edited Jan 14th 2014)

    As David Corfield noticed in another thread, the amount of discussion of monadicity on the nnLab had been rather lacking.

    As a first-aid means, I have

    More deserves to be done here, eventually.

    • CommentRowNumber12.
    • CommentAuthorDavid_Corfield
    • CommentTimeJan 14th 2014

    Is there something ’modal’ going on here, an adjunction suspended between two moments - the Eilenberg-Moore adjunction and the Kleisli adjunction?

    • CommentRowNumber13.
    • CommentAuthorMatt Earnshaw
    • CommentTimeAug 4th 2016

    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.

    • CommentRowNumber14.
    • CommentAuthorTodd_Trimble
    • CommentTimeAug 4th 2016

    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).

    • CommentRowNumber15.
    • CommentAuthorMike Shulman
    • CommentTimeAug 5th 2016

    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).

    • CommentRowNumber16.
    • CommentAuthorAli Caglayan
    • CommentTimeDec 25th 2018

    Added some more examples and reorganised. I will write out some in due time. I will appreciate any help fleshing out the examples, especially detailing what the unit and multiplication does. The monad examples in Riehl’s Category theory in context would be nice to include.

    diff, v74, current

  1. Changed link to monoid in monoidal category rather than monoid the set with operation.


    diff, v77, current

  2. Corrected typo (“principle ultrafilter” to “principal…”) and added missing parenthesis.

    Baptiste Loreau

    diff, v86, current

    • CommentRowNumber19.
    • CommentAuthorMorgan Rogers
    • CommentTimeMay 7th 2021
    I have just discovered that the nLab page on monads contains only the bicategorical definition, with only a vague allusions to monads in Cat. Yet all of the examples are of monads in CAT (notably of monads *on* Set rather than *in* Set, as the subsection heading misleading notes), and this is true in CT more generally in my experience. Is there any reason why a definition so inaccessible to students, and more importantly to computer scientists whose existing understanding of CT might be limited, is prioritised? Shouldn't the "monad in a bicategory" definition be in a later section, for example?
    I figured I should get some input before going ahead and making such changes.
    • CommentRowNumber20.
    • CommentAuthorUrs
    • CommentTimeMay 7th 2021

    Absolutely. If you do have the energy to improve this entry, please do.

    • CommentRowNumber21.
    • CommentAuthorJohn Baez
    • CommentTimeMay 8th 2021

    Pointed computer scientists to the page “monads in computer science”.

    diff, v89, current

    • CommentRowNumber22.
    • CommentAuthorJohn Baez
    • CommentTimeMay 8th 2021

    The section mis-titled “Monads in Set” is really about monads on Set! I fixed that.

    diff, v89, current

    • CommentRowNumber23.
    • CommentAuthorDmitri Pavlov
    • CommentTimeJun 22nd 2021


    An elementary proof of the equivalence between infinitary Lawvere theories and monads on the category of sets is given in Appendix A of

    diff, v90, current

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