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In linear and homological algebra, a monad is a 3-term complex

A → B → C

of objects in some abelian category whose middle term B is projective and whose first map A → B is injective and whose second map B → C is surjective. Equivalently a monad is a projective object together with a 3-step filtration (B ⊃ ker(B → C) ⊃ im(A → B)). In practice A, B, and C are often vector bundles over some space, and there are several minor extra conditions that some authors add to the definition. Monads were introduced by Horrocks (1964, p.698).
See also

ADHM construction

References
Barth, Wolf; Hulek, Klaus (1978), "Monads and moduli of vector bundles", Manuscripta Mathematica, 25 (4): 323–347, doi:10.1007/BF01168047, ISSN 0025-2611, MR 0509589, Zbl 0395.14007
Horrocks, G. (1964), "Vector bundles on the punctured spectrum of a local ring", Proceedings of the London Mathematical Society, Third Series, 14 (4): 689–713, doi:10.1112/plms/s3-14.4.689, ISSN 0024-6115, MR 0169877, Zbl 0126.16801

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