In
linear
Linearity is the property of a mathematical relationship ('' function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear ...
and
homological algebra, a monad is a 3-term complex
: ''A'' → ''B'' → ''C''
of objects in some
abelian category
In mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have desirable properties. The motivating prototypical example of an abelian category is the category of ...
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 .
See also
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ADHM construction
In mathematical physics and gauge theory, the ADHM construction or monad construction is the construction of all instantons using methods of linear algebra by Michael Atiyah, Vladimir Drinfeld, Nigel Hitchin, Yuri I. Manin in their paper "Constru ...
References
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Vector bundles
Homological algebra
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