In
linear and
homological algebra
Homological algebra is the branch of mathematics that studies homology (mathematics), homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology (a precurs ...
, 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
In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contrapositiv ...
and whose second map ''B'' → ''C'' is
surjective
In mathematics, a surjective function (also known as surjection, or onto function) is a function that every element can be mapped from element so that . In other words, every element of the function's codomain is the image of one element of i ...
. 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 "Const ...
References
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Vector bundles
Homological algebra
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