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 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 ab ...

whose middle term ''B'' is projective, 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 Filtration is a physical separation process that separates solid matter and fluid from a mixture using a ''filter medium'' that has a complex structure through which only the fluid can pass. Solid particles that cannot pass through the filter ...

''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 .

References

* * Vector bundles Homological algebra {{algebra-stub

See also

References