In
category theory, an abstract branch of
mathematics,
distributive law
In mathematics, the distributive property of binary operations generalizes the distributive law, which asserts that the equality
x \cdot (y + z) = x \cdot y + x \cdot z
is always true in elementary algebra.
For example, in elementary arithmetic, ...
s between monads are a way to express abstractly that two algebraic structures distribute one over the other one.
Suppose that
and
are two
monads on a
category
Category, plural categories, may refer to:
Philosophy and general uses
*Categorization, categories in cognitive science, information science and generally
* Category of being
* ''Categories'' (Aristotle)
* Category (Kant)
* Categories (Peirce) ...
C. In general, there is no natural monad structure on the composite functor ''ST''. However, there is a natural monad structure on the functor ''ST'' if there is a distributive law of the monad ''S'' over the monad ''T''.
Formally, a distributive law of the monad ''S'' over the monad ''T'' is a
natural transformation
In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i.e., the composition of morphisms) of the categories involved. Hence, a natur ...
:
such that the diagrams
:
:
commute.
This law induces a composite monad ''ST'' with
* as multiplication:
,
* as unit:
.
See also
*
distributive law
In mathematics, the distributive property of binary operations generalizes the distributive law, which asserts that the equality
x \cdot (y + z) = x \cdot y + x \cdot z
is always true in elementary algebra.
For example, in elementary arithmetic, ...
References
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
Adjoint functors
{{categorytheory-stub