distributive law between monads

In category theory, an abstract branch of mathematics, distributive laws between monads are a way to express abstractly that two algebraic structures distribute one over the other one.

Suppose that $(S,\mu^S,\eta^S)$ and $(T,\mu^T,\eta^T)$ are two monads on a category 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
:$l:TS\to ST$
such that the diagrams
:         
:         
commute.

This law induces a composite monad ''ST'' with
* as multiplication: $STST\xrightarrowSSTT\xrightarrowST$,
* as unit: $1\xrightarrowST$.

See also

* distributive law

References

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Adjoint functors

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