Distributive law between monads
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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 and 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
such that the diagrams
commute.
This law induces a composite monad ST with
- as multiplication: ,
- as unit: .
See also
References
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- Distributive law at the nLab
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- Škoda, Z. (2004). "Distributive laws for monoidal categories". arXiv:math/0406310.
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suggested) (help)CS1 maint: DOI inactive as of March 2020 (link) - Wisbauer, R. (2008). "Algebras versus coalgebras". Appl. Categ. Structures. 16 (1–2): 255–295. doi:10.1007/s10485-007-9076-5.