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| In [[category theory]], an abstract branch of [[mathematics]], [[distributive law]]s between monads are a way to express abstractly that two algebraic structures distribute one over the other one.
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| Suppose that <math>(S,\mu^S,\eta^S)</math> and <math>(T,\mu^T,\eta^T)</math> are two [[monad (category theory)|monads]] on a [[category theory|category]] '''C'''. In general, there is no natural monad structure on the composite functor ''ST''. On the other hand, there is a natural monad structure on the functor ''ST'' if there is a distributive law of the monad ''S'' over the monad ''T''.
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| Formally, a '''distributive law''' of the monad ''S'' over the monad ''T'' is a [[natural transformation]]
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| :<math>l:TS\to ST</math>
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| such that the diagrams
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| :[[Image:Distributive law monads mult1.png]] [[Image:Distributive law monads mult2.png]]
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| :[[Image:Distributive law monads unit1.png]] and [[Image:Distributive law monads unit2.png]]
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| commute.
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| This law induces a composite monad ''ST'' with
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| * as multiplication: <math>S\mu^T\cdot\mu^STT\cdot SlT</math>,
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| * as unit: <math>\eta^ST\cdot\eta^T</math>.
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| == See also ==
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| * [[distributive law]]
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| == References ==
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| * {{cite journal
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| | author = Jon Beck
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| | authorlink = Jonathan Mock Beck
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| | year = 1969
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| | title = Distributive laws
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| | journal = Lecture Notes in Mathematics
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| | volume = 80
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| | pages = 119–140
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| | doi = 10.1007/BFb0083084
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| | series = Lecture Notes in Mathematics
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| | isbn = 978-3-540-04601-1
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| }}
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| * {{cite book
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| | author = [[Michael Barr (mathematician)|Michael Barr]] and Charles Wells
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| | title = Toposes, Triples and Theories
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| | url = http://www.case.edu/artsci/math/wells/pub/pdf/ttt.pdf
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| | publisher = [[Springer-Verlag]]
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| | year = 1985
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| | isbn = 0-387-96115-1
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| }}
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| * {{nlab|id=distributive+law|title=Distributive law}}
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| * G. Böhm, Internal bialgebroids, entwining structures and corings, AMS Contemp. Math. 376 (2005) 207–226; [http://arxiv.org/abs/math.QA/0311244 arXiv:math.QA/0311244]
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| * T. Brzeziński, S. Majid, Coalgebra bundles, Comm. Math. Phys. 191 (1998), no. 2, 467–492 [http://arxiv.org/abs/q-alg/9602022 arXiv].
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| * T. Brzeziński, R. Wisbauer, Corings and comodules, London Math. Soc. Lec. Note Series 309, Cambridge 2003.
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| * T. F. Fox, M. Markl, Distributive laws, bialgebras, and cohomology. Operads: Proceedings of Renaissance Conferences (Hartford, CT/Luminy, 1995), 167–205, Contemp. Math. 202, AMS 1997.
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| * S. Lack, Composing PROPS, [http://www.tac.mta.ca/tac/volumes/13/9/13-09abs.html Theory Appl. Categ.] 13 (2004), No. 9, 147–163.
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| * S. Lack, R. Street, The formal theory of monads II, Special volume celebrating the 70th birthday of Professor Max Kelly. J. Pure Appl. Algebra 175 (2002), no. 1-3, 243–265.
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| * M. Markl, Distributive laws and Koszulness. Ann. Inst. Fourier (Grenoble) 46 (1996), no. 2, 307–323 ([http://www.numdam.org/numdam-bin/fitem?id=AIF_1996__46_2_307_0 numdam])
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| * R. Street, The formal theory of monads, J. Pure Appl. Alg. 2, 149–168 (1972)
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| * Z. Škoda, Distributive laws for monoidal categories ([http://front.math.ucdavis.edu/math.CT/0406310 arXiv:0406310]); Equivariant monads and equivariant lifts versus a 2-category of distributive laws ([http://front.math.ucdavis.edu/0707.1609 arXiv:0707.1609]); Bicategory of entwinings [http://arxiv.org/abs/0805.4611 arXiv:0805.4611]
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| * Z. Škoda, Some equivariant constructions in noncommutative geometry, Georgian Math. J. 16 (2009) 1; 183–202 [http://front.math.ucdavis.edu/0811.4770 arXiv:0811.4770]
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| * R. Wisbauer, Algebras versus coalgebras. Appl. Categ. Structures 16 (2008), no. 1-2, 255–295.
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| [[Category:Adjoint functors]]
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| {{categorytheory-stub}}
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