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,μ^S,η^S) and (T,μ^T,η^T) are two monads on a 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.

Formally, a distributive law of the monad S over the monad T is a natural transformation

$l:TS\to ST$

such that the diagrams

        

and

commute.

This law induces a composite monad ST with

• as multiplication: $S\mu^T\cdot\mu^STT\cdot SlT$,
• as unit: $\eta^ST\cdot\eta^T$.

See also

• distributive law

References

• Jon Beck (1969). "Distributive laws". Lecture Notes in Mathematics. Lecture Notes in Mathematics 80: 119–140. doi:10.1007/BFb0083084. ISBN 978-3-540-04601-1. 

• Michael Barr and Charles Wells (1985). Toposes, Triples and Theories. Springer-Verlag. ISBN 0-387-96115-1. http://www.case.edu/artsci/math/wells/pub/pdf/ttt.pdf. 
• Distributive law at nLab

• G. Böhm, Internal bialgebroids, entwining structures and corings, AMS Contemp. Math. 376 (2005) 207–226; arXiv:math.QA/0311244

• T. Brzeziński, S. Majid, Coalgebra bundles, Comm. Math. Phys. 191 (1998), no. 2, 467–492 arXiv.

• T. Brzeziński, R. Wisbauer, Corings and comodules, London Math. Soc. Lec. Note Series 309, Cambridge 2003.

• 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.

• S. Lack, Composing PROPS, Theory Appl. Categ. 13 (2004), No. 9, 147–163.

• 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.

• M. Markl, Distributive laws and Koszulness. Ann. Inst. Fourier (Grenoble) 46 (1996), no. 2, 307–323 (numdam)

• R. Street, The formal theory of monads, J. Pure Appl. Alg. 2, 149–168 (1972)

• Z. Škoda, Distributive laws for monoidal categories (arXiv:0406310); Equivariant monads and equivariant lifts versus a 2-category of distributive laws (arXiv:0707.1609); Bicategory of entwinings arXiv:0805.4611

• Z. Škoda, Some equivariant constructions in noncommutative geometry, Georgian Math. J. 16 (2009) 1; 183–202 arXiv:0811.4770

• R. Wisbauer, Algebras versus coalgebras. Appl. Categ. Structures 16 (2008), no. 1-2, 255–295.