nLab weak distributive law

Idea

The notion of a weak distributive law between two monads is a generalisation of that of a distributive law, in which forming the composite monad requires splitting an idempotent on the underlying composite endofunctor.

References

See also weak bimonad.

Weak distributive laws among monads:

• Gabriella Böhm, The weak theory of monads, Adv. in Math. 225:1 (2010) 1–32 doi

For the weak mixed distributive law (monad and comonad) version see

• Ross Street, Weak distributive laws, Theory and Appl. of Categ. 22 (2009) 313–320 [tac:22-12]

2-categorical context in the sense of formal theory of monads is also exposed in

• Gabriella Böhm, Stephen Lack, Ross Street, On the 2-categories of weak distributive laws, Comm. Alg. 39:12 (2011) 4567–4583 doi

• Daniela Petrisan, Ralph Sarkis. Semialgebras and weak distributive laws, Proceedings 37th Conference on

  Mathematical Foundations of Programming Semantics, EPTCS 351 (2021) 218–241. (doi:10.4204/EPTCS.351.14)

An application of weak distributive laws to explain Street’s weak wreath products (comparable to the treatment of wreaths in bicategories) and also related bilinear factorization structures

• Gabriella Böhm, On the iteration of weak wreath products, Theory and Appl. of Categories 26:2 (2012) 30–59 arXiv:1110.0652
• Gabriella Böhm, José Gómez-Torrecillas, Bilinear factorization of algebras, Bull. Belg. Math. Soc. Simon Stevin 20(2): 221-.244 (may 2013) doi