Semialgebras and Weak Distributive Laws
arXiv:2106.13489 · doi:10.4204/EPTCS.351.14
Abstract
Motivated by recent work on weak distributive laws and their applications to coalgebraic semantics, we investigate the algebraic nature of semialgebras for a monad. These are algebras for the underlying functor of the monad subject to the associativity axiom alone-the unit axiom from the definition of an Eilenberg-Moore algebras is dropped. We prove that if the underlying category has coproducts, then semialgebras for a monad M are in fact the Eilenberg-Moore algebras for a suitable monad structure on the functor id + M , which we call the semifree monad M^s. We also provide concrete algebraic presentations for semialgebras for the maybe monad, the semigroup monad and the finite distribution monad. A second contribution is characterizing the weak distributive laws of the form M T => T M as strong distributive laws M^s T => T M^s subject to an additional condition.
In Proceedings MFPS 2021, arXiv:2112.13746