Semialgebras and Weak Distributive Laws

TL;DR

This paper explores the algebraic structure of semialgebras for monads, showing their relation to Eilenberg-Moore algebras via a new semifree monad, and characterizes weak distributive laws in algebraic terms.

Contribution

It introduces the semifree monad M^s to characterize semialgebras and provides algebraic presentations for specific monads, advancing the understanding of weak distributive laws.

Findings

Semialgebras for a monad are Eilenberg-Moore algebras for a new monad M^s.

Concrete algebraic presentations for the maybe, semigroup, and finite distribution monads.

Characterization of weak distributive laws as strong laws under certain conditions.

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.