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Pré-Publication, Document De Travail Année : 2021

Semialgebras and Weak Distributive Laws

Résumé

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.
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Dates et versions

hal-03261093 , version 1 (18-06-2021)
hal-03261093 , version 2 (08-09-2021)

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Daniela Petrişan, Ralph Sarkis. Semialgebras and Weak Distributive Laws. 2021. ⟨hal-03261093v2⟩
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