Monad and Comonad Objects through 2-adjunctions of the type Adj-Mnd

TL;DR

This paper explores the relationships between distributive laws, monads, and comonads using 2-adjunctions, providing clearer proofs and naturality in the equivalences within category theory.

Contribution

It recasts classical monad and comonad theories through 2-adjunctions, establishing new equivalences and clarifying existing ones with enhanced naturality.

Findings

Equivalence between distributive laws and monad objects in a 2-category.

Correspondence of comonad objects to comonad liftings on Eilenberg-Moore algebras.

A new theorem relating mixed distributive laws with Eilenberg-Moore liftings.

Abstract

In this article, the author analyses distributive and mixed distributive laws and some of their equivalences through the use of 2-adjunctions of the type $\Adj$-$\Mnd$. As far as the distributive laws are concerned, the equivalence between this structures and monads objects in the 2-category $\Adj_{\R}(Cat)$ is analysed, where these monad objects will correspond to liftings to the category of algebras of Eilenberg-Moore. Second, the equivalence between these structures and a pair consisting of a Eilenberg-Moore lifting and a Kleisli extension is analysed too, according to E. Manes and P. Mulry (2010), where the author was able to recast the theorem with an additional naturality on the involved monads. On the other hand, the equivalence between mixed distributive laws and comonad objects in the same 2-category $\Adj_{\R}(Cat)$ was analysed also within the context of a 2-adjunction. This comonad object will correspond to a comonad lifting structure on the category of algebras of Eilenberg-Moore. Finally, a similar theorem relating mixed distributive laws with a pair of Eilenberg-Moore liftings on algebras and coalgebras is proved, with a naturality for the equivalence on the monad and comonad involved. The same stated objective in a previous installment, by the author and company, is followed. That is to say, using 2-adjunctions to analyse classical monad theory in order to provide clarity in the proofs and naturality on the equivalences.