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

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 -. As far as the distributive laws are concerned, the equivalence between this structures and monads objects in the 2-category 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 was analysed also within the context of a 2-adjunction. This…
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Taxonomy
TopicsHomotopy and Cohomology in Algebraic Topology · Algebraic structures and combinatorial models · Advanced Topics in Algebra
