On coalgebras over algebras

TL;DR

This paper generalizes Barr's characterization of final coalgebras to the Eilenberg-Moore category for a monad, introducing commuting pairs of endofunctors and relating final coalgebras to free algebras.

Contribution

It extends Barr's result to monad-based categories and introduces commuting pairs of endofunctors, linking final coalgebras to initial algebras via free algebra constructions.

Findings

Final coalgebras can be characterized in Eilenberg-Moore categories.

Introduction of commuting pairs of endofunctors with respect to a monad.

Final coalgebra of one endofunctor can be obtained as a free algebra over the initial algebra of another.

Abstract

We extend Barr's well-known characterization of the final coalgebra of a $Set$-endofunctor as the completion of its initial algebra to the Eilenberg-Moore category of algebras for a $Set$-monad $\mathbf{M}$ for functors arising as liftings. As an application we introduce the notion of commuting pair of endofunctors with respect to the monad $\mathbf{M}$ and show that under reasonable assumptions, the final coalgebra of one of the endofunctors involved can be obtained as the free algebra generated by the initial algebra of the other endofunctor.