On Coalgebras over Algebras
We extend Barr’s well-known characterization of the final coalgebra of a Set-endofunctor H as the completion of its initial algebra to the Eilenberg–Moore category Alg(M) of algebras associated to a Set-monad M, if H can be lifted to Alg(M). As further analysis, we introduce the notion of a commuting pair of endofunctors (T,H) with respect to a monad M and show that under reasonable assumptions, the final H-coalgebra can be obtained as the completion of the free M-algebra on the initial T-algebra.
A. Balan and A. Kurz, “On coalgebras over algebras,” Theoretical Computer Science, vol. 412, no. 38, pp. 4989–5005, Sep. 2011. DOI: 10.1016/j.tcs.2011.03.021