On coalgebras over algebras

A. Balan, A. Kurz
2011 Theoretical Computer Science  
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
more » ... ebra. for the lifted functor, with respect to the usual ultrametric inherited from the final sequence. Moreover, the corresponding topology is compatible with the M-algebra structure of both objects involved, in the sense that the algebra structure maps are continuous. To provide examples, we need to understand better the initial algebra of the lifted functor. This is the purpose of the second part of the paper, where the special case of an initial algebra that is free (as an M-algebra) is exhibited. Namely, for two endofunctors H, T and a monad M on Set, we call (T , H) an M-commuting pair if there is a natural isomorphism HM ∼ = MT , where M is the functor part of the monad. This notion is motivated by the fact that if both the algebra lift of H and the Kleisli lift of T exist, then mild requirements ensure that  H, the algebra lifted functor of H, is equivalent with the extension of T to Alg(M) if and only if they form a commuting pair (T , H). If this is the case, then one can recover the initial algebra for the lifted endofunctor  H as the free M-algebra built on the initial T -algebra. Consequently, the final  H-coalgebra can be realized as the completion of a free M-algebra. An earlier version of this paper appeared as [6] . In the present article, Section 2.4 is extended with a detailed analysis of the completion result. Section 2.5, devoted only to examples, is new; in Appendix A, we start with a monad M and a functor H having a Kleisli liftingĤ and show how to extendĤ from free algebras to all algebras, in the form of a left Kan extension. Finally, due to space limitations, an example of constructing a commuting pair is detailed in Appendix B.
doi:10.1016/j.tcs.2011.03.021 fatcat:7k5ih7fj6remln7z4fyln2jn7i