Induction, Coinduction, and Adjoints

Abstract

We investigate the reasons for which the existence of certain right adjoints implies the existence of some final coalgebras, and vice-versa. In particular we prove and discuss the following theorem which has been partially available in the literature: let F ⊣ G be a pair of adjoint functors, and suppose that an initial algebra F̂(X) of the functor H(Y) = X + F(Y) exists; then a right adjoint G̃(X) to F̂(X) exists if and only if a final coalgebra Ǧ(X) of the functor K(Y) = X × G(Y) exists. Motivated by the problem of understanding the structures that arise from initial algebras, we show the following: if F is a left adjoint with a certain commutativity property, then an initial algebra of H(Y) = X + F(Y) generates a subcategory of functors with inductive types where the functorial composition is constrained to be a Cartesian product.