## Induction, Coinduction, and Adjoints (2002)

Citations: | 3 - 2 self |

### BibTeX

@MISC{Cockett02induction,coinduction,,

author = {Robin Cockett and Luigi Santocanale},

title = {Induction, Coinduction, and Adjoints},

year = {2002}

}

### OpenURL

### Abstract

We investigate the reasons for which the existence of certain right adjoints implies the existence of some nal coalgebras, and vice-versa. In particular we prove and discuss the following theorem which has been partially available in the literature: let F a G be a pair of adjoint functors, and suppose that an initial algebra F (X) of the functor H(Y ) = X + F (Y ) exists; then a right adjoint G(X) to F (X) exists if and only if a nal coalgebra G(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.