On Corecursive Algebras for Functors Preserving Coproducts

TL;DR

This paper characterizes corecursive algebras for certain functors on hyper-extensive categories, showing their structure as coproducts of final coalgebras and free algebras, and identifies conditions under which functors are cia functors.

Contribution

It provides a structural description of free corecursive algebras for functors preserving coproducts and characterizes when such functors are cia functors, especially for finitary set functors.

Findings

Free corecursive algebra as coproduct of final coalgebra and free algebra

H is a cia functor if and only if it has a specific form for finitary set functors

All functors of the form H(-)+Y are cia functors

Abstract

For an endofunctor $H$ on a hyper-extensive category preserving countable coproducts we describe the free corecursive algebra on $Y$ as the coproduct of the final coalgebra for $H$ and the free $H$-algebra on $Y$. As a consequence, we derive that $H$ is a cia functor, i.e., its corecursive algebras are precisely the cias (completely iterative algebras). Also all functors $H(-) + Y$ are then cia functors. For finitary set functors we prove that, conversely, if $H$ is a cia functor, then it has the form $H = W \times (-) + Y$ for some sets $W$ and $Y$.