11 - Sufficient Conditions for Initial Algebras and Terminal Coalgebras

Summary

This chapter presents a number of sufficient conditions to guarantee that an endofunctor has an initial algebra or a terminal coalgebra. We generalize Kawahara and Mori’s notion of a bounded set functor and prove that for a cocomplete and co-well-powered category with a terminal object, every endofunctor bounded by a generating set has a terminal coalgebra. We use this to show that every accessible endofunctor on a locally presentable category has an initial algebra and a terminal coalgebra. We introduce pre-accessible functors and prove that on a cocomplete and co-well-powered category, the initial-algebra chain of a pre-accessible functor converges, and so the initial algebra exists. If the base category is locally presentable and the functor preserves monomorphisms, then the terminal coalgebra exists.