Final Coalgebras And a Solution Theorem for Arbitrary Endofunctors

Every endofunctor F of Set has an initial algebra and a final coalgebra, but they are classes in general. Consequently, the endofunctor F∞ of the category of classes that F induces generates a completely iterative monad T . And solutions of arbitrary guarded systems of iterative equations w.r.t. F exist, and can be found in naturally defined subsets of the classes TY . More generally, starting from any category K, we can form a free cocompletion K∞ of K under small-filtered colimits (e.g., Set∞ is the category of classes), and we give sufficient conditions to obtain analogous results for arbitrary endofunctors of K.