Infinite trees and completely iterative theories: a coalgebraic view

Infinite trees form a free completely iterative theory over any given signature—this fact, proved by Elgot, Bloom and Tindell, turns out to be a special case of a much more general categorical result exhibited in the present paper. We prove that whenever an endofunctor H of a category has final coalgebras for all functors H( _ )+X, then those coalgebras, TX, form a monad. This monad is completely iterative, i.e., every guarded system of recursive equations has a unique solution. And it is a free completely iterative monad on H. The special case of polynomial endofunctors of the category is the above mentioned theory, or monad, of infinite trees.

This procedure can be generalized to monoidal categories satisfying a mild side condition: if, for an object H, the endofunctor H_+I has a final coalgebra, T, then T is a monoid. This specializes to the above case for the monoidal category of all endofunctors.