Infinite trees and completely iterative theories: a coalgebraic view

In#nite 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 #nal 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 Set is the above mentioned theory, or monad, of in#nite 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 #nal coalgebra, T , then T is a monoid. This specializes to the above case for the monoidal category of all endofunctors. c © 2002 Elsevier Science B.V. All rights reserved.