 
Summary: INFINITE TREES AND COMPLETELY ITERATIVE THEORIES: A COALGEBRAIC
VIEW
PETER ACZEL, JI
R I AD
AMEK, STEFAN MILIUS, AND JI
R I VELEBIL
Abstract. Innite 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 innite 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, and it also yields Kleene algebras for the category
of all formal languages.
1. Introduction
