This paper studies a category X with an endofunctor T : X \to X. A T-algebra is given by a morphism Tx \to x in X. We examine the related questions of when T freely generates a triple (or monad) on X; when an object x in X freely generates a T-algebra; and when the category of T-algebras has coequalizers and other colimits. The paper defines a category of ``T-horns'' which effectively contains X as well as all T-algebras. It is assume that Xs is cocomplete and has a factorization system (E,M) satisfying reasonable properties. An ordinal-indexed sequence of T-horns is then defined which provides successive approximations to a free T-algebra generated by an object x in X, as well as approximations to coequalizers and other colimits for the category of T-algebras. Using the notions of an M-cone and a separated T-horn it is shown that if X is M-well-powered, then the ordinal sequence stabilizes at the desired free algebra or coequalizer or other colimit whenever they exist. This paper is a successor to a paper written by the first author in 1970 that showed that T generates a free triple when every x in X generates a free T-algebra. We also consider colimits in triple algebras and give some examples of functors T for which no x in X generates a free T-algebra.
Keywords: Free triples, coequalizers, T-horns, ordinal sequences
2010 MSC: 18A30, 18A32, 18C15
Theory and Applications of Categories, Vol. 34, 2019, No. 23, pp 662-683.