Abstract
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.
Joint work by
Michael Barr, John Kennison, and R. Raphael
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.
Published 2019-08-16.
http://www.tac.mta.ca/tac/volumes/34/23/34-23.pdf