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Coequalizers and free triples, II

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Michael Barr, John Kennison, and R. Raphael

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.

Published 2019-08-16.

http://www.tac.mta.ca/tac/volumes/34/23/34-23.pdf

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