It is well known that for any monad, the associated Kleisli category is embedded in the category of Eilenberg-Moore algebras as the free ones. We discovered some interesting examples in which this embedding is reflective; that is, it has a left adjoint. To understand this phenomenon we introduce and study a class of monads arising from factorization systems, and thereby termed factorization monads. For them we show that under some simple conditions on the factorization system the free algebras are a full reflective subcategory of the algebras. We provide various examples of this situation of a combinatorial nature.
Keywords: factorization systems, monads, Kleisli categories, Schanuel topos, Joyal species, combinatorial structures, power series
2000 MSC: 18A25, 18A40, 18C20, 05A10
Theory and Applications of Categories,
Vol. 15, CT2004,
No. 2, pp 40-65.