Reflective Kleisli subcategories of the category of Eilenberg-Moore algebras for factorization monads

Marcelo Fiore and Matias Menni

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.

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