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Reflective Kleisli subcategories of the category of Eilenberg-Moore algebras for factorization monads
Theory and applications of categories, CT2004, Tome 15 (2005), pp. 40-65 Cet article a éte moissonné depuis la source Theory and Applications of Categories website

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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.

Classification : 18A25, 18A40, 18C20, 05A10
Keywords: factorization systems, monads, Kleisli categories, Schanuel topos, Joyal species, combinatorial structures, power series 
@article{TAC_2005_15_a1,
     author = {Marcelo Fiore and Matias Menni},
     title = {Reflective {Kleisli} subcategories of the category {of
Eilenberg-Moore} algebras for factorization monads},
     journal = {Theory and applications of categories},
     pages = {40--65},
     year = {2005},
     volume = {15},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/TAC_2005_15_a1/}
}
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Marcelo Fiore; Matias Menni. Reflective Kleisli subcategories of the category of
Eilenberg-Moore algebras for factorization monads. Theory and applications of categories, CT2004, Tome 15 (2005), pp. 40-65. http://geodesic.mathdoc.fr/item/TAC_2005_15_a1/