Geodesic
Natural weak factorization systems
Archivum mathematicum, Tome 42 (2006) no. 4, pp. 397-408.

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In order to facilitate a natural choice for morphisms created by the (left or right) lifting property as used in the definition of weak factorization systems, the notion of natural weak factorization system in the category $\mathcal {K}$ is introduced, as a pair (comonad, monad) over $\mathcal {K}^{\bf 2}$. The link with existing notions in terms of morphism classes is given via the respective Eilenberg–Moore categories.
Classification : 18C15 
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     author = {Grandis, Marco and Tholen, Walter},
     title = {Natural weak factorization systems},
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     url = {http://geodesic.mathdoc.fr/item/ARM_2006__42_4_a4/}
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Grandis, Marco; Tholen, Walter. Natural weak factorization systems. Archivum mathematicum, Tome 42 (2006) no. 4, pp. 397-408. http://geodesic.mathdoc.fr/item/ARM_2006__42_4_a4/