Weak Reflections and Weak Factorization Systems

Abstract

We describe a one-to-one correspondence between saturated weak factorization systems and weak reflections in categories \(\mathcal{C}\) with finite products. This actually extends to an adjunction between the category of natural weak factorization systems on \(\mathcal{C}\) (in the sense of Grandis and Tholen, Arch Math 42:397–408, 2006, and Garner, arXiv preprint, 2007) and the category of monads on \(\mathcal{C}\). Explicit comparisons are made with the parallel result of Cassidy et al. (J Aust Math Soc 38:287–329, 1985), linking factorization systems and reflective subcategories.