Functorial factorization, well-pointedness and separability
Functorial factorization systems are one of the main topics in category theory. In the classical setting they are presented by two classes \({\mathcal E}\) and \({\mathcal M}\) of morphisms with appropriate closure properties that ensure a unique factorization \(f=m_f\cdot e_f\), with \(e_f\in {\mathcal E}\), \(m_f\in {\mathcal M}\), of every morphism \(f\) in the category. Weak factorization systems were introduced and studied by \textit{M. Korostenski} and \textit{W. Tholen} [J. Pure Appl. Algebra 85, No.~1, 57-72 (1993; Zbl 0778.18001)]. In the present paper the authors study the relation between weak factorization systems and well-pointed endofunctors (here well-pointedness is understood in the sense of \textit{G. M. Kelly} [Bull. Aust. Math. Soc. 22, 1-83 (1980; Zbl 0437.18004)]). The final part of the paper is dedicated to the factorization theory derived from taking separable and purely inseparable morphisms with respect to a pointed endofunctor.
scientific article; zbMATH DE number 1371460
scientific article