Factorization systems

A factorization system on a category is, loosely speaking, two wide subcategories \(L\) and \(R\), both containing all isomorphisms, such that every morphism factors as an $L$-morphism followed by an $R$-morphism, subject to some lifting or uniqueness condition. Some care in reading the literature is required since the term “factorization system” without qualification is not used consistently by all authors. For many, it mean orthogonal factorization system (OFS); for others, it means something weaker, such as a weak factorization system (WFS).

The ur-example of an orthogonal factorization system is the surjective-injective factorization in \(\mathsf{Set}\) or, more generally, epi-mono factorization in a topos.