definition

A factorization system on a category \(\cal X\) is a pair of subcategories \((\cal L, R)\) (the left class and the right class) such that

1. Both \(\cal L\) and \(\cal R\) contain all isomorphisms,
2. “\(\cal X =\cal L ;\cal R\)”: Every morphism in \(\cal X\) factors as a morphism of \(\cal L\) followed by a morphism in \(\cal R\) and this factorization is unique up to unique isomorphism: