Factorization systems and saturated classes

Factorization systems turn a class of morphisms into a canonical two-step decomposition of every map. Orthogonality supplies the uniqueness of this decomposition and gives reflective subcategories in every slice. These formal properties underlie the effective epimorphism–monomorphism and connected–truncated factorizations used throughout the text.

In a presentable category, factorization systems can be generated from classes of morphisms. A saturated class of small generation is the left class of a factorization system, while a strongly saturated class of small generation is the kernel of an accessible localization. The distinction between these two closure operations is essential later: modalities arise from saturated classes stable under base change, whereas congruences arise from strongly saturated classes stable under base change. This appendix collects the required existence and closure results.