factorization system over a subcategory
k-ary factorization system, ternary factorization system
factorization system in a 2-category
factorization system in an (∞,1)-category
An epi-mono factorization system is an orthogonal factorization system in which the left class is the class of epimorphisms and the right class is the class of monomorphisms. Such a factorization system exists on any (elementary) topos, and indeed on any pretopos. It provides the factorization through the image of any morphism.
Note that any category which admits an epi-mono factorization system is necessarily balanced. This excludes many commonly occurring categories. More common are (strong epi, mono) and (epi, strong mono) factorization systems; the former exists in any regular category and the latter in any quasitopos, as well as in other categories such as Top.
The epi-mono factorization system in a topos is the special case of the n-connected/n-truncated factorization system in an (∞,1)-topos for the case that $(n = -1)$ and restricted to 0-truncated objects.
(epi, mono) factorization system