nLab (epi, mono) factorization system

Definition

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.

Properties

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.

Related concepts

• (epi, mono) factorization system

• (eso+full, faithful) factorization system

• n-connected/n-truncated factorization system.

References

For instance:

• Francis Borceux, vol 1, section 4.4. of: Handbook of Categorical Algebra, Cambridge University Press (1994)