In a (weak) 2-category, the appropriate notion of an orthogonal factorization system is suitably weakened up to isomorphism. Specifically, a factorization system in a 2-category consists of two classes of 1-morphisms in such that:
Every 1-morphism in is isomorphic to a composite where and , and
For any in and in , the following square
(which commutes up to isomorphism) is a 2-pullback in .
This second property is a “2-categorical orthogonality.” In particular, it implies that any square
which commutes up to specified isomorphism, where and , has a diagonal filler making both triangles commute up to isomorphisms that are coherent with the given one. It also implies an additional factorization property for 2-cells.
The following are all factorization systems in the 2-category . Many of them have analogues in more general 2-categories.
essentially surjective functors, fully faithful functors. This is the “ur-example,” and it generalizes to enriched category theory, internal category theory, etc. See (eso, fully faithful) factorization system.
functors such that every object of is a retract of an object in the image of , and fully faithful functors whose image is closed under retracts.
The 2-category Topos admits several interesting factorization systems.
If instead is a strict 2-category and we require that
Every 1-morphism in is equal to a composite of a morphism in and a morphism in , and
The above square (which commutes strictly when is a strict 2-category) is a strict 2-pullback (i.e. a -enriched pullback).
then we obtain the notion of a -enriched, or strict 2-categorical, factorization system.
It is important to note that in general, the strict and weak notions of 2-categorical factorization system are incomparable; neither is a special case of the other. For example, on there is a weak 2-categorical factorization system where essentially surjective functors and fully faithful functors, and a strict 2-categorical factorization system where bijective on objects functors and fully faithful functors.
factorization system in a 2-category