factorization system over a subcategory
k-ary factorization system, ternary factorization system
factorization system in a 2-category
factorization system in an (∞,1)-category
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 $K$ consists of two classes $(E,M)$ of 1-morphisms in $K$ such that:
Every 1-morphism $f:x\to y$ in $K$ is isomorphic to a composite $e\circ m$ where $e\in E$ and $m\in M$, and
For any $e:a\to b$ in $E$ and $m:x\to y$ in $M$, the following square
(which commutes up to isomorphism) is a 2-pullback in $Cat$.
This second property is a “2-categorical orthogonality.” In particular, it implies that any square
which commutes up to specified isomorphism, where $e\in E$ and $m\in M$, has a diagonal filler $b\to x$ 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 $Cat$. Many of them have analogues in more general 2-categories.
$E=$ essentially surjective functors, $M=$ 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.
$E=$ functors $e\colon a\to b$ such that every object of $b$ is a retract of an object in the image of $a$, and $M=$ fully faithful functors whose image is closed under retracts.
$E=$ essentially surjective and full functors, $M=$ faithful functors. See (eso+full, faithful) factorization system.
$E=$ (possibly transfinite) composites of localizations, $M=$ conservative functors.
The 2-category Topos admits several interesting factorization systems.
If instead $K$ is a strict 2-category and we require that
Every 1-morphism in $K$ is equal to a composite of a morphism in $E$ and a morphism in $M$, and
The above square (which commutes strictly when $K$ is a strict 2-category) is a strict 2-pullback (i.e. a $Cat$-enriched pullback).
then we obtain the notion of a $Cat$-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 $Cat$ there is a weak 2-categorical factorization system where $E=$ essentially surjective functors and $M=$ fully faithful functors, and a strict 2-categorical factorization system where $E=$ bijective on objects functors and $M=$ fully faithful functors.
factorization system in a 2-category
Factorization systems in a 2-category play an important role in the construction of a proarrow equipment out of codiscrete cofibrations.
Combining the (eso,ff) and (eso+full, faithful) factorization systems into a ternary factorization system has connections with the theory of stuff, structure, property.
For instance