nLab
factorization system in a 2-category

Definition

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:

1. 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

2. For any $e:a\to b$ in $E$ and $m:x\to y$ in $M$, the following square

   $$\begin{array}{ccc}K(b,x)& \to & K(b,y)\\ \downarrow & \cong & \downarrow \\ K(a,x)& \to & K(a,y)\end{array}$$\array{ K(b,x) & \to & K(b,y) \\ \downarrow & \cong & \downarrow \\ K(a,x) & \to & K(a,y)}

   (which commutes up to isomorphism) is a 2-pullback in $\mathrm{Cat}$.

This second property is a “2-categorical orthogonality.” In particular, it implies that any square

$$\begin{array}{ccc}a& \to & x\\ {}^e\downarrow & \cong & {\downarrow }^m\\ b& \to & y\end{array}$$\array{a & \to & x \\ ^e\downarrow & \cong & \downarrow^m \\ b & \to & y}

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.

Examples

• The following are all factorization systems in the 2-category $\mathrm{Cat}$. 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 $e:a\to b$ 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.

  • The (geometric surjection, embedding) factorization system.

  • $E=$ hyperconnected geometric morphisms, $M=$ localic geometric morphisms.

Cat-enriched factorization systems

If instead $K$ is a strict 2-category and we require that

1. Every 1-morphism in $K$ is equal to a composite of a morphism in $E$ and a morphism in $M$, and

2. The above square (which commutes strictly when $K$ is a strict 2-category) is a strict 2-pullback (i.e. a $\mathrm{Cat}$-enriched pullback).

then we obtain the notion of a $\mathrm{Cat}$-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 $\mathrm{Cat}$ 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.

Related concepts

• factorization system

  • weak factorization system

  • orthogonal factorization system

• factorization system in a 2-category

• factorization system in an (∞,1)-category

  • orthogonal factorization system in an (∞,1)-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.

References

For instance

• Mathieu Dupont, Enrico Vitale, Proper factorization systems in 2-categories (pdf)