# nLab factorization system in a 2-category

### Context

#### 2-Category theory

2-category theory

# Contents

## 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 $\left(E,M\right)$ 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

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

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

## 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.

## References

For instance

Revised on November 21, 2011 00:15:46 by Urs Schreiber (89.204.154.71)