nLab
(bo, ff) factorization system

Definition

There is an orthogonal factorization system on the category Cat, whose left class is the class of bijective-on-objects functors, or “bo functors” and whose right class is the class of full and faithful functors, or “ff functors”.

This means that each functor $f$ decomposes as a composition of the form $je$, where $e$ is bijective on objects and $j$ fully faithful; and if

$$\begin{array}{ccc}A& \stackrel{u}{⟶}& C\\ e\downarrow & & \downarrow j\\ B& \stackrel{v}{⟶}& D\end{array}$$\array{ A &\stackrel{u}\longrightarrow& C \\ e\downarrow &&\downarrow j \\ B &\stackrel{v}\longrightarrow& D }

is a commutative diagram with $e$ bijective on objects and $s$ essentially surjective, then there is a unique functor $h:B\to C$ such that $he=u$ and $jh=v$.

In fact, this can be generalized to a square commuting up to invertible natural transformation, in which case one still concludes that $he=u$ but that $jh\cong v$, with the isomorphism composing with $e$ to give the original isomorphism. This means that this is an enhanced factorization system?.

Properties

This factorization system can be constructed using generalized kernels.

For essentially surjective functors, one can relax both the commuting and the uniqueness to obtain a factorization system in a 2-category.