category theory

# Contents

## 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 $j e$, where $e$ is bijective on objects and $j$ fully faithful; and if

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

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

In fact, this can be generalized to a square commuting up to invertible natural transformation, in which case one still concludes that $h e = u$ but that $j h \cong v$, 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.

Revised on November 21, 2011 00:19:34 by Urs Schreiber (89.204.154.71)