nLab
(bo, ff) factorization system

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 ff decomposes as a composition of the form jej e, where ee is bijective on objects and jj fully faithful; and if

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

is a commutative diagram with ee bijective on objects and ss essentially surjective, then there is a unique functor h:BCh \colon B\to C such that he=uh e = u and jh=vj h = v.

In fact, this can be generalized to a square commuting up to invertible natural transformation, in which case one still concludes that he=uh e = u but that jhvj h \cong v, with the isomorphism composing with ee 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)