nLab (bo, ff) factorization system




There is an orthogonal factorization system on the category StrCat, 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 &\overset{u}{\longrightarrow}& C \\ \mathllap{{}^{e}}\big\downarrow &&\big\downarrow \mathrlap{{}^{j}} \\ B &\underset{v}\longrightarrow& D }

is a commutative diagram with ee bijective on objects and jj fully faithful, then there is a unique functor h:BCh \colon B\to C such that he=uh e = u and jh=vj h = v. The object through which ff factors is called the full image of ff.

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.


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.

Last revised on August 26, 2022 at 09:02:55. See the history of this page for a list of all contributions to it.