category theory

# Factorization categories

## Idea

The factorization category (also called the interval category) $\mathrm{Fact}\left(f\right)$ of a morphism $f$ in a category $C$ is a way of organizing its binary factorizations $f=p\circ q$ into a category.

## Definition

The objects of $\mathrm{Fact}\left(f\right)$ are factorizations

(1)$\begin{array}{ccccc}X& & \stackrel{f}{\to }& & Y\\ & {}_{{p}_{1}}↘& & {↗}_{{p}_{2}}\\ & & D\end{array}$\begin{matrix} X &&\stackrel{f}{\to}&& Y \\ & {}_{p_1} \searrow && \nearrow_{p_2} \\ && D \end{matrix}

so that $f={p}_{2}{p}_{1}$, and a morphism from $\left({p}_{1},D,{p}_{2}\right)$ to $\left({q}_{1},E,{q}_{2}\right)$ is a morphism $h:D\to E$ making everything in sight commute. There’s an obvious projection functor

(2)${P}_{f}:\mathrm{Fact}\left(f\right)\to C$P_f \colon Fact(f) \to C

which maps $\left({p}_{1},D,{p}_{2}\right)$ to $D$ and $h:\left({p}_{1},D,{p}_{2}\right)\to \left({q}_{1},E,{q}_{2}\right)$ to $h$.

### As iterated comma categories

In terms of slice categories, a morphism $f:A\to B$ can be viewed as

1. an object in $C/B$
2. or an object in $A/C$

Now, taking over/under slices again yields only one new thing; it is easy to see that

• $\left(C/B\right)/f\cong C/A$, and
• $f/\left(A/C\right)\cong B/C$

the cool fact is that the two other options yield $\mathrm{Fact}\left(f\right)$

###### Lemma

$\mathrm{Fact}\left(f\right)\cong f/\left(C/B\right)\cong \left(A/C\right)/f$, and the following diagram commutes

(3)$\begin{array}{ccccc}\left(A/C\right)/f& \stackrel{\cong }{\to }& \mathrm{Fact}\left(f\right)& \stackrel{\cong }{←}& f/\left(C/B\right)\\ {\pi }_{f}^{A}↓& & {P}_{f}↓& & {\pi }_{f}^{B}\\ A/C& \underset{{\pi }_{A}}{\to }& C& \underset{{\pi }_{B}}{←}& C/B\end{array}$\array{ (A / C)/f &\stackrel{\cong}{\to}& Fact(f) & \stackrel{\cong}{\leftarrow} & f/(C/B) \\ \pi^A_f \downarrow && P_f \downarrow && \pi^B_f \\ A / C & \underset{\pi_A}{\to} & C & \underset{\pi_B}{\leftarrow} & C/B }

Eduardo Pareja-Tobes?: This should follow from properties of comma objects; I could add here the proof from Lawvere-Menni paper below, but I think it would be better to have more conceptual proof

## Properties

### Characterization in terms of initial and terminal objects

There is a fairly simple characterization of the categories arising as factorization categories of some $f$ in a category $C$. First of all, note that $\mathrm{Fact}\left(f\right)$ always has

• an initial object $f=f\circ \mathrm{id}$
• a terminal object $f=\mathrm{id}\circ f$

conversely, for any category $D$ with initial and terminal objects $0,1$ denoting the unique morphism $!:0\to 1$ we have that

(4)${\pi }_{!}:\mathrm{Fact}\left(!\right)\to D$\pi_! \colon Fact(!) \to D

is an equivalence. We get then

a category is equivalent to some $\mathrm{Fact}\left(f\right)$ iff it has initial and terminal objects

### Factorization categories vs the category of factorizations

We can view $\mathrm{Fact}\left(f\right)$ as a full reflective subcategory of the over-category $f/\mathrm{tw}\left(C\right)$; here $f$ is viewed as an object of the category of factorizations $\mathrm{tw}\left(C\right)$ of its ambient category $C$. There’s a functor

(5)${U}_{f}:\mathrm{Fact}\left(f\right)\to \mathrm{tw}\left(C\right)$U_f \colon Fact(f) \to tw(C)

which on objects is

(6)${U}_{f}\left({p}_{1},{p}_{2}\right)=\begin{array}{ccc}X& \stackrel{{1}_{X}}{←}& X\\ {p}_{1}↓& & ↓f\\ D& \underset{{p}_{2}}{\to }& Y\end{array}$U_f(p_1, p_2) = \begin{matrix} X & \overset{1_X}{\leftarrow} & X \\ p_1\downarrow & & \downarrow f \\ D & \underset{p_2}{\to} & Y \end{matrix}

and on arrows $U\left(h\right)=\left(h,\mathrm{id}\right)$.

This functor has a left adjoint

(7)${F}_{f}:\mathrm{tw}\left(C\right)/f\to \mathrm{Fact}\left(f\right)$F_f \colon tw(C)/f \to Fact(f)
• ${F}_{f}$ on objects:

(8)${F}_{f}\left(\phantom{\rule{thinmathspace}{0ex}}\begin{array}{ccc}A& \stackrel{h}{←}& X\\ g↓& & ↓f\\ D& \underset{q}{\to }& Y\end{array}\phantom{\rule{thinmathspace}{0ex}}\right)\phantom{\rule{thinmathspace}{0ex}}=\phantom{\rule{1em}{0ex}}\begin{array}{ccccc}X& & \stackrel{f}{\to }& & Y\\ & {}_{\mathrm{gh}}↘& & {↗}_{q}\\ & & D\end{array}$F_f\left(\, \begin{matrix} A & \overset{h}{\leftarrow} & X \\ g \downarrow & & \downarrow f \\ D & \underset{q}{\to} & Y \end{matrix} \, \right) \, = \quad \begin{matrix} X &&\stackrel{f}{\to}&& Y \\ & {}_{gh} \searrow && \nearrow_{q} \\ && D \end{matrix}
• ${F}_{f}$ on arrows: picks the morphism which goes between $D$ and $D\prime$.

It is immediate to check that ${F}_{f}\circ {U}_{f}={1}_{\mathrm{Fact}\left(f\right)}$.

## References

• Bill Lawvere, Matias Menni? The Hopf algebra of Möbius intervals Theory and applications of categories 2010
• B. Klin, Vladimiro Sassone, P. Sobocinski, Labels from reductions: Towards a general theory, Algebra and coalgebra in computer science: first international conference, CALCO 2005

category: computer science

Revised on September 8, 2012 13:48:15 by Tim Porter (95.147.236.244)