nLab factorization category

Idea

The factorization category (also called the interval category) $Fact(f)$ 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 $Fact(f)$ are factorizations

$$\begin{matrix} X &&\stackrel{f}{\to}&& Y \\ & {}_{p_1} \searrow && \nearrow_{p_2} \\ && D \end{matrix}$$

so that $f = p_2 p_1$, and a morphism from $(p_1, D, p_2)$ to $(q_1, E, q_2)$ is a morphism $h \colon D \to E$ making everything in sight commute. There’s an obvious projection functor

$$P_f \colon Fact(f) \to C$$

which maps $(p_1, D, p_2)$ to $D$ and $h\colon (p_1, D, p_2) \to (q_1, E, q_2)$ 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

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

the cool fact is that the two other options yield $Fact(f)$

Lemma

$Fact(f) \cong f/(C/B) \cong (A / C)/f$, and the following diagram commutes

$$\array{ (A / C)/f &\stackrel{\cong}{\to}& Fact(f) & \stackrel{\cong}{\leftarrow} & f/(C/B) \\ \pi^A_f \downarrow && P_f \downarrow && \pi^B_f \\ A / C & \underset{\pi_A}{\to} & C & \underset{\pi_B}{\leftarrow} & C/B }$$

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 $Fact(f)$ always has

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

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

$$\pi_! \colon Fact(!) \to D$$

is an equivalence. We get then

a category is equivalent to some $Fact(f)$ iff it has initial and terminal objects

Factorization categories vs the category of factorizations

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

$$U_f \colon Fact(f) \to tw(C) / f$$

which on objects is

$$U_f(p_1, p_2) = \begin{matrix} X & \overset{1_X}{\leftarrow} & X \\ p_1\downarrow & & \downarrow f \\ D & \underset{p_2}{\to} & Y \end{matrix}$$

and on arrows $U(h) = (h, id)$.

This functor has a left adjoint

$$F_f \colon tw(C)/f \to Fact(f)$$

• $F_f$ on objects:

  $$F_f\left(\, \begin{matrix} A & \overset{h}{\leftarrow} & X \\ g \downarrow & & \downarrow f \\ D & \underset{q}{\to} & Y \end{matrix} \, \right) \, = \quad \begin{matrix} X &&\stackrel{f}{\to}&& Y \\ & {}_{gh} \searrow && \nearrow_{q} \\ && D \end{matrix}$$

• $F_f$ on arrows: picks the morphism which goes between $D$ and $D'$.

It is immediate to check that $F_f \circ U_f = 1_{Fact(f)}$.

References

• Bill Lawvere, Matias Menni, The Hopf algebra of Möbius

  intervals, Theory and Applications of Categories, 24:10 (2010), 221-265. (tac)

• 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