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.
The objects of $Fact(f)$ are factorizations
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
which maps $(p_1, D, p_2)$ to $D$ and $h\colon (p_1, D, p_2) \to (q_1, E, q_2)$ to $h$.
In terms of slice categories, a morphism $f: A \to B$ can be viewed as
Now, taking over/under slices again yields only one new thing; it is easy to see that
the cool fact is that the two other options yield $Fact(f)$
$Fact(f) \cong f/(C/B) \cong (A / C)/f$, and the following diagram commutes
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
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
conversely, for any category $D$ with initial and terminal objects $0, 1$ denoting the unique morphism $! \colon 0 \to 1$ we have that
is an equivalence. We get then
a category is equivalent to some $Fact(f)$ iff it has initial and terminal objects
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
which on objects is
and on arrows $U(h) = (h, id)$.
This functor has a left adjoint
$F_f$ on objects:
$F_f$ on arrows: picks the morphism which goes between $D$ and $D'$.
It is immediate to check that $F_f \circ U_f = 1_{Fact(f)}$.