The factorization category (also called the interval category) of a morphism in a category is a way of organizing its binary factorizations into a category.
The objects of are factorizations
so that , and a morphism from to is a morphism making everything in sight commute. There’s an obvious projection functor
which maps to and to .
As iterated comma categories
In terms of slice categories, a morphism can be viewed as
- an object in
- or an object in
Now, taking over/under slices again yields only one new thing; it is easy to see that
- , and
the cool fact is that the two other options yield
, 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
Characterization in terms of initial and terminal objects
There is a fairly simple characterization of the categories arising as factorization categories of some in a category . First of all, note that always has
conversely, for any category with initial and terminal objects denoting the unique morphism we have that
is an equivalence. We get then
a category is equivalent to some iff it has initial and terminal objects
Factorization categories vs the category of factorizations
We can view as a full reflective subcategory of the over-category ; here is viewed as an object of the category of factorizations of its ambient category . There’s a functor
which on objects is
and on arrows .
This functor has a left adjoint
It is immediate to check that .
- 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