factorization category

Factorization categories


The factorization category (also called the interval category) Fact(f)Fact(f) of a morphism ff in a category CC is a way of organizing its binary factorizations f=pqf = p\circ q into a category.


The objects of Fact(f)Fact(f) are factorizations

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

so that f=p 2p 1f = p_2 p_1, and a morphism from (p 1,D,p 2)(p_1, D, p_2) to (q 1,E,q 2)(q_1, E, q_2) is a morphism h:DEh \colon D \to E making everything in sight commute. There’s an obvious projection functor

(2)P f:Fact(f)C P_f \colon Fact(f) \to C

which maps (p 1,D,p 2)(p_1, D, p_2) to DD and h:(p 1,D,p 2)(q 1,E,q 2)h\colon (p_1, D, p_2) \to (q_1, E, q_2) to hh.

As iterated comma categories

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

  1. an object in C/BC/B
  2. or an object in A/CA / C

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

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

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


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

(3)(A/C)/f Fact(f) f/(C/B) π f A P f π f B A/C π A C π B C/B \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


Characterization in terms of initial and terminal objects

There is a fairly simple characterization of the categories arising as factorization categories of some ff in a category CC. First of all, note that Fact(f)Fact(f) always has

conversely, for any category DD with initial and terminal objects 0,10, 1 denoting the unique morphism !:01! \colon 0 \to 1 we have that

(4)π !:Fact(!)D \pi_! \colon Fact(!) \to D

is an equivalence. We get then

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

Factorization categories vs the category of factorizations

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

(5)U f:Fact(f)tw(C)/f U_f \colon Fact(f) \to tw(C) / f

which on objects is

(6)U f(p 1,p 2)=X 1 X X p 1 f D p 2 Y 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)U(h) = (h, id).

This functor has a left adjoint

(7)F f:tw(C)/fFact(f) F_f \colon tw(C)/f \to Fact(f)
  • F fF_f on objects:

    (8)F f(A h X g f D q Y)=X f Y gh q D 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 fF_f on arrows: picks the morphism which goes between DD and DD'.

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


  • 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

Revised on November 23, 2015 14:40:35 by Noam Zeilberger (