weak factorization system



Let KK be a category and write arr(K)arr(K) for the arrow category of KK: the category with arrows (= morphisms) afba \stackrel{f}{\to} b of KK as objects and commutative squares gu=vfg u=v f

a u c f g b v d \array{ a &\stackrel{u}{\to}& c \\ \downarrow^f && \downarrow^g \\ b &\stackrel{v}{\to}& d }

as morphisms (u,v):fg(u,v) : f \rightarrow g. We may also refer to a commutative square gu=vfg u=v f as a lifting problem between ff and gg.

We say a morphism ff has the left lifting property with respect to a morphism gg or equivalently that gg has the right lifting property with respect to ff, if for every commutative square (u,v):fg(u,v) :f \rightarrow g as above, there is an arrow γ\gamma

a u c f γ g b v d \array{ a &\stackrel{u}{\to}& c \\ \downarrow^f &{}^{\exists \gamma}\nearrow& \downarrow^g \\ b &\stackrel{v}{\to}& d }

from the codomain bb of ff to the domain cc of gg such that both triangles commute. We call such an arrow γ\gamma a lift or a solution to the lifting problem (u,v)(u,v).

(If this lift is unique, we say that ff is orthogonal fgf \perp g to gg.)


A weak factorization system on a category KK is a pair (L,R)(L, R) of classes of morphisms such that

(i) Every morphism ff of KK can be factored as f=rlf=r l with lLl \in L and rRr \in R.

(ii) LL is the class of morphisms which have the left lifting property with respect to every morphism of RR.

(iii) RR is the class of morphisms which have the right lifting property with respect to every morphism of LL.

See the Catlab for the theory.

Orthogonal Factorization Systems

An orthogonal factorization system is a weak factorization system where we additionally require that the solutions to each lifting problem be unique.

While every OFS is evidently a WFS, the primary examples of each are different. A “basic example” of an OFS is (epi, mono) in Set (meaning LL is the collection of epimorphisms and RR that of monomorphisms), while a “basic example” of a WFS is (mono, epi) in SetSet. The superficial similarity of these two examples masks the fact that they generalize in very different ways. The OFS (epi, mono) generalizes to any topos or pretopos, and in fact to any regular category if we replace “epi” with regular epi. Likewise it generalizes to any quasitopos if we instead replace “mono” with regular mono.

On the other hand, saying that (mono,epi) is a WFS in SetSet is equivalent to the axiom of choice. A less loaded statement is that (L,R)(L,R) is a WFS, where LL is the class of inclusions AABA\hookrightarrow A\sqcup B into a binary coproduct and RR is the class of split epis. In this form the statement generalizes to any extensive category; see also weak factorization system on Set.


  • Model categories provide many examples of weak factorization systems. In fact, most applications of WFS involve model categories or model-categorical ideas.

  • The existence of certain WFS on Set is related to the axiom of choice.

  • See the Catlab for more examples.


  • The classes (L,R)(L,R) of a weak factorization system enjoy many good closure properties. Both are closed under retracts and contain all isomorphisms. LL is closed under pushouts and RR is closed under pullbacks. LL is closed under arbitrary coproducts and RR is closed under arbitrary products. LL is also closed under transfinite composition. The closure properties for LL can be summarized by saying that LL is saturated, which means precisely this.

  • However, LL is not closed under all colimits in arr(K)arr(K) and similarly RR is not closed under all limits in arr(K)arr(K); they are not necessarily closed under (co)equalizers. However, if (L,R)(L,R) is an orthogonal factorzation system, then LL is closed under all colimits and RR is closed under all limits.

We give the details of the proof that morphisms defined by a right lifting property are stable under pullback.


RR is preserved under pullback.

This is for instance lemma 7.2.11 in

  • Hirschhorn, Model categories and their localization .

Let p:XYp : X \to Y be in RR and and let

Z× fX X f *p p Z f Y \array{ Z \times_f X &\to& X \\ \downarrow^{f^* p} && \downarrow^p \\ Z &\stackrel{f}{\to} & Y }

be a pullback diagram. We need to show that f *pf^* p has the right lifting property with respect to all i:ABi : A \to B in LL. So let

A Z× fX i f *p B g Z \array{ A &\to& Z \times_f X \\ \downarrow^i && \downarrow^{f^* p} \\ B &\stackrel{g}{\to}& Z }

be any commuting square. We need to construct a diagonal lift of that square. To that end, first compose with the pullback square from above to obtain the commuting diagram

A Z× fX X i f *p p B g Z f Y. \array{ A &\to& Z \times_f X &\to& X \\ \downarrow^i && \downarrow^{f^* p} && \downarrow^p \\ B &\stackrel{g}{\to}& Z &\stackrel{f}{\to}& Y } \,.

By the right lifting property of pp, there is a diagonal lift of the total outer diagram

A X i (fg)^ p B fg Y. \array{ A &\to& X \\ \downarrow^i &{}^{\hat {(f g)}}\nearrow& \downarrow^p \\ B &\stackrel{f g}{\to}& Y } \,.

By the pullback property this gives rise to the lift g^\hat g in

Z× fX X g^ f *p p B g Z f Y. \array{ && Z \times_f X &\to& X \\ &{}^{\hat g} \nearrow& \downarrow^{f^* p} && \downarrow^p \\ B &\stackrel{g}{\to}& Z &\stackrel{f}{\to}& Y } \,.

In order for g^\hat g to qualify as the intended lift of the total diagram, it remains to show that

A Z× fX i g^ B \array{ A &\to& Z \times_f X \\ \downarrow^i & {}^{\hat g}\nearrow \\ B }

commutes. To do so we notice that we obtain two pullback cones with tip AA:

  • one is given by the morphisms

    1. AZ× fXXA \to Z \times_f X \to X
    2. AiBgZA \stackrel{i}{\to} B \stackrel{g}{\to} Z

    with universal morphism into the pullback being

    • AZ× fXA \to Z \times_f X
  • the other by

    1. AiBg^Z× fXXA \stackrel{i}{\to} B \stackrel{\hat g}{\to} Z \times_f X \to X
    2. AiBgZA \stackrel{i}{\to} B \stackrel{g}{\to} Z.

    with universal morphism into the pullback being

    • AiBg^Z× fXA \stackrel{i}{\to} B \stackrel{\hat g}{\to} Z \times_f X.

The commutativity of the diagrams that we have established so far shows that the first and second morphisms here equal each other, respectively. By the fact that the universal morphism into a pullback diagram is unique this implies the required identity of morphisms.

Functorial Factorization

The precise requirements for a factorization of morphisms to be functorial are frequently misstated. What follows is a fairly uncommon (but correct) definition:

Write [2][2] and [3][3] for the ordinal numbers, regarded as categories. So arr(K)arr(K) is isomorphic to the functor category [[2],K][[2],K]. There are three injective functors [2][3][2] \rightarrow [3]; let d 1d_1 be the functor that sends the objects {0,1}\{0,1\} of [2][2] to the objects {0,2}\{0,2\} of [3][3]. This induces a functor c:[[3],K][[2],K]c : [[3],K] \rightarrow [[2],K] which can be thought of as “composition.”

A functorial factorization is a functor F:[[2],K][[3],K]F : [[2],K] \rightarrow [[3],K] such that cFc F is the identity on arr(K)arr(K). Not all weak factorization systems are functorial, although most (including those produced by the small object argument) are, but all orthogonal ones are automatically functorial.


Revised on March 23, 2012 07:20:41 by Urs Schreiber (