factorization system over a subcategory
k-ary factorization system, ternary factorization system
factorization system in a 2-category
factorization system in an (∞,1)-category
Let $K$ be a category and write $arr(K)$ for the arrow category of $K$: the category with arrows (= morphisms) $a \stackrel{f}{\to} b$ of $K$ as objects and commutative squares $g u=v f$
as morphisms $(u,v) : f \rightarrow g$. We may also refer to a commutative square $g u=v f$ as a lifting problem between $f$ and $g$.
We say a morphism $f$ has the left lifting property with respect to a morphism $g$ or equivalently that $g$ has the right lifting property with respect to $f$, if for every commutative square $(u,v) :f \rightarrow g$ as above, there is an arrow $\gamma$
from the codomain $b$ of $f$ to the domain $c$ of $g$ such that both triangles commute. We call such an arrow $\gamma$ a lift or a solution to the lifting problem $(u,v)$.
(If this lift is unique, we say that $f$ is orthogonal $f \perp g$ to $g$.)
A weak factorization system on a category $K$ is a pair $(L, R)$ of classes of morphisms such that
(i) Every morphism $f$ of $K$ can be factored as $f=r l$ with $l \in L$ and $r \in R$.
(ii) $L$ is the class of morphisms which have the left lifting property with respect to every morphism of $R$.
(iii) $R$ is the class of morphisms which have the right lifting property with respect to every morphism of $L$.
See the Catlab for the theory.
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 $L$ is the collection of epimorphisms and $R$ that of monomorphisms), while a “basic example” of a WFS is (mono, epi) in $Set$. 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 $Set$ is equivalent to the axiom of choice. A less loaded statement is that $(L,R)$ is a WFS, where $L$ is the class of inclusions $A\hookrightarrow A\sqcup B$ into a binary coproduct and $R$ 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 $(L,R)$ of a weak factorization system enjoy many good closure properties. Both are closed under retracts and contain all isomorphisms. $L$ is closed under pushouts and $R$ is closed under pullbacks. $L$ is closed under arbitrary coproducts and $R$ is closed under arbitrary products. $L$ is also closed under transfinite composition. The closure properties for $L$ can be summarized by saying that $L$ is saturated, which means precisely this.
However, $L$ is not closed under all colimits in $arr(K)$ and similarly $R$ is not closed under all limits in $arr(K)$; they are not necessarily closed under (co)equalizers. However, if $(L,R)$ is an orthogonal factorzation system, then $L$ is closed under all colimits and $R$ is closed under all limits.
We give the details of the proof that morphisms defined by a right lifting property are stable under pullback.
$R$ is preserved under pullback.
This is for instance lemma 7.2.11 in
Let $p : X \to Y$ be in $R$ and and let
be a pullback diagram. We need to show that $f^* p$ has the right lifting property with respect to all $i : A \to B$ in $L$. So let
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
By the right lifting property of $p$, there is a diagonal lift of the total outer diagram
By the pullback property this gives rise to the lift $\hat g$ in
In order for $\hat g$ to qualify as the intended lift of the total diagram, it remains to show that
commutes. To do so we notice that we obtain two pullback cones with tip $A$:
one is given by the morphisms
with universal morphism into the pullback being
the other by
with universal morphism into the pullback being
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.
The precise requirements for a factorization of morphisms to be functorial are frequently misstated. What follows is a fairly uncommon (but correct) definition:
Write $[2]$ and $[3]$ for the ordinal numbers, regarded as categories. So $arr(K)$ is isomorphic to the functor category $[[2],K]$. There are three injective functors $[2] \rightarrow [3]$; let $d_1$ be the functor that sends the objects $\{0,1\}$ of $[2]$ to the objects $\{0,2\}$ of $[3]$. This induces a functor $c : [[3],K] \rightarrow [[2],K]$ which can be thought of as “composition.”
A functorial factorization is a functor $F : [[2],K] \rightarrow [[3],K]$ such that $c F$ is the identity on $arr(K)$. 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.