factorization system over a subcategory
k-ary factorization system, ternary factorization system
factorization system in a 2-category
factorization system in an (∞,1)-category
A weak factorization system on a category is a pair $(\mathcal{L},\mathcal{R})$ of classes of morphisms (“projective morphisms” and “injective morphisms”) such that 1) every morphism of the category factors as the composite of one in $\mathcal{L}$ followed by one in $\mathcal{R}$, and 2) $\mathcal{L}$ and $\mathcal{R}$ are closed under having the lifting property against each other.
If the liftings here are unique, then one speaks of an orthogonal factorization system. A classical example of an orthogonal factorization system is the (epi,mono)-factorization system on the category Set or in fact on any tops.
Non-orthogonal weak factorization systems are the key ingredient in model categories, which by definition carry a weak factorization system called ($\mathcal{L} =$ cofibrations,$\mathcal{R} =$ acyclic fibrations) and another one called ($\mathcal{L} =$ acyclic cofibrations, $\mathcal{R} =$ fibrations). Indeed most examples of non-orthogonal weak factorization systems arise in the context of model category theory. A key tool for constructing these, or verifying their existence, is the small object argument.
There are other properties which one may find or impose on a weak factorization system, for instance functorial factorization. There is also extra structure which one may find or impose, such as for algebraic weak factorization systems. For more variants see at factorization system.
A weak factorization system (WFS) on a category $\mathcal{C}$ is a pair $(\mathcal{L},\mathcal{R})$ of classes of morphisms of $\mathcal{C}$ such that
Every morphism $f \colon X\to Y$ of $\mathcal{C}$ may be factored as the composition of a morphism in $\mathcal{L}$ followed by one in $\mathcal{R}$
The classes are closed under having the lifting property against each other:
$\mathcal{L}$ is precisely the class of morphisms having the left lifting property against every morphisms in $\mathcal{R}$;
$\mathcal{R}$ is precisely the class of morphisms having the right lifting property against every morphisms in $\mathcal{L}$.
For $\mathcal{C}$ a category, a functorial factorization of the morphisms in $\mathcal{C}$ is a functor
which is a section of the composition functor $d_1 \;\colon \;\mathcal{C}^{\Delta[2]}\to \mathcal{C}^{\Delta[1]}$.
In def. 2 we are using the following standard notation, see at simplex category and at nerve of a category:
Write $[1] = \{0 \to 1\}$ and $[2] = \{0 \to 1 \to 2\}$ for the ordinal numbers, regarded as posets and hence as categories. The arrow category $Arr(\mathcal{C})$ is equivalently the functor category $\mathcal{C}^{\Delta[1]} \coloneqq Funct(\Delta[1], \mathcal{C})$, while $\mathcal{C}^{\Delta[2]}\coloneqq Funct(\Delta[2], \mathcal{C})$ has as objects pairs of composable morphisms in $\mathcal{C}$. There are three injective functors $\delta_i \colon [1] \rightarrow [2]$, where $\delta_i$ omits the index $i$ in its image. By precomposition, this induces functors $d_i \colon \mathcal{C}^{\Delta[2]} \longrightarrow \mathcal{C}^{\Delta[1]}$. Here
$d_1$ sends a pair of composable morphisms to their composition;
$d_2$ sends a pair of composable morphisms to the first morphisms;
$d_0$ sends a pair of composable morphisms to the second morphisms.
A weak factorization system, def. 1, is called a functorial weak factorization system if the factorization of morphisms may be chosen to be a functorial factorization $fact$, def. 2, i.e. such that $d_2 \circ fact$ lands in $\mathcal{L}$ and $d_0\circ fact$ in $\mathcal{R}$.
Not all weak factorization systems are functorial, although most (including those produced by the small object argument, with due care) are. But all orthogonal factorization systems, def. 4, automatically are functorial.
An orthogonal factorization system (OFS) is a weak factorization system $(\mathcal{L},\mathcal{R})$, def. 1 such that the lifts of elements in $\mathcal{L}$ against elements in $\mathcal{R}$ are unique.
While every OFS (def. 4) is a WFS (def. 1), the primary examples of each are different:
A “basic example” of an OFS is (epi,mono)-factorization 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.
Let $(\mathcal{L},\mathcal{R})$ be a weak factorization system, def. 1 on some category $\mathcal{C}$. Then
Both classes contain the class of isomorphism of $\mathcal{C}$.
Both classes are closed under composition in $\mathcal{C}$.
$\mathcal{L}$ is also closed under transfinite composition.
Both classes are closed under forming retracts in the arrow category $\mathcal{C}^{\Delta[1]}$ (see remark 5).
$\mathcal{L}$ is closed under forming pushouts of morphisms in $\mathcal{C}$ (“cobase change”).
$\mathcal{R}$ is closed under forming pullback of morphisms in $\mathcal{C}$ (“base change”).
$\mathcal{L}$ is closed under forming coproducts in $\mathcal{C}^{\Delta[1]}$.
$\mathcal{R}$ is closed under forming products in $\mathcal{C}^{\Delta[1]}$.
We go through each item in turn.
containing isomorphisms
Given a commuting square
with the left morphism an isomorphism, the a lift is given by using the inverse of this isomorphism ${}^{{f \circ i^{-1}}}\nearrow$. Hence in particular there is a lift when $p \in \mathcal{R}$ and so $i \in \mathcal{L}$. The other case is formally dual.
closure under composition
Given a commuting square of the form
consider its pasting decomposition as
Now the bottom commuting square has a lift, by assumption. This yields another pasting decomposition
and now the top commuting square has a lift by assumption. This is now equivalently a lift in the total diagram, showing that $p_1\circ p_1$ has the right lifting property against $\mathcal{L}$ and is hence in $\mathcal{R}$. The case of composing two morphisms in $\mathcal{L}$ is formally dual. From this the closure of $\mathcal{L}$ under transfinite composition follows since the latter is given by colimits of sequential composition and successive lifts against the underlying sequence as above constitutes a cocone, whence the extension of the lift to the colimit follows by its universal property.
closure under retracts
Let $j$ be the retract of an $i \in \mathcal{L}$, i.e. let there be a commuting diagram of the form.
Then for
a commuting square, it is equivalent to its pasting composite with that retract diagram
Now the pasting composite of the two squares on the right has a lift, by assumption,
By composition, this is also a lift in the total outer rectangle, hence in the original square. Hence $j$ has the left lifting property against all $p \in \mathcal{R}$ and hence is in $\mathcal{L}$. The other case is formally dual.
closure under pushout and pullback
Let $p \in \mathcal{R}$ and and let
be a pullback diagram in $\mathcal{C}$. We need to show that $f^* p$ has the right lifting property with respect to all $i \in \mathcal{L}$. So let
be a commuting square. We need to construct a diagonal lift of that square. To that end, first consider the pasting composite 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 universal property of the pullback 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 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 other case is formally dual.
closure under (co-)products
Let $\{(A_s \overset{i_s}{\to} B_s) \in \mathcal{L}\}_{s \in S}$ be a set of elements of $\mathcal{L}$. Since colimits in the presheaf category $\mathcal{C}^{\Delta[1]}$ are computed componentwise, their coproduct in this arrow category is the universal morphism out of the coproduct of objects $\underset{s \in S}{\coprod} A_s$ induced via its universal property by the set of morphisms $i_s$:
Now let
be a commuting square. This is in particular a cocone under the coproduct of objects, hence by the universal property of the coproduct, this is equivalent to a set of commuting diagrams
By assumption, each of these has a lift $\ell_s$. The collection of these lifts
is now itself a compatible cocone, and so once more by the universal property of the coproduct, this is equivalent to a lift $(\ell_s)_{s\in S}$ in the original square
This shows that the coproduct of the $i_s$ has the left lifting property against all $f\in \mathcal{R}$ and is hence in $\mathcal{L}$. The other case is formally dual.
Beware, in the situation of prop. 1, that $\mathcal{L}$ is not in general closed under all colimits in $\mathcal{C}^{\Delta[1]}$, and similarly $\mathcal{R}$ is not in general closed under all limits in $\mathcal{C}^{\Delta[1]}$. Also $\mathcal{L}$ is not in general closed under forming coequalizers in $\mathcal{C}$, and $\mathcal{R}$ is not in general closed under forming equalizers in $\mathcal{C}$. However, if $(\mathcal{L},\mathcal{R})$ is an orthogonal factorization system, def. 4, then $\mathcal{L}$ is closed under all colimits and $\mathcal{R}$ is closed under all limits.
Here by a retract of a morphism $X \stackrel{f}{\longrightarrow} Y$ in some category $\mathcal{C}$ is meant a retract of $f$ as an object in the arrow category $\mathcal{C}^{\Delta[1]}$, hence a morphism $A \stackrel{g}{\longrightarrow} B$ such that in $\mathcal{C}^{\Delta[1]}$ there is a factorization of the identity on $g$ through $f$
This means equivalently that in $\mathcal{C}$ there is a commuting diagram of the form
Then:
If $f$ has the left lifting property against $p$, then $f$ is a retract of $i$.
If $f$ has the right lifting property against $i$, then $f$ is a retract of $p$.
We discuss the first statement, the second is formally dual.
Write the factorization of $f$ as a commuting square of the form
By the assumed lifting property of $f$ against $p$ there exists a diagonal filler $g$ making a commuting diagram of the form
By rearranging this diagram a little, it is equivalent to
Completing this to the right, this yields a diagram exhibiting the required retract according to remark 5:
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.