A weak factorization system on a category is a pair of classes of morphisms (“projective morphisms” and “injective morphisms”) such that 1) every morphism of the category factors as the composite of one in followed by one in , and 2) and 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 topos.
Non-orthogonal weak factorization systems are the key ingredient in model categories, which by definition carry a weak factorization system called ( cofibrations, acyclic fibrations) and another one called ( acyclic cofibrations, 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.
The classes are closed under having the lifting property against each other:
Write and for the ordinal numbers, regarded as posets and hence as categories. The arrow category is equivalently the functor category , while has as objects pairs of composable morphisms in . There are three injective functors , where omits the index in its image. By precomposition, this induces functors . Here
sends a pair of composable morphisms to their composition;
sends a pair of composable morphisms to the first morphisms;
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 , def. 2, i.e. such that lands in and in .
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.
A “basic example” of an OFS is (epi,mono)-factorization in Set (meaning is the collection of epimorphisms and that of monomorphisms), while a “basic example” of a WFS is (mono, epi) in . 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 is a WFS, where is the class of inclusions into a binary coproduct and is the class of split epis. In this form the statement generalizes to any extensive category; see also weak factorization system on Set.
Both classes contain the class of isomorphism of .
Both classes are closed under composition in .
is also closed under transfinite composition.
is closed under forming coproducts in .
is closed under forming products in .
We go through each item in turn.
Given a commuting square
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 has the right lifting property against and is hence in . The case of composing two morphisms in is formally dual. From this the closure of 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
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 has the left lifting property against all and hence is in . The other case is formally dual.
closure under pushout and pullback
Let and and let
By the right lifting property of , there is a diagonal lift of the total outer diagram
In order for 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 :
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 be a set of elements of . Since colimits in the presheaf category are computed componentwise, their coproduct in this arrow category is the universal morphism out of the coproduct of objects induced via its universal property by the set of morphisms :
By assumption, each of these has a lift . The collection of these lifts
This shows that the coproduct of the has the left lifting property against all and is hence in . The other case is formally dual.
Beware, in the situation of prop. 1, that is not in general closed under all colimits in , and similarly is not in general closed under all limits in . Also is not in general closed under forming coequalizers in , and is not in general closed under forming equalizers in . However, if is an orthogonal factorization system, def. 4, then is closed under all colimits and is closed under all limits.
This means equivalently that in there is a commuting diagram of the form
We discuss the first statement, the second is formally dual.
Write the factorization of as a commuting square 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.
See the Catlab for more examples.