related by the Dold-Kan correspondence
This construction is notably used in the theory of model categories and in particular cofibrantly generated model categories in order to demonstrate the existence of the required factorization of morphisms into composites of (acyclic) cofibrations following by (acyclic) fibrations, and in order to find such factorization choices functorially.
To say that a weak factorization system is cofibrantly generated by is to say that the right class of the system consists of precisely those maps which have the right lifting property with respect to (the -injective morphisms)
The left class is then necessarily the class of maps who have the left lifting property with respect to the right class (the -cofibrations)
When a weak factorization system is cofibrantly generated, another consequence of Quillen’s small object argument is that the left class is the smallest saturated class of maps containing .
Given that the classes of a cofibrantly generated weak factorization system are determined by lifting properties, the content of the small object argument is to produce the required factorizations. With care, this construction is functorial, so the result is a functorial weak factorization system.
If the category is just assumed to have all colimits then the domains of the maps in are required to satisfy a smallness condition that says that any morphism from these objects to a sufficiently-large-directed colimit will factor through the base of the colimiting diagram. (See the reference by Hovey below.) If the category is required to be a locally presentable category then no further condition is required (see the other references below).
every morphism has a factorization of the form
One sometimes says (e.g. Hirschhorn) that a collection of morphisms admits a small object argument if all domains are small relative to transfinite composites of pushouts of elements of .
Given a morphism , we would like to factor as followed by , where has the right lifting property with respect to all arrows in . The arrow will be constructed to be a transfinite composite of pushouts of coproducts of maps in . The left class of a weak factorization system is closed under all of these constructions, so will be in the left class cofibrantly generated by .
For convenience, suppose our category is locally small. We can then consider the set of lifting problems between (on the right) and elements (on the left), i.e. the set of commuting diagrams
Form the coproduct morphism
over of the corresponding elements of ; the squares of then specify a canonical morphism
from the domain of this morphism to . The pushout
of this diagram defines an object and morphisms and factoring . Intuitively (following the example of simplicial boundary inclusions) if we think of the morphisms as being inclusions of spheres into balls, we have formed by sphere attachments for every attaching map from a domain of into .
Now, we iterate this construction with in place of and taking colimits to construct for limit ordinals .
This construction does not converge. So we choose instead to stop at a sufficiently large ordinal , chosen so that the domains of the maps in will satisfy the smallness property assumed in the theorem. Define to be the transfinite composite of the and to be the induced map from the colimit to , so that
It is clear from the construction that is in the left class of the weak factorization system, so it remains to show that has the right lifting property with respect to each . Given a lifting problem,
the map from to factors through some , with , since is a filtered colimit and using the assumed smallnes of (see compact object). Because was defined to be a pushout over squares including this one, we have a map , which is the desired lift:
One of the important conclusions of the small object argument is that it is functorial, hence that it produces functorial factorizations. But since (in its ordinary form) the process does not “converge” (in the up-to-isomorphism sense) but rather is merely stopped when it has gone far enough along, for functoriality we have to take care to terminate the construction at the same ordinal for every input.
Additionally, in an enriched situation, ideally one would like the factorizations to be an enriched functor. The version of the small object argument given above does not produce an enriched functor, since it takes coproducts over maps in an ordinary category. It can be modified to produce an enriched functor by replacing these coproducts by copowers, but the resulting factorizations are only rarely homotopically well-behaved (in a model category, for instance). One important special case when they are well-behaved is when all objects of the enriching category are cofibrant, as is the case for simplicial sets and for the folk model structure on Cat.
A modified version of Quillen’s small object argument due to Richard Garner produces not just functorial factorization but those of an algebraic weak factorization system. Unlike Quillen’s construction, his converges. Details are contained in Understanding the small object argument.
Standard textbook references are for instance
theorem 2.1.14 in
or section 10.5 in
Based on this a good quick reference is the first two pages of
See also the appendix of HTT.
For more conceptual background see