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Quillen’s small object argument is a transfinite functorial construction of a weak factorization system (on some category $C$) that is cofibrantly generated by a set of morphisms $I \subset Mor(C)$.
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 $I$ is to say that the right class $R$ of the system consists of precisely those maps which have the right lifting property with respect to $I$ (the $I$-injective morphisms)
The left class $L$ is then necessarily the class of maps who have the left lifting property with respect to the right class (the $I$-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 $I$.
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 $C$ is just assumed to have all colimits then the domains of the maps in $I$ 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).
Let $I \subset Mor(C)$ be a set of morphisms in a category $C$.
Let $C$ be
or, more generally, such that it has all colimits and each domain of morphisms in $I$ is a small object.
or, yet more generally, such that it has all colimits and each domain of morphisms in $I$ is small relative to transfinite composites of pushouts of maps in $I$.
Then
every morphism $f$ has a factorization of the form
where
$rlp(I)$ is the set of morphisms with right lifting property with respect to $I$
$cell(I)$ is the set of transfinite compositions of pushouts of morphisms in $I$;
One sometimes says (e.g. Hirschhorn) that a collection $I$ of morphisms admits a small object argument if all domains are small relative to transfinite composites of pushouts of elements of $I$.
Examples of categories where the argument applies that are not presentable include the category Top of topological spaces.
Given a morphism $f: X \rightarrow Y$, we would like to factor $f$ as $a : X \rightarrow Z$ followed by $q : Z \rightarrow Y$, where $q$ has the right lifting property with respect to all arrows in $I$. The arrow $a$ will be constructed to be a transfinite composite of pushouts of coproducts of maps in $i$. The left class of a weak factorization system is closed under all of these constructions, so $a$ will be in the left class cofibrantly generated by $i$.
For convenience, suppose our category is locally small. We can then consider the set $S_1$ of lifting problems between $f$ (on the right) and elements $i \in I$ (on the left), i.e. the set of commuting diagrams
Form the coproduct morphism
over $S_1$ of the corresponding elements of $I$; the squares of $S_1$ then specify a canonical morphism
from the domain of this morphism to $Z_0:=X$. The pushout
of this diagram defines an object $Z_1$ and morphisms $a_1 : Z_0 \rightarrow Z_1$ and $q_1 : Z_1 \rightarrow Y$ factoring $f$. Intuitively (following the example of simplicial boundary inclusions) if we think of the morphisms $i \in I$ as being inclusions of spheres into balls, we have formed $Z_1$ by sphere attachments for every attaching map from a domain of $I$ into $X=Z_0$.
Now, we iterate this construction with $q_1 : Z_1 \rightarrow Y$ in place of $f$ and taking colimits to construct $Z_{\alpha}$ for limit ordinals $\alpha$.
This construction does not converge. So we choose instead to stop at a sufficiently large ordinal $\beta$, chosen so that the domains of the maps in $I$ will satisfy the smallness property assumed in the theorem. Define $a$ to be the transfinite composite of the $a_{\alpha}$ and $q$ to be the induced map from the colimit $Z_{\beta}$ to $Y$, so that
It is clear from the construction that $a$ is in the left class of the weak factorization system, so it remains to show that $q$ has the right lifting property with respect to each $i \in I$. Given a lifting problem,
the map from $K$ to $Z_\beta$ factors through some $Z_{\alpha}$, with $\alpha \lt \beta$, since $Z_\beta$ is a filtered colimit and using the assumed smallnes of $K$ (see compact object). Because $Z_{\alpha+1}$ was defined to be a pushout over squares including this one, we have a map $L \rightarrow Z_{\alpha +1} \rightarrow Z_{\beta} = colim_\alpha Z_{\alpha}$, 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 $\beta$ 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
A reference with an eye towards combinatorial model categories and Smith's theorem is
Based on this a good quick reference is the first two pages of
See also the appendix of HTT.
For more conceptual background see