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small object argument

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Contents

Idea

Quillen’s small object argument is a transfinite functorial construction of a weak factorization system (on some category CC) that is cofibrantly generated by a set of morphisms IMor(C)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.

Statement

To say that a weak factorization system is cofibrantly generated by II is to say that the right class RR of the system consists of precisely those maps which have the right lifting property with respect to II

R=rlp(I). R = rlp(I) \,.

The left class LL is then necessarily the class of maps who have the left lifting property with respect to the right class

L=llp(R)=llp(rlp(I)). L = llp(R) = llp(rlp(I)) \,.

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 II.

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 CC is just assumed to have all colimits then the domains of the maps in II 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).

Theorem (small object argument)

Let IMor(C)I \subset Mor(C) be a set of morphisms in a category CC.

Let CC be

Then

every morphism ff has a factorization of the form

f:gcell(I)hrlp(I) f : \stackrel{g \in cell(I)}{\to} \stackrel{h \in rlp(I) }{\to}

where

Remark

One sometimes says (e.g. ModLoc ) that a collection II of morphisms admits a small object argument if all domains are small relative to transfinite composites of pushouts of elements of II.

Remark

Examples of categories where the argument applies that are not presentable include the category Top of topological spaces.

The Construction

Given a morphism f:XYf: X \rightarrow Y, we would like to factor ff as a:XZa : X \rightarrow Z followed by q:ZYq : Z \rightarrow Y, where qq has the right lifting property with respect to all arrows in II. The arrow aa will be constructed to be a transfinite composite of pushouts of coproducts of maps in ii. The left class of a weak factorization system is closed under all of these constructions, so aa will be in the left class cofibrantly generated by ii.

For convenience, suppose our category is locally small. We can then consider the set S 1S_1 of lifting problems between ff (on the right) and elements iIi \in I (on the left), i.e. the set of commuting diagrams

S 1={K X i f L Y|iI}. S_1 = \left\{ \array{ K &\to& X \\ \downarrow^{\mathrlap{i}} && \downarrow^{\mathrlap{f}} \\ L &\to& Y } \;\; | \;\; i \in I \right\} \,.

Form the coproduct morphism

(K (I/f)L (I/f)):= iS 1(KiL) ( K_{(I/f)} \to L_{(I/f)} ) := \coprod_{i \in S_1} (K \stackrel{i}{\to} L)

over S 1S_1 of the corresponding elements of II; the squares of S 1S_1 then specify a canonical morphism

K (I/f)X=:Z 0 K_{(I/f)} \to X =: Z_0

from the domain of this morphism to Z 0:=XZ_0:=X. The pushout

K (I/f) Z 0 a 1 L (I/f) Z 0 L (I/f)K (I/f)=:Z 1 q 1 Y \array{ K_{(I/f)} &\to& Z_0 \\ \downarrow && \downarrow^{\mathrlap{a_1}} \\ L_{(I/f)} &\to& Z_0 \coprod_{L_{(I/f)}} K_{(I/f)} =: Z_1 \\ &&& \searrow^{\mathrlap{q_1}} \\ &&&& Y }

of this diagram defines an object Z 1Z_1 and morphisms a 1:Z 0Z 1a_1 : Z_0 \rightarrow Z_1 and q 1:Z 1Yq_1 : Z_1 \rightarrow Y factoring ff. Intuitively (following the example of simplicial boundary inclusions) if we think of the morphisms iIi \in I as being inclusions of spheres into balls, we have formed Z 1Z_1 by sphere attachments for every attaching map from a domain of II into X=Z 0X=Z_0.

Now, we iterate this construction with q 1:Z 1Yq_1 : Z_1 \rightarrow Y in place of ff and taking colimits to construct Z αZ_{\alpha} for limit ordinals α\alpha.

This construction does not converge. So we choose instead to stop at a sufficiently large ordinal β\beta, chosen so that the domains of the maps in II will satisfy the smallness property assumed in the theorem. Define aa to be the transfinite composite of the a αa_{\alpha} and qq to be the induced map from the colimit Z βZ_{\beta} to YY, so that

(XfY)=(XaZ βqY). (X \stackrel{f}{\to} Y) = (X \stackrel{a}{\to} Z_\beta \stackrel{q}{\to} Y) \,.

It is clear from the construction that aa is in the left class of the weak factorization system, so it remains to show that qq has the right lifting property with respect to each iIi \in I. Given a lifting problem,

K Z β i a L Y \array{ K &\to& Z_\beta \\ \downarrow^{\mathrlap{i}} && \downarrow^{\mathrlap{a}} \\ L &\to& Y }

the map from KK to Z βZ_\beta factors through some Z αZ_{\alpha}, with α<β\alpha \lt \beta, since Z βZ_\beta is a filtered colimit and using the assumed smallnes of KK (see compact object). Because Z α+1Z_{\alpha+1} was defined to be a pushout over squares including this one, we have a map LZ α+1Z β=colim αZ αL \rightarrow Z_{\alpha +1} \rightarrow Z_{\beta} = colim_\alpha Z_{\alpha}, which is the desired lift:

K Z α Z β Z α+1 i L Y \array{ K &\to& Z_\alpha &\to& Z_\beta \\ \downarrow && \downarrow &\nearrow& \downarrow \\ && Z_{\alpha+1} \\ \downarrow^{\mathrlap{i}} & \nearrow & && \downarrow \\ L &\to& &\to& Y }

A note on Functoriality

One of the important conclusions of the small object argument is that it is functorial. 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 β\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 variant

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.

References

Standard textbook references are for instance

theorem 2.1.14 in

  • Mark Hovey, Model categories, volume 63 of Mathematical Surveys and Monographs, American Mathematical Society, (2007),

or section 10.5 in

  • Philip Hirschhorn, Model categories and their localization, volume 99 of Mathematical Surveys and Monographs, American Math- ematical Society, cited here as ModLoc

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

  • Richard Garner, Understanding the small object argument, Applied Categorical Structures, 17 (3):247–285, 2009 (pdf)
Revised on January 7, 2014 02:29:59 by Tim Porter (2.26.23.142)