Small Object Argument

The small object argument is a fundamental construction technique in model category theory that allows us to factor morphisms with desired lifting properties. It provides a systematic way to construct factorizations required by the model category axioms.

TheoremSmall Object Argument

Let $\mathcal{M}$ be a cocomplete category and $I$ a set of morphisms in $\mathcal{M}$ . Assume:

The domains of morphisms in $I$ are small relative to $I$ -cell complexes
$\mathcal{M}$ has functorial factorizations

Then every morphism $f: X \to Y$ factors as $f = p \circ i$ where:

$i: X \to Z$ is an $I$ -cell complex (a transfinite composition of pushouts of coproducts of maps in $I$ )
$p: Z \to Y$ has the right lifting property with respect to all maps in $I$

Moreover, this factorization can be made functorial.

DefinitionSmallness and Relative Cell Complexes

An object $A$ is small relative to a class $\mathcal{K}$ if the covariant hom-functor $\text{Hom}(A, -)$ preserves $\lambda$ -filtered colimits of sequences from $\mathcal{K}$ for some regular cardinal $\lambda$ .

An $I$ -cell complex is a morphism obtained as a (possibly transfinite) composition: $X = X_0 \to X_1 \to X_2 \to \cdots \to X_\alpha \to \cdots \to \text{colim}_\alpha X_\alpha = Z$ where each $X_\alpha \to X_{\alpha+1}$ is a pushout of a coproduct of maps from $I$ .

ExampleFactorizations in sSet

In $\mathbf{sSet}$ , the small object argument with $I = \{\Lambda^n_k \hookrightarrow \Delta^n\}$ (horn inclusions) produces factorizations: $f = p \circ i$ where $i$ is a cofibration and $p$ is a trivial fibration (has the right lifting property against horn inclusions, hence is both a Kan fibration and a weak equivalence).

This is precisely one of the factorizations required by the model category axioms.

RemarkTransfinite Composition

The construction proceeds by transfinite induction:

At successor stages, attach cells to solve all lifting problems that have appeared
At limit stages, take colimits
The process terminates when no new lifting problems appear

This systematic approach ensures the factorization exists and has the desired properties.

ExampleCW Approximation

In topological spaces, the small object argument applied to the generating cofibrations $\{S^{n-1} \to D^n\}$ produces the CW approximation: every map factors through a relative CW complex followed by a trivial fibration.

This recovers the classical construction from algebraic topology in an abstract categorical framework.

RemarkFunctoriality

A key feature of the small object argument is that the factorization can be made functorial. This means there exists a functor factorizing morphisms, not just a factorization of each individual morphism.

Functoriality is essential for many applications, particularly in constructing model structures and verifying coherence conditions.

DefinitionRelative $I$-cell Complexes

Given a class $I$ of morphisms, the class of relative $I$ -cell complexes is the smallest class containing $I$ and closed under:

Pushouts
Transfinite composition
Retracts (for the saturation)

These are exactly the morphisms built by the small object argument.

The small object argument is the main technical tool for verifying that a category satisfies the factorization axioms of a model category. It transforms the abstract requirement of factorizations into a concrete construction, making model category theory applicable to a wide range of examples.