Small object argument

In mathematics, especially in category theory, Quillen’s small object argument, when applicable, constructs a factorization of a morphism in a functorial way. In practice, it can be used to show some class of morphisms constitutes a weak factorization system in the theory of model categories.

The argument was introduced by Quillen to construct a model structure on the category of (reasonable) topological spaces.^[1] The original argument was later refined by Garner.^[2]

Statement

Let $C$ be a category that has all small colimits. We say an object $x$ in it is compact with respect to an ordinal $\omega$ if $\operatorname {Hom} (x,-)$ commutes with an $\omega$ -filterted colimit. In practice, we fix $\omega$ and simply say an object is compact if it is so with respect to that fixed $\omega$ .

If $F$ is a class of morphismms, we write $l(F)$ for the class of morphisms that satisfy the left lifting property with respect to $F$ . Similarly, we write $r(F)$ for the right lifting property. Then

Theorem—^[3]^[4] Let $F$ be a class of morphisms in $C$ . If the source (domain) of each morphism in $F$ is compact, then each morphism $f$ in $C$ admits a functorial factorization $f=p\circ i$ where $i,p$ are in $l(r(F)),r(F)$ .

Example: presheaf

Here is a simple example of how the argument works in the case of the category $C$ of presheaves on some small category.^[5]

Let $I$ denote the set of monomorphisms of the form $K\to L$ , $L$ a quotient of a representable presheaf. Then $l(r(I))$ can be shown to be equal to the class of monomorphisms. Then the small object argument says: each presheaf morphism $f$ can be factored as $f=p\circ i$ where $i$ is a monomorphism and $p$ in $r(I)=r(l(r(I))$ ; i.e., $p$ is a morphism having the right lifting property with respect to monomorphisms.

Proof

For now, see.^[6] But roughly the construction is a sort of successive approximation.

References

^ D. G. Quillen. Homotopical algebra. Lecture Notes in Mathematics, No. 43. Springer-Verlag, Berlin, 1967
^ Richard Garner, Understanding the small object argument, Applied Categorical Structures 17 3 247-285 (2009) [arXiv:0712.0724, doi:10.1007/s10485-008-9137-4]
^ Cisinski 2023, Proposition 2.1.9.
^ Riehl 2014, Theorem 12.2.2.
^ Cisinski 2023, Example 2.1.11. Second method
^ Riehl 2014, § 12.2. and § 12.5.

Mark Hovey, Model categories, volume 63 of Mathematical Surveys and Monographs, American Mathematical Society, (2007),
Emily Riehl, Categorical Homotopy Theory, Cambridge University Press (2014) [1]
Cisinski, Denis-Charles (2023). Higher Categories and Homotopical Algebra (PDF). Cambridge University Press. ISBN 978-1108473200.

Small object argument

Statement

Example: presheaf

Proof

See also

References

Further reading