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. The original argument was later refined by Garner.

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 $\mathrm{Hom}(x,-)$ 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 morphisms, 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 —  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.

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. But roughly the construction is a sort of successive approximation.

See also

• Anodyne extension

References

• 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) (in en). Higher Categories and Homotopical Algebra. Cambridge University Press. ISBN 978-1108473200. https://cisinski.app.uni-regensburg.de/CatLR.pdf. 

Further reading

• https://ncatlab.org/nlab/show/small+object+argument