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

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

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

Example: presheaf

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

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

See also

• Anodyne extension