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 ω if Hom⁡(x,−) commutes with an ω-filtered colimit. In practice, we fix ω and simply say an object is compact if it is so with respect to that fixed ω.

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∘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→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∘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