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 topological spaces. The original argument was later refined by Garner.
Statement
Let be a category that has all small colimits. We say an object in it is compact with respect to an ordinal if commutes with an -filtered colimit. In practice, we fix and simply say an object is compact if it is so with respect to that fixed.If is a class of morphisms, we write for the class of morphisms that satisfy the left lifting property with respect to. Similarly, we write for the right lifting property. Then
Example: presheaf
Here is a simple example of how the argument works in the case of the category of presheaves on some small category.Let denote the set of monomorphisms of the form, a quotient of a representable presheaf. Then can be shown to be equal to the class of monomorphisms. Then the small object argument says: each presheaf morphism can be factored as where is a monomorphism and in ; i.e., is a morphism having the right lifting property with respect to monomorphisms.