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.D. G. Quillen. Homotopical algebra. Lecture Notes in Mathematics, No. 43. Springer-Verlag, Berlin, 1967 The original argument was later refined by Garner.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]

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.

Proof

For now, see: But roughly the construction is a sort of successive approximation.

Small object argument

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