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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.cite-ref-1[1] The original argument was later refined by Garner.cite-ref-2[2]

• Statement

• Example: presheaf

• Proof

• See also

• References

• Further reading

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

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

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

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

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

For now, see.cite-ref-6[6] But roughly the construction is a sort of successive approximation.

• Anodyne extension

cite-note-11. ↑ D. G. Quillen. Homotopical algebra. Lecture Notes in Mathematics, No. 43. Springer-Verlag, Berlin, 1967

cite-note-22. ↑ 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]

cite-note-33. ↑ Cisinski 2023, Proposition 2.1.9.

cite-note-44. ↑ Riehl 2014, Theorem 12.2.2. harvnb error: no target: CITEREFRiehl2014 (help)

cite-note-55. ↑ Cisinski 2023, Example 2.1.11. Second method

cite-note-66. ↑ Riehl 2014, § 12.2. and § 12.5. harvnb error: no target: CITEREFRiehl2014 (help)

• 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]

• citerefcisinski2023Cisinski, Denis-Charles (2023). Higher Categories and Homotopical Algebra (PDF). Cambridge University Press. ISBN 978-1108473200.

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