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.<ref>D. G. Quillen. Homotopical algebra. Lecture Notes in Mathematics, No. 43. Springer-Verlag, Berlin, 1967</ref> The original argument was later refined by Garner.<ref>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]</ref>
== Statement == Let <math>C</math> be a category that has all small colimits. We say an object <math>x</math> in it is compact with respect to an ordinal <math>\omega</math> if <math>\operatorname{Hom}(x, -)</math> commutes with an <math>\omega</math>-filtered colimit. In practice, we fix <math>\omega</math> and simply say an object is compact if it is so with respect to that fixed <math>\omega</math>.
If <math>F</math> is a class of morphisms, we write <math>l(F)</math> for the class of morphisms that satisfy the left lifting property with respect to <math>F</math>. Similarly, we write <math>r(F)</math> for the right lifting property. Then
{{math_theorem|math_statement=<ref>{{harvnb|Cisinski|2023|loc=Proposition 2.1.9.}}</ref><ref>{{harvnb|Riehl|2014|loc=Theorem 12.2.2.}}</ref> Let <math>F</math> be a class of morphisms in <math>C</math>. If the source (domain) of each morphism in <math>F</math> is compact, then each morphism <math>f</math> in <math>C</math> admits a functorial factorization <math>f = p \circ i</math> where <math>i, p</math> are in <math>l(r(F)), r(F)</math>.}}
== Example: presheaf == Here is a simple example of how the argument works in the case of the category <math>C</math> of presheaves on some small category.<ref>{{harvnb|Cisinski|2023|loc=Example 2.1.11. Second method}}</ref>
Let <math>I</math> denote the set of monomorphisms of the form <math>K \to L</math>, <math>L</math> a quotient of a representable presheaf. Then <math>l(r(I))</math> can be shown to be equal to the class of monomorphisms. Then the small object argument says: each presheaf morphism <math>f</math> can be factored as <math>f = p \circ i</math> where <math>i</math> is a monomorphism and <math>p</math> in <math>r(I) = r(l(r(I))</math>; i.e., <math>p</math> is a morphism having the right lifting property with respect to monomorphisms.
== Proof == {{expand section|date=March 2025|small=no}}
For now, see:<ref>{{harvnb|Riehl|2014|loc=§ 12.2. and § 12.5.}}</ref> But roughly the construction is a sort of successive approximation.
== See also == *Anodyne extension
== References == {{reflist}} * Mark Hovey, Model categories, volume 63 of Mathematical Surveys and Monographs, American Mathematical Society, (2007), * Emily Riehl, Categorical Homotopy Theory, Cambridge University Press (2014) [http://www.math.jhu.edu/~eriehl/cathtpy.pdf] * {{cite book |last=Cisinski |first=Denis-Charles |author-link=Denis-Charles Cisinski |url=https://cisinski.app.uni-regensburg.de/CatLR.pdf |title=Higher Categories and Homotopical Algebra |date=2023|publisher=Cambridge University Press |isbn=978-1108473200 |location= |language=en |authorlink=}}
== Further reading == * https://ncatlab.org/nlab/show/small+object+argument
<!-- {{categorytheory-stub}} -->
Category:Category theory Category:Factorization