# Small object argument

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In mathematics, especially in [category theory](/source/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](/source/weak_factorization_system) in the theory of [model categories](/source/Model_category).

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](/source/compact_object_(mathematics)) 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](/source/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](/source/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](/source/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](/source/Cambridge_University_Press) |isbn=978-1108473200 |location= |language=en |authorlink=}}

== Further reading ==
* https://ncatlab.org/nlab/show/small+object+argument

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Category:Category theory
Category:Factorization

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