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