## Return to Answer

1 [made Community Wiki]

The small object argument. Essentially this states that if you have a collection of maps $f_\alpha$ in a presentable category (actually, you only need the domains of the $f_\alpha$ to be compact, along with cocompleteness of the category), then any map in the category can be functorially factored as the composite of two maps:

1. A map which is a transfinite pushout of coproducts of the $f_\alpha$.
2. A map which has the right lifting property with respect to the $f_\alpha$.

This was first used by Grothendieck to show (in his Tohoku paper) that a Grothendieck abelian category always has enough injectives (which, as far as I know, is not directly obvious for abelian sheaves on a site, for instance). Later it became the main tool in constructing model structures on categories, because it lets you show that the factorizations needed in the definition exist.