Subfunctor - Open Subfunctors

Open Subfunctors

Subfunctors are also used in the construction of representable functors on the category of ringed spaces. Let F be a functor from the category of ringed spaces to the category of sets, and let G ⊆ F. Suppose that this inclusion morphism G→F is representable by open immersions, i.e., for any representable functor Hom(−, X) and any morphism Hom(−, X)→F, the fibered product G×_FHom(−, X) is a representable functor Hom(−, Y) and the morphism Y→X defined by the Yoneda lemma is an open immersion. Then G is called an open subfunctor of F. If F is covered by representable open subfunctors, then, under certain conditions, it can be shown that F is representable. This is a useful technique for the construction of ringed spaces. It was discovered and exploited heavily by Alexandre Grothendieck, who applied it especially to the case of schemes. For a formal statement and proof, see Grothendieck, Éléments de Géométrie Algébrique, vol. 1, 2nd ed., chapter 0, section 4.5.