In category theory, a branch of mathematics, a subfunctor is a special type of functor that is an analogue of a subset.
Subfunctors are also used in the construction of representable functors on the category of ringed spaces.
Let F be a contravariant 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.
It was discovered and exploited heavily by Alexander Grothendieck, who applied it especially to the case of schemes.
For a formal statement and proof, see Grothendieck, Éléments de géométrie algébrique, vol.