In the philosophy of the Grothendieck school, one starts with some category of “local models” of spaces, equips it with a subcanonical Grothendieck topology, , and enlarges to some category of sheaves of sets on the site playing the role of spaces. There are further generalizations to stacks and so on.
The main role of properties of spaces have to be done in a relative setup, that is, the emphasis is on properties of morphisms. Thus one of the main steps in the construction of the theory is to extend good classes of morphisms of local models to the category of spaces. Grothendieck axiomatizes the situation, actually for general presheaves.
A morphism of presheaves of sets on is said to be representable by a morphism in if for every morphism from a representable presheaf , the projection from the pullback is (the image under the Yoneda embedding of) a morphism in .
When is the class of all morphisms in , we simply say that is representable.
In geometrical contexts, we usually assume that is itself closed under pullbacks in , i.e. if is in and a morphism in , then the pullback exists and the projection is in . If has all pullbacks, then the class of all morphisms in satisfies this property.
If is closed under pullback, then a morphism between representable presheaves is representable by a morphism in if and only if it is itself (the image under the Yoneda embedding of) a morphism in . In this way, the class of morphisms in is extended to a class of morphisms in the category of presheaves of sets .