Yoneda Lemma - Generalities

Generalities

The Yoneda lemma suggests that instead of studying the (locally small) category C, one should study the category of all functors of C into Set (the category of sets with functions as morphisms). Set is a category we understand well, and a functor of C into Set can be seen as a "representation" of C in terms of known structures. The original category C is contained in this functor category, but new objects appear in the functor category which were absent and "hidden" in C. Treating these new objects just like the old ones often unifies and simplifies the theory.

This approach is akin to (and in fact generalizes) the common method of studying a ring by investigating the modules over that ring. The ring takes the place of the category C, and the category of modules over the ring is a category of functors defined on C.