Generator

Summary

A generator, also called a separator, of a category is an object $G$ with the following property. Its morphisms into other objects are enough to tell any two different parallel morphisms apart. Equivalently, the hom functor $\Hom(G, -)$ is faithful. Generators and the dual cogenerators appear as hypotheses in the special adjoint functor theorem, and generators are part of the definition of a Grothendieck category.