Generator (category theory)

Examples

• In the category of abelian groups, the group of integers is a generator: If f and g are different, then there is an element, such that. Hence the map suffices.
• Similarly, the one-point set is a generator for the category of sets. In fact, any nonempty set is a generator.
• In the category of sets, any set with at least two objects is a cogenerator.
• In the category of modules over a ring R, a generator in a finite direct sum with itself contains an isomorphic copy of R as a direct summand. Consequently, a generator module is faithful, i.e. has zero annihilator.