Generator (category theory) explained

is a collection

lG\subseteqOb(lC)

of objects in

, such that for any two distinct morphisms

f,g:X\toY

l{C}

, that is with

f ≠ g

, there is some

and some morphism

h:G\toX

such that

f\circh ≠ g\circh.

If the collection consists of a single object

, we say it is a generator (or separator).

Generators are central to the definition of Grothendieck categories.

The dual concept is called a cogenerator or coseparator.

Examples

is a generator: If f and g are different, then there is an element

x\inX

, such that

f(x) ≠ g(x)

. Hence the map

Z → X,

n\mapston ⋅ x

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 elements 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.