nLab
cogenerator

Definition

A cogenerator in a category $C$ is an object $S$ such that the functor $h_S=C(-,S):C^{\mathrm{op}}\to \mathrm{Set}$ is faithful. This means that for any pair $g_1,g_2\in C(X,Y)$, if they are indistinguishable by morphisms to $S$ in the sense that

$$\forall (\theta :Y\to S),\theta \circ g_1=\theta \circ g_2,$$\forall (\theta: Y \to S),\; \theta \circ g_1 = \theta \circ g_2 ,

then $g_1=g_2$.

One often extends this notion to a cogenerating family of objects, which is a (usually small) set $𝒮=\{S_a,a\in A\}$ of objects in $C$ such that the family $C(-,S_a)$ is jointly faithful. This means that for any pair $g_1,g_2\in C(X,Y)$, if they are indistinguishable by morphisms to $𝒮$ in the sense that

$$\forall (a:A),\forall (\theta :Y\to S_a),\theta \circ g_1=\theta \circ g_2,$$\forall (a: A),\; \forall (\theta: Y \to S_a),\; \theta \circ g_1 = \theta \circ g_2 ,

then $g_1=g_2$.

Examples

In Set, the set of truth values is a cogenerator. More generally, in any well-pointed topos, the subobject classifier is a cogenerator.

The existence of a small (co)generating family is one of the conditions in one version of the adjoint functor theorem.

Terminology

The concept of cogenerator is dual to that of separator, so it can also be refereed to as a coseparator.

Related concepts

• generator

• separator

• cogenerator