category theory

# Contents

## Definition

A cogenerator in a category $C$ is an object $S$ such that the functor ${h}_{S}=C\left(-,S\right):{C}^{\mathrm{op}}\to \mathrm{Set}$ is faithful. This means that for any pair ${g}_{1},{g}_{2}\in C\left(X,Y\right)$, if they are indistinguishable by morphisms to $S$ in the sense that

$\forall \left(\theta :Y\to S\right),\phantom{\rule{thickmathspace}{0ex}}\theta \circ {g}_{1}=\theta \circ {g}_{2},$\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 $𝒮=\left\{{S}_{a},a\in A\right\}$ of objects in $C$ such that the family $C\left(-,{S}_{a}\right)$ is jointly faithful. This means that for any pair ${g}_{1},{g}_{2}\in C\left(X,Y\right)$, if they are indistinguishable by morphisms to $𝒮$ in the sense that

$\forall \left(a:A\right),\phantom{\rule{thickmathspace}{0ex}}\forall \left(\theta :Y\to {S}_{a}\right),\phantom{\rule{thickmathspace}{0ex}}\theta \circ {g}_{1}=\theta \circ {g}_{2},$\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.

Revised on January 24, 2013 06:05:59 by David Roberts (192.43.227.18)