cogenerator

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

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

then $g_1 = g_2$.

One often extends this notion to a **cogenerating family** of objects, which is a (usually small) set $\mathcal{S} = \lbrace S_a, a\in A\rbrace$ 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 $\mathcal{S}$ in the sense that

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

then $g_1 = g_2$.

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

Much more generally, we have

Every topos with a small set of generators (e.g., a well-pointed topos, or a Grothendieck topos), and that has products of objects indexed over sets no larger in cardinality than the generating set, admits an injective cogenerator.

Let $C$ be a set of generators for the topos; as usual, let $\Omega$ be the subobject classifier. We claim that a product

$\prod_{c \in C} \Omega^c$

is a cogenerator. For suppose $f, g \colon X \stackrel{\to}{\to} Y$ are distinct morphisms. The contravariant power object functor $\Omega^-$ is faithful (a familiar fact, since it is monadic), so that $\Omega^f, \Omega^g: \Omega^Y \stackrel{\to}{\to} \Omega^X$ are distinct. Since the objects $c$ form a generating set, there is some $h \colon c \to \Omega^Y$ such that the composites

$c \stackrel{h}{\to} \Omega^Y \stackrel{\overset{\Omega^f}{\to}}{\underset{\Omega^g}{\to}} \Omega^X$

are distinct. The map $h$ may be transformed to a map $\tilde{h}: Y \to \Omega^c$, and it follows that the two composites

$X \stackrel{\overset{f}{\to}}{\underset{g}{\to}} Y \stackrel{\tilde{h}}{\to} \Omega^c$

are distinct. For any other $c' \in C$, we may uniformly define $Y \to \Omega^{c'}$ to be the map classifying the maximal subobject of $c' \times Y$, so that these maps together with $\tilde{h}$ collectively induce a map

$Y \to \prod_{c \in C} \Omega^c$

that yields distinct results when composed with $f$ and $g$. This proves the claim.

The object $\prod \Omega^c$ is injective because already $\Omega$ is injective (see Mac Lane-Moerdijk, IV.10), and it is a general fact that in a cartesian closed category (or more generally a closed monoidal category), an exponential (or internal Hom) $X^Y$ whose base $X$ is injective is also injective, and products of injective objects are injective.

Notice also that the existence of a small (co)generating family is one of the conditions in one version of the adjoint functor theorem. We may conclude, for example, that Grothendieck toposes are cototal (q.v.).

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

Revised on June 24, 2013 22:09:21
by Todd Trimble
(67.81.93.26)