# Set of generators in an abelian category - two definitions

Let $\mathcal C$ be a category. We say that $\mathcal C$ has a set of generators $\{ G_i\}_{i \in I}$ if whenever we take two distinct morphisms $f, g \colon A \to B$ in $\mathcal C$ there exists some $i \in I$ and a morphism $h \colon G_i \to A$ such that $fh \not= gh$.

However, I have also read the following definition of a set of generators in an abelian category - If $\mathcal C$ is abelian, we say that $\mathcal C$ has a set of generators $\{ G_i \}_{i \in I}$ if whenever $B$ is a subobject of $A$ such that $B \not = A$ then there exists some $i \in I$ and a morphism $h \colon G_i \to A$ such that $\operatorname{Im}h$ is not a subobject of $B$.

Are these two definitions equivalent in an abelian category? If so, why?

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The two definitions are equivalent in any cocomplete abelian category, or more generally, any locally small category with equalisers and small coproducts in which all epimorphisms are extremal.

Recall: An extremal epimorphism is an epimorphism $f : A \to B$ with the following property:

• If $f = m \circ f'$ for some monomorphism $m : B' \to B$, then $m$ is an isomorphism.

Remarks. In a category with equalisers, any morphism with the above property is automatically an epimorphism. In an abelian category, every epimorphism is normal, hence regular, strong, and extremal a fortiori.

Let's also rephrase your two definitions in a more constructive way without pesky negations.

Definition. Let $\mathcal{C}$ be a category and let $\mathcal{G} = \{ G_i : i \in I \}$ be a family of objects in $\mathcal{C}$. We say $\mathcal{G}$ is a separating family for $\mathcal{C}$ just if the following holds:

• If $f \circ h = g \circ h$ for all $h : G_i \to A$ and all $G_i$ in $\mathcal{G}$, then $f = g$.

And we say $\mathcal{G}$ is an extremal separating family for $\mathcal{C}$ when this holds:

• If $m : A' \to A$ is a monomorphism such that every morphism $G_i \to A$ factors through $m$ for all $G_i$ in $\mathcal{G}$, then $m$ is an isomorphism.

Proposition. Let $\mathcal{C}$ be a locally small category with equalisers and small coproducts, and let $\mathcal{G} = \{ G_i : i \in I \}$ be a small family of objects in $\mathcal{C}$. Let $G = \coprod_{i \in I} G_i$.

• $\mathcal{G}$ is a separating family if and only if, for all objects $A$ in $\mathcal{C}$, the canonical morphism $\coprod_{f \in \mathcal{C}(G_i, A)} G_i \to A$ is an epimorphism.

• $\mathcal{G}$ is an extremal separating family if and only if, for all objects $A$ in $\mathcal{C}$, the canonical morphism $\coprod_{f \in \mathcal{C}(G_i, A)} G_i \to A$ is a extremal epimorphism.

Proof. This is a straightforward exercise.

Corollary. If $\mathcal{C}$ is a locally small category with equalisers and small coproducts, and all epimorphisms in $\mathcal{C}$ are extremal, then every small separating family for $\mathcal{C}$ is also a extremal separating family.　◼

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Would it be to correct to say that in a category where every mono is regular, that every epi is extremal? I think so, because if we factor $f$ as $f = f' m$ for some mono m, and suppose that $st = tm$, we get $smf' = sf = tf = tmf'$ which implies that $t = s$ which tells us $m$ is an epi. Since it is a regular mono too it is an iso – Paul Slevin Dec 14 '12 at 14:05
Regular epimorphisms are strong, and strong epimorphisms are extremal. The nLab page has a nice chart. – Zhen Lin Dec 14 '12 at 14:32