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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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up vote 3 down vote accepted

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
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Regular epimorphisms are strong, and strong epimorphisms are extremal. The nLab page has a nice chart. –  Zhen Lin Dec 14 '12 at 14:32
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