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The title tells everything: an abelian group object in a category $\mathbf C$ with finite products is a triple $(G,m,e)$, $m\colon G\times G\to G$, $e\colon 1\to G$ such that the well known diagrams commute and such that $m\circ \sigma_{GG}=m$, where $\sigma_{AB}\colon A\times B\to B\times A$ is the map turning $\mathbf C$ into a symmetric monoidal cat.

Thanks in advance!

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I wouldn't know, but does the tag [topos-theory] fit in here? –  Asaf Karagila Jan 22 '12 at 19:23

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I will only assume that $\mathcal{E}$ is an elementary topos; this includes the case of toposes of sheaves over a site. Let $\textrm{Ab}(\mathcal{E})$ be the category of abelian group objects in $\mathcal{E}$. It is indeed true that $\textrm{Ab}(\mathcal{E})$ is an abelian category: to show this, one merely repeats in the internal logic of $\mathcal{E}$ the usual argument that $\textrm{Ab}(\mathbf{Set})$ is an abelian group. The only thing to beware of is that the internal logic of $\mathcal{E}$ is intuitionistic, i.e. the law of excluded middle does not necessarily hold in the internal logic $\mathcal{E}$. But this is no great obstacle as the usual arguments are constructive (as one would hope for such elementary facts)!

For the case where $\mathcal{E}$ is a more general category, I refer you to this MathOverflow question.

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I'm a noob in logic. Can you please be more specific about one merely repeats in the internal logic of $\cal E$ the usual argument that Ab(Set) is an abelian group? –  tetrapharmakon Jan 23 '12 at 17:30
    
The usual argument that $\textrm{Ab}(\textbf{Set})$ is an abelian category is a proof which can be written down in formal logic, and this can be interpreted in any elementary topos. The interpretation is valid so long as the proof only uses rules of inference which are valid in every elementary topos, and this basically means that the proof has to be constructive. If this is unfamiliar I advise you to look at the answers provided on MO instead. –  Zhen Lin Jan 23 '12 at 17:55

A topos is a category of sheaves on some site. The abelian group objects are then the category of sheaves of abelian groups on this site. The category of sheaves of abelian groups on a site is an abelian category.

[I wouldn't use the phrasing "abelian subcategory of $\mathcal E$" since $\mathcal E$ itself has no abelian category structure, and the abelian group objects are not a full subcategory (not all morphisms of sheaves will respect the abelian group structure on the sheaves of abelian groups).]

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Sorry, I forgot to specify that I meant to consider an elementary topos. I agree with the bracketed sentence, thanks. –  tetrapharmakon Jan 23 '12 at 17:29

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