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This question is mostly about a reference request. Let $\mathcal{E}$ be a Grothendieck topos. I am looking for a reference of the following two facts. I am aware that $(2) \Rightarrow (1)$ by Gabriel-Popescu. I was sure I had seen $(1)$ in Bourceux's encyclopedia, but I can't find it anymore.

  1. The category of internal abelian group objects $\mathsf{Ab}(\mathcal{E})$ is a Grothendieck category.
  2. Call $\mathsf{Set}[\mathsf{Ab}]$ the classifying topos of abelian groups, and let $\mathcal{E} \simeq \mathsf{Sh}(C,J)$. Then $$\mathsf{Ab}(\mathcal{E}) \simeq \mathsf{Cocontlex(\mathsf{Set}[\mathsf{Ab}], \mathcal{E})} \simeq \mathsf{lex}(\mathsf{Ab}_\omega,\mathcal{E}) \simeq \mathsf{lex}(\mathsf{Ab}_\omega,\mathsf{Sh}(C,J)) \simeq \mathsf{Sh}(C,\mathsf{Ab}).$$
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    $\begingroup$ I don't understand which part of (2) is non-trivial ? The equivalence between the first and the last is pretty obvious if you just write what it means to be an abelian group in Sh(C,J) vs a sheaf of abelian group (it works for any algebraic theory) $\endgroup$
    – Simon Henry
    Commented Oct 24, 2020 at 12:42
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    $\begingroup$ @SimonHenry, I agree with you. I was asking because I might want to cite it. $\endgroup$
    – Ivan Di Liberti
    Commented Oct 24, 2020 at 12:44
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    $\begingroup$ Like @SimonHenry says, the equivalence between the first and last of (2) holds for any algebraic theory and, if I am not mistaken, appears as Proposition 6.3.1 in SGA 4 (although of course they do not use the terminology "algebraic theory"). But maybe you are looking for a reference mentioning the whole chain of equivalences? For what concerns (1), again I may be mistaken but I think it is (a special case of) Proposition 6.7 of SGA 4. $\endgroup$
    – Damiano Mazza
    Commented Oct 25, 2020 at 21:01

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Johnstone's "Topos theory" has a chapter on cohomology; your (1) is the first result the author proves. I don't remember about (2), but maybe it's also there?

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    $\begingroup$ Thanks. $(1)$ is Thm. 8.11 in Topos Theory. (2) does not seems to be there. $\endgroup$
    – Ivan Di Liberti
    Commented Oct 24, 2020 at 12:22

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