# From Topoi to Grothendieck categories

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 $$\mathsf{Ab}(\mathcal{E}) \simeq \mathsf{Cocontlex(\mathsf{Set}[\mathsf{Ab}], \mathcal{E})} \simeq \mathsf{lex}(\mathsf{Ab}_\omega,\mathcal{E}) \simeq \mathsf{lex}(\mathsf{Ab}_\omega,\mathsf{Sh}(C,J)) \simeq \mathsf{Sh}(C,\mathsf{Ab}).$$
• 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) – Simon Henry Oct 24 at 12:42
• @SimonHenry, I agree with you. I was asking because I might want to cite it. – Ivan Di Liberti Oct 24 at 12:44
• 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. – Damiano Mazza Oct 25 at 21:01

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