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Suppose that $\mathcal{C}$ is a cartesian closed category. When is the category of abelian group objects $\mathcal{Ab}(\mathcal{C})$ a symmetric monoidal closed with respect to something substituting tensor product in the classical case $\mathcal{C}=\mathcal{Set}$?

I need reference or sketch of an argument in the case when $\mathcal{C}$ is a topos.

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up vote 3 down vote accepted

I previously indicated how to construct the tensor product for $\textbf{Ab}(\mathcal{V})$ when $\mathcal{V}$ is a sufficiently nice cartesian closed category. (The comments are probably more helpful than the post itself.) A Grothendieck topos is certainly nice enough, but so is an elementary topos with a natural numbers object. (See Theorem D5.3.5 in Sketches of an elephant.)

Basically, we mimic the standard construction of tensor products by generators and relations; the only stumbling block is the construction of free internal abelian groups. Once this is done, $\textbf{Ab}(\mathcal{V}) \to \mathcal{V}$ will be monadic, say with left adjoint $F : \mathcal{V} \to \textbf{Ab}(\mathcal{V})$, and $\textbf{Ab}(\mathcal{V})$ will have coequalisers for reflexive pairs. (See Lemma D5.3.2 in Sketches of an elephant.) One can then define a certain object $I$ in $\mathcal{V}$ for each pair of internal abelian groups $A$ and $B$ and a pair of morphisms $I \rightrightarrows F (A \times B)$ in $\mathcal{V}$ such that $$F I \rightrightarrows F (A \times B) \to A \otimes B$$ presents $A \otimes B$ as a coequaliser of a reflexive pair in $\textbf{Ab}(V)$. (We could take $$I = (A \times A \times B) \amalg (A \times B \times B) \amalg (F1 \times A \times B) \amalg (A \times F1 \times B)$$ so that we can code the four equations \begin{align} (a + a') \otimes b & = a \otimes b + a' \otimes b \newline a \otimes (b + b') & = a \otimes b + a \otimes b' \newline (r a) \otimes b & = r (a \otimes b) \newline a \otimes (r b) & = r (a \otimes b) \end{align} but one can probably get away with less.)

It turns out to be much easier to construct internal homs for $\textbf{Ab}(\mathcal{V})$: you just need $\mathcal{V}$ to be cartesian closed with equalisers. Of course, once you have both the tensor product and internal homs, then they make $\textbf{Ab}(\mathcal{V})$ into a symmetric monoidal closed category.

One can also carry out this construction in much greater generality for any commutative monad on any sufficiently nice symmetric monoidal closed category $\mathcal{V}$. This, I think, is an old result of Kock [1971, 1972] generalising Linton [1966].

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