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For a finite group denote by $\mathbf{Ch}^G$ the dg-category of $G$-representations in chain complexes over a field.

Is there a tensor product $\otimes$ of dg-categories (similar to the Deligne tensor product for Abelian categories) such that for another finite group $H$ \begin{align} (*) \quad \quad \mathbf{Ch}^{G\times H} \cong \mathbf{Ch}^G \otimes \mathbf{Ch}^H \quad \quad ?\end{align}

Thanks for any hints.

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    $\begingroup$ Yes, at least for presentable $dg$-categories. The idea is to take $A \hat \otimes B$ to be the category of co-continuous functors from the naive tensor product $A \otimes B$ to $\rm Ch$. This paper might have more info arxiv.org/pdf/math/0408337.pdf. $\endgroup$
    – Phil Tosteson
    Commented Aug 29, 2018 at 15:14
  • $\begingroup$ Thanks for your comment. Can you explain why this tensor product will give me equivalences like (*)? $\endgroup$
    – Lukas Woike
    Commented Aug 30, 2018 at 11:39

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