# $k$-linear $\infty$ stable categories and dg categories

This question is related to this question, where I asked about the relation between the derived category of a fiber product $$Y \times_Z W$$ and the push out of the diagram of derived categories one obtains considering the pullback functors. It was pointed out to me that it was proven by Ben-Zvi, Francis and Nadler, arXiv:0805.0157, that if we consider the derived fiber product, then (under some assumptions) we have an equivalence $$QC(Y \times_Z W) \simeq QC(Y) \otimes_{QC(Z)} QC(W)$$ where $$QC(-)$$ denotes the $$\infty$$ stable category of quasi coherent sheaves and the tensor product is computed in the $$\infty$$ category of presentable $$\infty$$ categories with morphisms given by left adjoints. As I am interested in characteristic zero, the formalism of $$k$$ linear $$\infty$$ stable categories is equivalent to that of pretriangulated dg categories. I would like to restate the above result in the formalism of pretriangulated dg categories, but I am having some problems doing it. The categories need to be substituted with dg enhancements of the triangulated categories of quasi coherent sheaves and functors of $$\infty$$ categories need to be substituted with quasi functors, but what is the counterpart of the tensor product? I tried having a look at the paper by Cohn, arXiv:1208.2587, where the equivalence between $$k$$ linear $$\infty$$ stable and pretriangulated dg is proven, and as far as I understand one needs to replace one of the two dg categories by a flat dg category. However, I don't understand what being flat for a dg category means, as in the cited paper the definition is given for spectral categories. Would anyone be able to shed some light on this tensor product for me? Thanks.

## 2 Answers

The derived tensor product of dg-categories was explored by Toën, see his article The homotopy theory of dg-categories and derived Morita theory, in particular, Section 4, where Toën explains how to derive the tensor product of dg-categories.

For a dg-category $$\mathcal C$$, being flat means that all enriched Hom's are flat (as k-modules). In other words, for every two objects $$a,b$$, one requires that the functor $$\mathcal C(x,y)\otimes-$$ preserve quasi-isomorphisms.

But you are interested in a relative tensor product? In this case I guess that the definition of $$\mathcal C$$ being flat over $$\mathcal A$$ (when one has a dg-functor $$\mathcal A\to\mathcal C$$) is that $$\mathcal C$$ is flat as an $$\mathcal A$$-module, that is $$\mathcal C\otimes_{\mathcal A}-:\mathcal A-mod\to\mathcal C-mod$$ preserves weak-equivalences and colimits.