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.
See e.g. http://www.mi-ras.ru/~akuznet/dgcat/Keller%20On%20differential%20graded%20categories.pdf
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.
