Given two small dg categories $C,D$, defined over a field, we introduce their (non-symmetric) twisted tensor product $C\sotimes D$. We show that $-\sotimes D$ is left adjoint to the functor $\mathcal {Coh}(D,-)$, where $\mathcal {Coh}(D,E)$ is the dg category of dg functors $D\to E$ and their coherent natural transformations. This adjunction holds in the category of small dg categories (not in the homotopy category of dg categories Hot). We show that for $C,D$ cofibrant, the adjunction descends to the corresponding adjunction in the homotopy category. Then comparison with a result of Toën [33] shows that, for $C,D$ cofibrant, $C\sotimes D$ is isomorphic to $C\otimes D$, as an object of the homotopy category Hot.

中文翻译：

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关于小dg类的扭曲张量积

给定在字段上定义的两个小dg类别$ C，D $，我们介绍它们的（非对称）扭曲张量积$ C \ sotimes D $。我们显示$-\ stimes D $与函子$ \ mathcal {Coh}（D，-）$相邻，其中$ \ mathcal {Coh}（D，E）$是dg函子$ D的dg类别。 \ E $及其连贯的自然变换。此附加条件属于小dg类别的类别（不属于dg类别Hot的同伦类别）。我们表明，对于$ C，D $共纤化剂，该附加词下降到同伦类别中的相应附加词。然后，与Toën[33]的结果进行比较，结果表明，对于$ C，D $共纤化剂，$ C \ sotimes D $与$ C \ otimes D $是同构的，是同构类别Hot的对象。