On the twisted tensor product of small dg categories

Abstract

Given two small dg categories $C,D$, defined over a field, we introduce their (non-symmetric) twisted tensor product $C\tilde{\otimes }D$. We show that $-\tilde{\otimes }D$ is left adjoint to the functor $\mathcal{C}oh(D,-)$, where $\mathcal{C}oh(D,E)$ 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, $C\tilde{\otimes }D$ is isomorphic to $C\otimes D$, as an object of the homotopy category Hot.