Abstract
Given two small dg categories C,D , defined over a field, we introduce their (non-symmetric) twisted tensor product C\widetilde \otimes D . We show that -\widetilde \otimes 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\widetilde \otimes D is isomorphic to C\otimes D , as an object of the homotopy category Hot.
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