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Given two small dg categories $C,D$, defined over a field, we introduce their (non-symmetric) twisted tensor product $Coverset{sim}{otimes} D$. We show that $-overset{sim}{otimes} D$ is left adjoint to the functor $Coh(D,-)$, where $Coh(D,E)$ is the dg category of dg functors $Dto E$ and their coherent natural transformations. This adjunction holds in the category of small dg categories (not in the homotopy category of dg categories $mathrm{Hot}$). We show that for $C,D$ cofibrant, the adjunction descends to the corresponding adjunction in the homotopy category. Then comparison with a result of To{e}n shows that, for $C,D$ cofibtant, $Coverset{sim}{otimes} D$ is isomorphic to $Cotimes D$, as an object of the homotopy category $mathrm{Hot}$.
It is well-known that the pre-2-category $mathscr{C}at_mathrm{dg}^mathrm{coh}(k)$ of small dg categories over a field $k$, with 1-morphisms defined as dg functors, and with 2-morphisms defined as the complexes of coherent natural transformations, fai
In this paper, which is subsequent to our previous paper [PS] (but can be read independently from it), we continue our study of the closed model structure on the category $mathrm{Cat}_{mathrm{dgwu}}(Bbbk)$ of small weakly unital dg categories (in the
We define a tensor product of linear sites, and a resulting tensor product of Grothendieck categories based upon their representations as categories of linear sheaves. We show that our tensor product is a special case of the tensor product of locally
One-sided exact categories are obtained via a weakening of a Quillen exact category. Such one-sided exact categories are homologically similar to Quillen exact categories: a one-sided exact category $mathcal{E}$ can be (essentially uniquely) embedded
Restriction categories were established to handle maps that are partially defined with respect to composition. Tensor topology realises that monoidal categories have an intrinsic notion of space, and deals with objects and maps that are partially def