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Register a new userOn the twisted tensor product of small dg categories
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Boris Shoikhet
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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 $mathrm{Hot}$.
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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, fails to be a strict 2-category. In [T2], D.Tamarkin constructed a contractible 2-operad in the sense of M.Batanin [Ba3], acting on $mathscr{C}at_mathrm{dg}^mathrm{coh}(k)$. According to Batanin loc.cit., it is a possible way to define a weak 2-category. In this paper, we provide a construction of {it another} contractible 2-operad $mathcal{O}$, acting on $mathscr{C}at_mathrm{dg}^mathrm{coh}(k)$. Our main tool is the {it twisted tensor product} of small dg categories, introduced in [Sh3]. We establish a one-side associativity for the twisted tensor product, making $(mathscr{C}at_mathrm{dg}^mathrm{coh}(k),overset{sim}{otimes})$ a skew monoidal category in the sense of [LS], and construct a {it twisted composition} $mathscr{C}oh_mathrm{dg}(D,E)overset{sim}{otimes}mathscr{C}oh_mathrm{dg}(C,D)tomathscr{C}oh_mathrm{dg}(C,E)$, and prove some compatibility between these two structures. Taken together, the two structures give rise to a 2-operad $mathcal{O}$, acting on $mathscr{C}at_mathrm{dg}^mathrm{coh}(k)$. Its contractibility is a consequence of a general result of [Sh3].
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