The twisted tensor product of dg categories and a contractible 2-operad

Abstract

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} $\mathscr{C}oh_\mathrm{dg}(D,E)\overset{\sim}{\otimes}\mathscr{C}oh_\mathrm{dg}(C,D)\to\mathscr{C}oh_\mathrm{dg}(C,E)$, 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].