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

Abstract

It is well-known that the “pre-2-category” Catdgcoh(k) of small dg categories over a field k, with 1-morphisms defined as dg functors, and 2-morphisms defined as the complexes of coherent natural transformations, fails to be a strict 2-category. The question “What do dg categories form?”, raised by V. Drinfeld in [14], is interpreted in this context as a question of finding a weak 2-category structure on Catdgcoh(k). In [32], D. Tamarkin proposed an answer to this question, by constructing a contractible 2-operad in the sense of M. Batanin [3], acting on Catdgcoh(k).In this paper, we construct another contractible 2-operad, acting on Catdgcoh(k). Our main tool is the twisted tensor product of small dg categories, introduced in [25]. We establish a one-side associativity for the twisted tensor product, making (Catdgcoh(k),⊗∼) a skew monoidal category in the sense of [30], and construct a twisted compositionCohdg(D,E)⊗∼Cohdg(C,D)→Cohdg(C,E), and prove some compatibility between these two structures. Taken together, the two structures give rise to a 2-operad O, acting on Catdgcoh(k). Its contractibility is a consequence of a general result of [25].