A proof of the contractibility of the 2-operad defined via the twisted tensor product

Abstract

In our recent papers [Sh1,2], we introduced a {\it twisted tensor product} of dg categories, and provided, in terms of it, {\it a contractible 2-operad $\mathcal{O}$}, acting on the category of small dg categories, in which the "natural transformations" are derived. We made use of some homotopy theory developed in [To] to prove the contractibility of the 2-operad $\mathcal{O}$. The contractibility is an important issue, in vein of the theory of Batanin [Ba1,2], according to which an action of a contractible $n$-operad on $C$ makes $C$ a weak $n$-category. In this short note, we provide a new elementary proof of the contractibility of the 2-operad $\mathcal{O}$. The proof is based on a direct computation, and is independent from the homotopy theory of dg categories (in particular, it is independent from [To] and from Theorem 2.4 of [Sh1]).