On comparison of the Tamarkin and the twisted tensor product 2-operads

Abstract

There are known two different constructions of contractible dg 2-operads, providing a weak 2-category structure on the following dg 2-quiver of small dg 2-categories. Its vertices are small dg 2-categories over a given field, arrows are dg functors, and the 2-arrows $F\Rightarrow G$ are defined as the Hochschild cochains of $C$ with coefficients in $C$-bimodule $D(F(-),G(=))$, where $F,G\colon C\to D$ are dg functors, $C,D$ small dg categories. It is known that such definition is correct homotopically, but, on the other hand, the corresponding dg 2-quiver fails to be a strict 2-category. The question ``What do dg categories form'' is the question of finding a weak 2-category structure on it, in an appropriate sense. One way of phrasing it out is to make it an algebra over a contractible 2-operad, in the sense of M.Batanin [Ba1,2] (in turn, there are many compositions of 2-arrows for a given diagram, but their totality forms a contractible complex) . In [T], D.Tamarkin proposed a contractible $\Delta$-colored 2-operad in Sets, whose dg condensation solves the problem. In our recent paper arXiv:1807.04305 we constructed contractible dg 2-operad, called the twisted tensor product operad, acting on the same 2-quiver (the construction uses the twisted tensor product of small dg categories introduced in arXiv:1803.01191). In this paper, we compare the two constructions.