2018/07/11 by Boris Shoikhet, Shoikhet, Boris
Mathematics · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Category Theory (math.CT) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Quantum Algebra (math.QA)
paper · pdf · doi:10.48550/arxiv.1807.04305
openalex publication_date 2018/07/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
It is well-known that the "pre-2-category" \mathscrCatdgcoh(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 \mathscrCatdgcoh(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 O, acting on \mathscrCatdgcoh(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 (\mathscrCatdgcoh(k),\overset∼⊗) a skew monoidal category in the sense of [LS], and construct a \it twisted composition \mathscrCohdg(D,E)\overset∼⊗\mathscrCohdg(C,D)→\mathscrCohdg(C,E), and prove some compatibility between these two structures. Taken together, the two structures give rise to a 2-operad O, acting on \mathscrCatdgcoh(k). Its contractibility is a consequence of a general result of [Sh3].