2019/10/27 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.1910.12371
openalex publication_date 2019/10/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 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 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 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]).