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تسجيل مستخدم جديدA proof of the contractibility of the 2-operad defined via the twisted tensor product
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تأليف
Boris Shoikhet
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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]).
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It is well-known that the pre-2-category $mathscr{C}at_mathrm{dg}^mathrm{coh}(k)$ 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, fai
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