English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

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 asthe complexes of coherent natural transformations, fails to be a strict2-category. In [T2], D.Tamarkin constructed a contractible 2-operad in thesense of M.Batanin [Ba3], acting on$\mathscr{C}at_\mathrm{dg}^\mathrm{coh}(k)$. According to Batanin loc.cit., itis a possible way to define a "weak 2-category".  In this paper, we provide a construction of {\it another} contractible2-operad $\mathcal{O}$, acting on $\mathscr{C}at_\mathrm{dg}^\mathrm{coh}(k)$.Our main tool is the {\it twisted tensor product} of small dg categories,introduced in [Sh3]. We establish a one-side associativity for the twistedtensor product, making$(\mathscr{C}at_\mathrm{dg}^\mathrm{coh}(k),\overset{\sim}{\otimes})$ a skewmonoidal category in the sense of [LS], and construct a {\it twistedcomposition}$\mathscr{C}oh_\mathrm{dg}(D,E)\overset{\sim}{\otimes}\mathscr{C}oh_\mathrm{dg}(C,D)\to\mathscr{C}oh_\mathrm{dg}(C,E)$,and prove some compatibility between these two structures. Taken together, thetwo structures give rise to a 2-operad $\mathcal{O}$, acting on$\mathscr{C}at_\mathrm{dg}^\mathrm{coh}(k)$. Its contractibility is aconsequence of a general result of [Sh3].

The twisted tensor product of dg categories and a contractible 2-operad