Twisted arrow categories of operads and Segal conditions

Speaker: Sergei Burkin – Pontifical Catholic University of Rio de Janeiro

Abstract: Several well-known categories, including simplex category $\Delta$, Joyal’s categories $\Theta_n$, Segal’s category $\Gamma$ and Moerdijk-Weiss dendroidal category $\Omega$, allow to encode homotopy coherent structures via Segal conditions. We show that most of these categories arise from operads in a canonical way, namely, via the only natural generalization of Quillen’s twisted arrow category construction to operads. This construction gives a good context for these categories. It is closely related to Baez-Dolan plus construction and to universal enveloping category construction of Ginzburg and Kapranov.