Twisted arrow categories, operads and Segal conditions

We introduce twisted arrow categories of operads and of algebras over operads. Up to equivalence of categories, the simplex category $\mathrm{\Delta }$, Segal’s category $\mathrm{\Gamma }$, Connes cyclic category $\mathrm{\Lambda }$, Moerdijk-Weiss dendroidal category $\mathrm{\Omega }$, and categories similar to graphical categories of Hackney-Robertson-Yau are twisted arrow categories of symmetric or cyclic operads. Twisted arrow categories of operads admit Segal presheaves and 2-Segal presheaves, or decomposition spaces. Twisted arrow category of an operad $P$ is the $(\mathrm{\infty },1)$-localization of the corresponding category $\mathrm{\Omega }\mathrm{/}P$ by the boundary preserving morphisms. Under mild assumptions, twisted arrow categories of operads, and closely related universal enveloping categories, are generalized Reedy. We also introduce twisted arrow operads, which are related to Baez–Dolan plus construction.