Twisted arrow categories, operads and Segal conditions

TL;DR

This paper introduces twisted arrow categories for operads and their algebras, connecting various known categories to operadic structures and exploring their properties as Segal presheaves, Reedy categories, and related constructs.

Contribution

It defines twisted arrow categories for operads and algebras, linking classical categories to operadic frameworks and analyzing their categorical and homotopical properties.

Findings

Twisted arrow categories of operads include classical categories like Δ, Γ, Λ, and Ω.

They admit Segal and 2-Segal presheaves, or decomposition spaces.

Under mild conditions, these categories are generalized Reedy.

Abstract

We introduce twisted arrow categories of operads and of algebras over operads. Up to equivalence of categories, the simplex category $\Delta$, Segal's category $\Gamma$, Connes cyclic category $\Lambda$, Moerdijk-Weiss dendroidal category $\Omega$, 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 $(\infty, 1)$-localization of the corresponding category $\Omega/P$ 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.