Integration of a categorical operad

TL;DR

This paper introduces a Grothendieck construction for non-symmetric operads valued in categories, establishing an equivalence with operadic 2-categories fibered over finite ordinals, thus offering a new framework for their analysis.

Contribution

It develops a novel Grothendieck construction and an inverse equivalence for non-symmetric categorical operads, providing a new characterization and tools for their study.

Findings

Established an equivalence between categorical operads and operadic 2-categories.

Provided a new characterization of non-symmetric categorical operads.

Developed tools to analyze operads via fibered 2-categories.

Abstract

We describe a Grothendieck construction for non-symmetric operads with values in categories, and hence in groupoids and posets. The construction produces a 2-category which is operadically fibered over the category D of finite non-empty ordinals and surjections. We describe an inverse for the construction, yielding an equivalence of constant-free non-symmetric categorical operads and operadic 2-categories (split-)fibered over D, which resembles the correspondence of categorical presheaves and fibered categories. The result provides a new characterization of non-symmetric categorical operads and tools to study them.