Locally constant n-operads as higher braided operads

TL;DR

This paper introduces a new category of locally constant n-operads, generalizing classical operads to higher braided structures, and shows their homotopy categories align with known operad types for specific n values.

Contribution

It defines locally constant n-operads as higher braided operads and establishes their homotopy equivalences with classical operads for n=1,2, and infinity.

Findings

Homotopy category of locally constant 1-operads is equivalent to nonsymmetric operads.

Homotopy category of locally constant 2-operads is equivalent to braided operads.

Homotopy category of locally constant ∞-operads is equivalent to symmetric operads.

Abstract

We introduce a category of locally constant $n$-operads which can be considered as the category of higher braided operads. For $n=1,2,\infty$ the homotopy category of locally constant $n$-operads is equivalent to the homotopy category of classical nonsymmetric, braided and symmetric operads correspondingly.