Homotopical operadic calculus in positive characteristic

TL;DR

This paper extends the homotopical operadic calculus from characteristic zero to positive characteristic by introducing quasi-planar cooperads, overcoming representation theory challenges of symmetric group actions.

Contribution

It develops a new framework for operadic calculus in positive characteristic using quasi-planar cooperads, enabling the study of algebraic structures in this setting.

Findings

Extended operadic calculus to positive characteristic

Introduced quasi-planar cooperads as key tools

Resolved representation theory issues in positive characteristic

Abstract

Algebraic operads provide a powerful tool to understand the homotopy theory of the types of (co)algebras they encode. So far, the principal results and methods that this theory provides were only available in characteristic zero. The reason is that operads carry an action of all the symmetric groups, whose representation theory becomes much more involved in positive characteristic. The goal of this paper is to extend these results and methods to a positive characteristic setting. We solve the main problems that appear in this new setting by using the notion of a quasi-planar cooperad as the building block of the theory.