Self-dual nonsymmetric operads with two binary operations

TL;DR

This paper classifies self-dual quadratic nonsymmetric operads with two binary operations, extending associativity concepts to a broader operadic framework and exploring their Koszul duality properties.

Contribution

It provides a complete classification of such operads, generalizing associativity to two operations and analyzing their duality structures.

Findings

Complete classification of self-dual quadratic nonsymmetric operads

Extension of associativity concepts to two binary operations

Analysis of Koszul duality in this operadic context

Abstract

We consider nonsymmetric operads with two binary operations satisfying relations in arity 3; hence these operads are quadratic, and so we can investigate Koszul duality. We first consider operations which are nonassociative (not necessarily associative) and then specialize to the associative case. We obtain a complete classification of self-dual quadratic nonsymmetric operads with two (associative or nonassociative) binary operations. These operads generalize associativity for one operation to the setting of two operations.