Compatible structures on unary binary nonsymmetric operads with quadratic and cubic relations

TL;DR

This paper provides an operadic framework to study various compatibility conditions among replicated algebraic operations, extending previous work from binary quadratic to unary-binary quadratic and cubic operads.

Contribution

It introduces a unified operadic approach to compatibility conditions in nonsymmetric operads with unary and binary operations, generalizing earlier binary quadratic operad studies.

Findings

Linear compatibility is Koszul dual to total compatibility.

Matching compatibility is self-dual.

Compatibility conditions can be expressed via Manin square products.

Abstract

Various compatibility conditions among replicated copies of operations in a given algebraic structure have appeared in broad contexts in recent years. Taking an uniform approach, this paper gives an operadic study of compatibility conditions for nonsymmetric operads with unary and binary operations, and homogeneous quadratic and cubic relations. This generalizes the previous studies for binary quadratic operads. We consider three compatibility conditions, namely the linear compatibility, matching compatibility and total compatibility, with increasingly strict restraints among the replicated copies. The linear compatibility is in Koszul dual to the total compatibility, while the matching compatibility is self dual. Further, each compatibility can be expressed in terms of either one or both of the two Manin square products.