Ribbon dioperads and modular ribbon properads

TL;DR

This paper introduces ribbon dioperads and modular ribbon properads, establishing algebraic structures and constructions, and applies these to higher Hochschild cochains and pre-Calabi-Yau categories.

Contribution

It defines new algebraic structures and develops their properties, including Lie algebra structures and cobar constructions, with applications to advanced categorical theories.

Findings

Lie algebra structures on colimit and limit objects

Norm map between colimit and limit objects

Cobar construction for dg ribbon co-dioperads

Abstract

We define and study the notions of ribbon dioperads and modular ribbon properads. We give a Lie algebra structure on the colimit total object and the limit total object of a ribbon dioperad, and we give a norm map between them. We give a cobar construction for dg ribbon co-dioperads. We also prove similar results for modular ribbon properads. These results are applied to the study of higher Hochschild cochain complexes and pre-Calabi-Yau categories.