Properads and Homotopy Algebras Related to Surfaces

TL;DR

This paper explores the structure of properads and their associated homotopy algebras, providing explicit descriptions and linking them to solutions of generalized master equations, with applications to surfaces and modular operads.

Contribution

It offers an explicit characterization of algebras over the cobar construction of properads and introduces new associative analogues called IBA-homotopy algebras.

Findings

Explicit description of properad algebras via cobar construction

Connection between solutions of master equations and properadic structures

Introduction of IBA-homotopy algebras as associative analogues

Abstract

Starting from a biased definition of a properad, we describe explicitly algebras over the cobar construction of a properad. Equivalent description in terms of solutions of generalized master equations, which can be interpreted as homological differential operators, are explained from the properadic point of view. This is parallel to the Barannikov's theory for modular operads. In addition to well known IBL-homotopy algebras, the examples include their associative analogues, which we call $IBA$-homotopy algebras, and a combination of the above two.