Properadic homotopical calculus

TL;DR

This paper extends operadic calculus to properadic calculus, enabling the study of homotopy gebras and generalizing key concepts like infinity-morphisms and the homotopy transfer theorem, with applications to involutive Lie bialgebras.

Contribution

It introduces a properadic calculus framework for homotopy theories, generalizing operadic concepts and providing new formulas and insights for homotopy involutive Lie bialgebras.

Findings

Recovered homotopy properties of involutive Lie bialgebras.

Generalized infinity-morphisms and homotopy transfer theorem.

Produced explicit formulas for homotopy involutive Lie bialgebras.

Abstract

In this paper, we initiate the generalisation of the operadic calculus which governs the properties of homotopy algebras to a properadic calculus which governs the properties of homotopy gebras over a properad. In this first article of a series, we generalise the seminal notion of infini-morphisms and the ubiquitous homotopy transfer theorem. As an application, we recover the homotopy properties of involutive Lie bialgebras developed by Cieliebak--Fukaya--Latschev and we produce new explicit formulas.