The integration theory of curved absolute homotopy Lie algebras

TL;DR

This thesis develops a new integration theory for curved homotopy Lie algebras using operadic calculus, enabling applications in deformation and rational homotopy theories.

Contribution

It introduces the curved operadic calculus and the concept of absolute algebras, advancing the understanding of curved homotopy Lie algebra integration.

Findings

Developed operadic calculus for non-conilpotent coalgebras

Formulated the theory of curved operads for curved algebras

Applied the framework to deformation and rational homotopy theory

Abstract

The main goal of this thesis is to develop the integration theory of curved homotopy Lie algebras. In the first chapter, we develop the operadic calculus needed: we encode non-necessarily conilpotent coalgebras with operads and introduce their dual notion of an algebra over a cooperad, which we call "absolute algebras". The first sections of this chapter are a state of the art, and the last three sections are made of original results. In the second chapter, we develop the new theory of curved operadic calculus, which allows us to encode curved types of algebras with curved operads. Finally, in the third chapter, we develop the integration theory of curved absolute homotopy Lie algebras using the tools introduced in the first two chapters. This new approach allows us to obtain applications to deformation theory and rational homotopy theory.