Models for classifying spaces and derived deformation theory

TL;DR

This paper develops rational homotopy models for classifying spaces of fibrations using L-infinity algebra extensions, linking them to classical cohomological functors and exploring their algebraic structures.

Contribution

It introduces a novel approach to model classifying spaces via L-infinity algebra extensions and analyzes their algebraic structures, including homotopy abelian properties.

Findings

Rational homotopy models are constructed using L-infinity algebra extensions.

Chevalley-Eilenberg complexes of L-infinity algebras have an L-infinity structure with a homotopy abelian property.

The models relate to classical homological functors like Chevalley-Eilenberg and Harrison cohomology.

Abstract

Using the theory of extensions of L-infinity algebras, we construct rational homotopy models for classifying spaces of fibrations, giving answers in terms of classical homological functors, namely the Chevalley-Eilenberg and Harrison cohomology. We also investigate the algebraic structure of the Chevalley-Eilenberg complexes of L-infinity algebras and show that they possess, along with the Gerstenhaber bracket, an L-infinity structure that is homotopy abelian.