Disconnected rational homotopy theory

TL;DR

This paper develops two algebraic frameworks for rational disconnected homotopy theory, using differential graded algebras and Lie algebras, and explores the structure of Maurer-Cartan spaces within these models.

Contribution

It introduces two novel algebraic models for rational disconnected homotopy theory, expanding the tools available for studying such spaces.

Findings

Construction of algebraic models based on differential graded algebras and Lie algebras.

Results on the structure of Maurer-Cartan spaces in these models.

Enhanced understanding of rational disconnected topological spaces.

Abstract

We construct two algebraic versions of homotopy theory of rational disconnected topological spaces, one based on differential graded commutative associative algebras and the other one on complete differential graded Lie algebras. As an application of the developed technology we obtain results on the structure of Maurer-Cartan spaces of complete differential graded Lie algebras.