Enriched Differential Lie Algebras in Topology

TL;DR

This paper develops a new algebraic framework called Edgl, linking differential graded Lie algebras to topology, enabling algebraic modeling of complex topological spaces and extending Sullivan's rationalization to all path connected spaces.

Contribution

It introduces the category of enriched differential graded Lie algebras (edgl), establishing a homotopy theory and minimal models that connect to Sullivan models for broader classes of spaces.

Findings

Unique minimal edgl models for connected spaces

Algebraic connection between edgl and Sullivan models

Extension of Sullivan rationalization to all path connected spaces

Abstract

This paper introduces a new category, Edgl, of enriched differential graded Lie algebras (edgl), directly related to the topology of all connected CW complexes and simplicial sets. It is equipped with a homotopy theory analogous to that developed by Sullivan for commutative differential graded algebras. Each connected space has a unique minimal edgl model, and an algebraic process connects this to the minimal Sullivan model. Minimal edgl models naturally represent cofibrations and, in particular cell attachments, and the interplay between edgl and Sullivan models permits the extension to all path connected spaces of results previously established only for simply connected spaces. This, in particular, provides applications and interesting examples of the classical Sullivan rationalization $X\to X_{\mathbb Q}$ of a path connected space.