A modern perspective on rational homotopy theory

TL;DR

This paper presents a modern approach to rational homotopy theory by reconstructing Quillen's model structures through left transfer and Bousfield localization, extending the theory to broader classes of spaces.

Contribution

It introduces a new method to recover and generalize Quillen's rational homotopy model structures using model category techniques.

Findings

Reconstruction of Quillen's model structures via transfer and localization.

Extension of rational homotopy theory beyond 1-connected spaces.

Identification of fibrant objects as rational spaces.

Abstract

In Quillen's paper on rational homotopy theory, the category of 1-reduced simplicial sets is endowed with a family of model structures, the most prominent of which is the one in which the weak equivalences are the rational homotopy equivalences and the fibrant objects are the rational Kan complexes. In this paper, we give a modern approach to this family of model structures. We recover Quillen's family of model structures by first left-transferring the model structure on pointed simplicial sets and then left Bousfield localizing at the rationalization maps of spheres. Applying this localization to the model category of all spaces yields a model category in which the weak equivalences are the rational homotopy equivalences in the extended sense of G\'omez-Tato, Halperin, and Tanr\'e and the fibrant objects are the rational spaces. Thus, we generalize Quillen's family of model structures beyond the rational homotopy theory of 1-connected spaces.