A modern perspective on rational homotopy theory

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 Gómez-Tato, Halperin, and Tanré 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.

💡 Research Summary

The paper revisits Quillen’s celebrated construction of a family of model structures on the category of 1‑reduced simplicial sets, but it does so with the modern toolbox of left‑induced model structures and left Bousfield localisation. The author’s program can be divided into three logical phases.

Phase 1 – Left transfer from pointed simplicial sets.
The standard Quillen model structure on pointed simplicial sets ((\mathbf{sSet}_*,; \text{weak homotopy eq.},; \text{cofibrations},; \text{Kan fibrations})) is transferred along the forgetful functor
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