A modern perspective on rational homotopy theory
Eleftherios Chatzitheodoridis
TL;DR
The paper advances Quillen's rational homotopy framework by presenting a modern, flexible approach that constructs a family of model structures on $1$-reduced simplicial sets via left transfer from pointed simplicial sets and left localization with respect to $M^{-1}\mathbb{Z}$. It provides two coherent descriptions of the whole family: (i) left transfer followed by left Bousfield localization and (ii) localization with respect to homology, together with a transfer to the $1$-reduced setting; both yield simplicial model categories in which fibrant objects are precisely the $M$-local spaces. It further extends the rational localization to all spaces and to simplicial/topological spaces, establishing Quillen equivalences that reflect the $M$-local homotopy theory beyond 1-connectedness. Finally, it shows that rational homotopy localization and rational homology localization diverge outside the 1-connected regime, underscoring fundamental distinctions between rational homotopy and rational homology in general. This work provides robust, modern tools for analyzing rationalization, localization, and their model-categorical consequences across simplicial sets and 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 whose weak equivalences are the rational homotopy equivalences. In this paper, we give a modern approach to this family of model structures.