Research

We introduce rational (∞, 1)-categories, which are (∞, 1)-categories enriched in spaces whose higher homotopy groups are rational vector spaces. We provide two models for rational (∞, 1)-categories, rational complete Segal spaces and rational Segal categories, and we show that they are equivalent.

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.

In this thesis, we introduce rational (∞, 1)-categories, which are (∞, 1)-categories enriched in spaces whose higher homotopy groups are rational vector spaces; such spaces are called rational. We provide two models for rational (∞, 1)-categories, rational complete Segal spaces and rational Segal categories, and we show that they are equivalent. Our methods work for (∞, 1)-categories enriched in general localizations of spaces, and we develop our arguments at that level of generality while occasionally touching base with our rational homotopy-theoretic case of special interest.

To develop our models for rational (∞, 1)-categories, we first produce a model category whose fibrant objects are the rational spaces. To that end, we give a modern perspective on Quillen's paper on rational homotopy theory, where Quillen provides a model category whose fibrant objects are the simply connected rational spaces. Specifically, we recover Quillen's model category as a left Bousfield localization, and we then apply the same localization to all spaces to get our desired generalization of Quillen's model category to non-simply connected spaces. Besides its usefulness for the purpose of modeling rational (∞, 1)-categories, our model category for rational spaces may be of independent interest in rational homotopy theory, for it encodes the rational homotopy theory of non-simply connected spaces developed by Gómez-Tato, Halperin, and Tanré.