Abstract
In this thesis, we introduce rational (infinity, 1)-categories, which are (infinity, 1)-categories enriched in spaces whose higher homotopy groups are rational vector spaces; such spaces are called rational. We provide two models for rational (infinity, 1)-categories, rational complete Segal spaces and rational Segal categories, and we show that they are equivalent. Our methods work for (infinity, 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 (infinity, 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 (infinity, 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.
