On univalence, Rezk Completeness and presentable quasi-categories

This thesis is concerned with constructions in fibration categories and model categories motivated by Homotopy Type Theory and the relationship between homotopical algebra and higher category theory in the sense of Joyal and Lurie. We present some general results on univalence in type theoretic fibration categories and type theoretic model categories, extending results of Shulman and generalizing results of Lumsdaine and Kapulkin. We then study the model structure for Bousfield-Segal spaces introduced by Bergner and relate the associated model structure for complete Bousfield-Segal spaces to the work of Rezk, Schwede and Shipley and of Cisinski, showing that it yields a model of Homotopy Type Theory. We further formulate and prove a strong relationship between Rezk's completeness condition of Segal objects and the univalence condition of fibrations in a large class of type theoretic model categories. We give a definition of combinatorial model categories with universal homotopy colimits and semi-left exact left Bousfield localizations. Building on results of Dugger, Rezk, Lurie and Gepner and Kock, we show that these notions relate to locally cartesian closed presentable quasi-categories and semi-left exact localizations in the sense of Gepner and Kock in the same way as model toposes and left exact Bousfield localizations in the sense of Rezk relate to Grothendieck 1-toposes and left exact localizations in the sense of Lurie. We further relate semi-left exactness to right properness. We show that relative compact maps in presentable quasi-categories are exactly those maps presented by small fibrations between fibrant objects in Dugger's model categorical "small presentation" and discuss generalizations of this comparison to simplicial presheaf categories over small simplicial categories.