Infinity category theory: Equivalences and Adjunctions

This lecture covers the definition and properties of equivalences and adjunctions in infinity category theory, focusing on the preservation of limits and colimits. It also explores the concept of co-natural transformations between categories and the establishment of relationships between so-categories. The lecture delves into the simplicial enrichment of functors and their role in preserving isofibrations. Additionally, it discusses the weak universal property characterizing infinity categories up to equivalence, emphasizing the importance of adjoints and the equivalence of hom-spaces. The lecture concludes with a detailed examination of groupoidal objects and their preservation by weak equivalences.