Towards an (∞,2)-category of homotopy coherent monads in an ∞-cosmos

This thesis is part of a program initiated by Riehl and Verity to study the category theory of (infinity,1)-categories in a model-independent way. They showed that most models of (infinity,1)-categories form an infinity-cosmos K, which is essentially a category enriched in quasi-categories with some additional structure reminiscent of a category of fibrant objects. Riehl and Verity showed that it is possible to formulate the category theory of (infinity,1)-categories directly with infinity-cosmos axioms. This should also help organize the category theory of (infinity,1)-categories with structure. Given a category K enriched in quasi-categories, we build via a nerve construction a stratified simplicial set N_Mnd(K) whose objects are homotopy coherent monads in K. If two infinity-cosmoi are weakly equivalent, their respective stratified simplicial sets of homotopy coherent monads are also equivalent. We also provide an (infinity,2)-category Adj_r(K) whose objects are homotopy coherent adjunctions in K, that we use to classify the 1-simplices of N_Mnd(K) up to homotopy.