On cofinal functors of ∞-bicategories

Abstract

In this thesis we investigate the notion of cofinal functor of ∞-bicategories and establish foundational results in the theory of ∞-bicategories along the way.

We start with an introductory section where we present and motivate the main results achieved to later move into the main body of the thesis which is structured as follows:

• In Chapter 1 we review the relevant (∞,1)-categorical theory that will be later generalized to the (∞,2)-categorical realm.

•In Chapter 2 we construct a model structure on the category of marked biscaled simplicial sets over a scaled simplicial set S which models outer 2-Cartesian fibrations: An (∞,2)-categorical upgrade of the notion of Cartesian fibration.

•In Chapter 3 we prove an ∞-bicategorical Grothendieck construction relating outer 2-Cartesian fibrations and contravariant functors with values in ∞-bicategories.

•In Chapter 4 we characterize cofinal functors of ∞-bicategories via generalizations of the conditions of Quillen's Theorem A.

•In Chapter 5 we provide applications of our cofinality criterion as well pointing out the next steps in the research programme of the author.