Theorem A for marked 2-categories

Abstract

In this work, we prove a generalization of Quillen's Theorem A to 2-categories equipped with a special set of morphisms which we think of as weak equivalences, providing sufficient conditions for a 2-functor to induce an equivalence on ( ∞ , 1 ) -localizations. When restricted to 1-categories with all morphisms marked, our theorem retrieves the classical Theorem A of Quillen. We additionally state and provide evidence for a new conjecture: the cofinality conjecture , which describes the relation between a conjectural theory of marked ( ∞ , 2 ) -colimits and our generalization of Theorem A.