  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 $(\infty,1)$-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
$(\infty,2)$-colimits and our generalization of Theorem A.
