Limits in 2-Categories of Locally-Presentable Categories

Abstract

This thesis has its origins in responding to some unpublished work of Ulmer [26], [27], [28]. There, Ulmer proves that certain constructions on locally-presentable categories yield locally-presentable categories. Let C be a small category and T a set of cones in C. The category [C,A] is the full subcategory of the functor category [C,A] given by those functors T such that each TY, where Y is in T* is a limit-cone. Gabriel and Ulmer [10] had already established that [C,A]j, is reflective in [C,A], and hence locally presentable, if A is locally presentable. The result about reflectivity was extended by Freyd and Kelly [9] to the case where A is a locally-bounded category and T is a (possibly) large set. Some results on the coreflectivity of subcategories determined by functors taking (inductive) cones to colimit-cones existed, but were unpublished, before the work of Ulmer. One major thrust of this work was to establish coreflectivity for the case of A being a locally-presentable category.