Decomposing the classifying diagram in terms of classifying spaces of groups

Osborne, Christina, Mathematics - Graduate School of Arts and Sciences, University of Virginia
Bergner, Julia, Department of Mathematics, University of Virginia

The classifying diagram was defined by Rezk and is a generalization of the nerve of a category; in contrast to the nerve, the classifying diagram of two categories is equivalent if and only if the categories are equivalent. In this thesis we prove that the classifying diagram of any category is characterized in terms of classifying spaces of stabilizers of groups. We also prove explicit decompositions of the classifying diagrams for the categories of finite ordered sets, finite dimensional vector spaces, and finite sets in terms of classifying spaces of groups. For the classification diagram, which was defined by Rezk to work with categories with weak equivalences, we prove analogous results that were previously known for the classifying diagram. We close by comparing the classifying and classification diagrams, highlighting the differences and challenges of working with categories that have weak equivalences.

PHD (Doctor of Philosophy)
classifying diagram, classifying space, classification diagram, simplicial space, complete Segal space, nerve
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