Decomposing the classifying diagram in terms of classifying spaces of groups

TL;DR

This paper characterizes the classifying diagram of any category using classifying spaces of stabilizers and provides explicit decompositions for categories like finite sets and vector spaces.

Contribution

It introduces a new characterization of the classifying diagram via classifying spaces of stabilizers and offers explicit decompositions for specific categories.

Findings

Classifying diagram of any category is characterized by classifying spaces of stabilizers.

Explicit decompositions for categories of finite ordered sets, vector spaces, and finite sets.

Establishes a connection between classifying diagrams and classifying spaces of groups.

Abstract

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 paper 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.