On the loop space of a 2-category

TL;DR

This paper investigates the classifying space of a 2-category and demonstrates how, under certain conditions, its loop space can be reconstructed from the endomorphisms of an object, linking categorical structures with homotopy theory.

Contribution

It establishes a method to recover the loop space of the classifying space of a 2-category from endomorphisms, extending classical results to higher categorical contexts.

Findings

Loop space of classifying space can be recovered from endomorphisms

Main theorem applies under specific conditions

Provides subsidiary results supporting the main proof

Abstract

Every small category $C$ has a classifying space $BC$ associated in a natural way. This construction can be extended to other contexts and set up a fruitful interaction between categorical structures and homotopy types. In this paper we study the classifying space $B_2C$ of a 2-category $C$ and prove that, under certain conditions, the loop space $\Omega_c B_2C$ can be recovered up to homotopy from the endomorphisms of a given object. We also present several subsidiary results that we develop to prove our main theorem.