Fat realization and Segal's classifying space

TL;DR

This paper provides a new proof demonstrating that the fat realization and Segal's classifying space of an internal category in topological spaces are homotopy equivalent, clarifying their relationship in algebraic topology.

Contribution

The paper introduces a novel proof of the homotopy equivalence between fat realization and Segal's classifying space for internal categories.

Findings

Fat realization and Segal's classifying space are homotopy equivalent.

The proof offers a new perspective on internal categories in topology.

Clarifies the relationship between two important constructions in algebraic topology.

Abstract

In this paper, we give a new proof of a well-known theorem due to tom Dieck that the fat realization and Segal's classifying space of an internal category in the category of topological spaces are homotopy equivalent.