Classification spaces of maps in model categories

TL;DR

This paper corrects a mistake in Dwyer-Kan's work and identifies homotopy function complexes in a model category with nerves of specific zig-zag categories, clarifying their structure.

Contribution

It provides a correction to prior work and establishes a new identification of homotopy function complexes with nerve constructions.

Findings

Corrected a small mistake in Dwyer-Kan's paper.

Identified homotopy function complexes with nerves of zig-zag categories.

Clarified the structure of classification spaces in model categories.

Abstract

This paper corrects a small mistake in a paper of Dwyer-Kan, and uses this to identify homotopy function complexes in a model category with the nerves of certain categories of zig-zags.