Crossed modules and the homotopy 2-type of a free loop space

TL;DR

This paper describes the homotopy 2-type of the free loop space of a space X, assuming X is a classifying space of a crossed module, using advanced categorical structures.

Contribution

It provides a complete description of the crossed module over a groupoid that determines the homotopy 2-type of the free loop space, extending previous understanding.

Findings

Complete characterization of the homotopy 2-type of LX

Use of crossed modules over groupoids for homotopy analysis

Application of monoidal closed structure on crossed complexes

Abstract

The question was asked by Niranjan Ramachandran: how to describe the fundamental groupoid of LX, the free loop space of a space X? We give an answer by assuming X to be the classifying space of a crossed module over a group, and then describe completely a crossed module over a groupoid determining the homotopy 2-type of LX. The method requires detailed information on the monoidal closed structure on the category of crossed complexes.