Freeness conditions for crossed squares and squared complexes

TL;DR

This paper explores the structure of totally free crossed squares and square complexes, linking simplicial groups, tensor products of crossed modules, and algebraic derivations, advancing the understanding of algebraic topology structures.

Contribution

It introduces a new interpretation of free simplicial groups in terms of totally free crossed squares and extends Ellis's results to tensor products of crossed modules.

Findings

Characterization of totally free crossed squares

Connection between simplicial groups and tensor products of crossed modules

Algebraic derivation of Brown and Loday's result

Abstract

Following Ellis, we investigate the notion of totally free crossed squares and related square complexes. It is shown how to interpret the information in a free simplicial group given with a choice of CW-basis, in terms of the data for a totally free crossed square. Results of Ellis then apply to give a description in terms of tensor products of crossed modules. The paper ends with a purely algebraic derivation of a result of Brown and Loday.