Computing 3-Dimensional Groups : Crossed Squares and Cat${}^2$-Groups

Abstract

The category $\mathbf{XSq}$ of crossed squares is equivalent to the category $Cat2$ of cat${}^2$-groups. Functions for computing with these structures have been developed in the package $\mathsf{XMod}$ written using the $\mathsf{GAP}$ computational discrete algebra programming language. This paper includes details of the algorithms used. It contains tables listing the $1,000$ isomorphism classes of cat${}^2$-groups on groups of order at most $30$.

15 pages, xypic