A -group is a double groupoid (or, equivalently, double category) internal to the category Grp of groups. It is a -object in the category of -groups. It is a group with two independent -group structures.
In the same ‘mode’ as the detailed algebraic definition of cat-1-group we have the following:
-groups are equivalent to crossed squares.
Note that to calculate a colimit it is better to work with crossed squares, but to prove something is a colimit, it is better to work with -groups.
The use of -fold groupoids in rather than -groupoids means that information contained in the ‘model’ is more ‘spread out’, but is often repeated in different forms in different parts of the model.
The homotopical example found by Loday can be seen as follows. Let be a pointed triad, so that . Let be the space of maps which map the faces of direction to the base point, the faces of direction into and the faces of direction into . Let . Then is certainly a group. The surprise is that the compositions of cubes in directions and are inherited by to make it a -group. The equivalent crossed square is the classical one involving triad homotopy groups.
Loday gives the more general description in terms of squares of spaces and the fundamental groups of the associated fibration sequences, and the proof of the -structure uses bisimplicial methods. This generalises nicely to the -fold case.
It is not completely clear how best to deal with the many-pointed case, but working within categories of groupoids with a given set, , of objects is one possibility. There are then change of ‘object’ functors between the categories, leading to a fibered category situation.
J.-L. Loday, Spaces with finitely many homotopy groups, J.Pure Appl. Alg., 24 (1982), 179–202.
R. Brown and J.-L. Loday, Van Kampen theorems for diagrams of spaces, Topology, 26 (1987), 311–337.