nLab cat-2-group

Idea

A $cat^2$-group is a double groupoid (or, equivalently, double category) internal to the category Grp of groups. It is a $cat^1$-object in the category of $cat^1$-groups. It is a group with two independent $cat^1$-group structures.

In the same ‘mode’ as the detailed algebraic definition of cat-1-group we have the following:

Algebraic definition

A $cat^2$-group is a quintuple, $(G,s_1,s_2, t_1,t_2)$, where $G$ is a group and $s_i,t_i$, $i = 1,2$, are endomorphisms of $G$ satisfying conditions

1. $s_i t_i = t_i$ and $t_i s_i = s_i$, for $i = 1,2$ whilst $s_i s_j = s_j s_i$, $t_i t_j = t_j t_i$ and $s_i t_j = t_j s_i$ for $i \neq j$;
2. $[Ker\,s_i, \,Ker\,t_i ] = 1$, for $i = 1,2$.

Remarks

$cat^2$-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 $cat^2$-groups.

• $cat^2$-groups are a special case of cat-n-groups, which are $n$-fold groupoids in the $Grp$.

• The use of $n$-fold groupoids in $Grp$ rather than $(n+1)$-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 $(X;A,B)$ be a pointed triad, so that $A,B \subseteq X$. Let $\Phi$ be the space of maps $I^3 \to X$ which map the faces of direction $1$ to the base point, the faces of direction $2$ into $A$ and the faces of direction $3$ into $B$. Let $G = \Pi'(X;A,B)= \pi_1(\Phi,*)$. Then $G$ is certainly a group. The surprise is that the compositions of cubes in directions $2$ and $3$ are inherited by $G$ to make it a $cat^2$-group. The equivalent crossed square is the classical one $\Pi(X;A,B)$ 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 $cat^2$-structure uses bisimplicial methods. This generalises nicely to the $n$-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, $O$, of objects is one possibility. There are then change of ‘object’ functors between the categories, leading to a fibered category situation.

References

• 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.