nLab
cat-2-group

Idea

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

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

Algebraic definition

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

  1. s it i=t is_i t_i = t_i and t is i=s it_i s_i = s_i, for i=1,2i = 1,2 whilst s is j=s js is_i s_j = s_j s_i, t it j=t jt it_i t_j = t_j t_i and s it j=t js is_i t_j = t_j s_i for iji \neq j;
  2. [Kers i,Kert i]=1[Ker\,s_i, \,Ker\,t_i ] = 1, for i=1,2i = 1,2.

Remarks

cat 2cat^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 2cat^2-groups.

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

  • The use of nn-fold groupoids in GrpGrp rather than (n+1)(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)(X;A,B) be a pointed triad, so that A,BXA,B \subseteq X. Let Φ\Phi be the space of maps I 3XI^3 \to X which map the faces of direction 11 to the base point, the faces of direction 22 into AA and the faces of direction 33 into BB. Let G=Π(X;A,B)=π 1(Φ,*)G = \Pi'(X;A,B)= \pi_1(\Phi,*). Then GG is certainly a group. The surprise is that the compositions of cubes in directions 22 and 33 are inherited by GG to make it a cat 2cat^2-group. The equivalent crossed square is the classical one Π(X;A,B)\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 2cat^2-structure uses bisimplicial methods. This generalises nicely to the nn-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, OO, 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.

Revised on June 27, 2009 01:20:06 by Toby Bartels (71.104.230.172)