nLab cat-1-group


Although just another way of writing down the axioms of a strict 2-group, the form of specification used for cat 1cat^1-groups, and, adapted for cat-n-groups, is very useful as it is in purely group theoretic terms and so is often easier to check than the more categorically phrased version, for instance, in deriving a cat 1cat^1-group structure from a simplicial group, or a cat ncat^n-group structure from an nn-fold simplicial group.

The inclusion of this entry is to help the user move between the various forms used in the literature: see references below.


A cat 1cat^1-group is a triple, (G,s,t)(G,s,t), where GG is a group and s,ts,t are endomorphisms of GG satisfying conditions

  1. st=ts t = t and ts=st s = s.

  2. [Kers,Kert]=1[Ker\,s, \,Ker\,t ] = 1 (where [Kers,Kert][Ker\,s, \,Ker\,t ] is the group generated by the commutators [x,y]=xyx 1y 1,xKers,yKert[x, y] =xyx^{-1} y^{-1}, x \in Ker\,s, y \in Ker\,t)


A cat1^1-group is just a reformulation of an internal category in Grp. (The interchange law is given by the kernel commutator condition.) As we know these latter objects are equivalent to crossed modules, we expect to be able to go between cat 1cat^1-groups and crossed modules without hindrance, and we can:


(Form of the Brown–Spencer theorem). The categories of cat 1cat^1-groups and crossed modules are equivalent.


Setting M=KersM = Ker s, N=ImsN = Im s and =t|M,\partial = t | M, then the action of NN on MM by conjugation within GG makes :MN\partial: M\to N into a crossed module. Conversely if :MN\partial: M\to N is a crossed module, then setting G=MNG = M \rtimes N and letting s,ts,t be defined by

s(m,n)=(1,n) s(m,n) = (1,n)


t(m,n)=(1,(m)n) t(m,n) = (1,\partial(m)n)

for mMm \in M, nN,n \in N, we have that (G,s,t)(G,s,t) is a cat 1cat^1-group.

See also


Loday introduced cat 1cat^1-groups in his work on the modelling of homotopy n-types.

  • J.-L. Loday, Spaces with finitely many homotopy groups, J.Pure Appl. Alg., 24, (1982), 179 – 202.

Last revised on June 21, 2019 at 07:01:30. See the history of this page for a list of all contributions to it.