nLab
cat-1-group

Idea

Although just another way of writing down the axioms of a strict 2-group, the form of specification used for ${\mathrm{cat}}^1$-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 ${\mathrm{cat}}^1$-group structure from a simplicial group, or a ${\mathrm{cat}}^n$-group structure from an $n$-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.

Definition

A ${\mathrm{cat}}^1$-group is a triple, $(G,s,t)$, where $G$ is a group and $s,t$ are endomorphisms of $G$ satisfying conditions

1. $st=t$ and $ts=s$.
2. $[\mathrm{Ker}s,\mathrm{Ker}t]=1$.

Discussion

A cat${}^1$-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 ${\mathrm{cat}}^1$-groups and crossed modules without hindrance, and we can:

Lemma

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

Proof

Setting $M=\mathrm{Ker}s$, $N=\mathrm{Im}s$ and $\partial =t\mid M,$ then the action of $N$ on $M$ by conjugation within $G$ makes $\partial :M\to N$ into a crossed module. Conversely if $\partial :M\to N$ is a crossed module, then setting $G=M⋊N$ and letting $s,t$ be defined by

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

and

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

for $m\in M$, $n\in N,$ we have that $(G,s,t)$ is a ${\mathrm{cat}}^1$-group.

References

Loday introduced ${\mathrm{cat}}^1$-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.