Whenever you have a group $G$ acting as automorphisms of an abelian group $H$, you get a 2-group with $G$ as the group of objects and $H$ as the group of endomorphisms of the identity object. Every strict 2-group where all morphisms are endomorphisms is of this form.

Can’t you do this trick for ANY group $G$ and abelian group $H$? I don’t see why $G$ needs to act as automorphisms of $H$.

I am still trying to find somebody interested in un-unwriting this unwritten article:

The canonical 2-representation (pdf, 11 unwritten pages)

Abstract: Every finite strict 2-group has a canonical 2-representation on Vect-module categories. This easily generalizes to strict Lie 2-groups and possibly to Fréchet Lie 2-groups.

Examples discussed are the 2-representations which yield the associated 2-bundles known as line 2-bundle (abelian gerbes) and String 2-bundles.

If I were to un-unwrite this myself now, I’d stop talking about Fréchet and start talking instead internal to sheaves on smooth test domains.

As Andrew Stacey pointed out to me: it is actually known that doing loop groups not as Freéchet manifolds but as things internal to sheaves on smooth test domains is possible and, more shockingly, yields the familiar representation theory.

“And there was me thinking that a 2-group was a 2-category with one object, one morphism, and all 2-morphisms invertible.”

Indeed, every such 2-category yields a 2-group, namely the “shifted” version of the ordinary (abelian) group of endomorphisms of that single morphism.

More generally, though, 2-groups come from 2-groupoids with a single object.

Even though it may seem backwards, that’s actually a good way to define $n$-groups: they are the Hom (n-1)-categories of one-object n-groupoids.

Depending on how strict or weak you chose your n-groupoid here, you get correspondingly a more or less strict or weak notion of n-group.

As John Armstrong mentioned, it makes sense to think of an n-group as a one-object n-groupoid, because we should think of it as (a sub n-group, possibly, of) the n-group of automorphisms (or autoequivalences, if we are working weakly) of some object. That’s the single object we keep seeing.

It might be an interesting exercise to try to reverse-engineer a given n-group to realize it as the automorphism n-group of some concrete object in some given n-category.

Whenever you have a group G acting as automorphisms of an abelian group H, you get a 2-group with G as the group of objects and H as the group of endomorphisms of the identity object. Every strict 2-group where all morphisms are endomorphisms is of this form.

In particular, whenever you have a group G and a representation of it on a vector space H, you get an example of this situation. So, any group representation gives a 2-group. I find this very cute, but I don’t feel I understand its profound significance (if there is one).

When I realized this fact (which surely had been known for a long time), I decided to take the Lorentz group and its representation on R^4, turn this stuff into a 2-group, and call it the Poincare 2-group.

The use of this thing, if any, is still quite mysterious to me. Derek Wise, Laurent Freidel, Aristide Baratin and I are about 2/3 done with a paper on representations of 2-groups, which began as a paper representations of the Poincare 2-group. Jeff was also involved in working on this paper. Baratin and Freidel have done some calculations relating the representation theory of the Poincare 2-group to some 4d TQFT-like models, but I don’t think anyone really understands what’s going on yet.

By the way, for any n you can build an (n+1)-group by starting with a group G acting as automorphisms of a group H, and taking G to be the objects and H to be the endo-n-morphisms of the identity object. Well, that’s almost true… actually the case n = 0 is a weird degenerate case which doesn’t quite work the same way. This case is the usual semidirect product!

More plainly, the Poincaré 2-group is a way of restoring the distinction between rotations/boosts on the one hand, and translations on the other which is lost when you look at the Poincaré group as one big group, rather than as a construct (usually the semidirect product) built from a pair of groups.

Is this the same connection between 2-groups and semidirect products as is illustrated in the last section of the 2nd edition of CWM?