More general 2-groups correspond to group objects in the 2-category incarnation of Grpd. For them associativity, inverses etc have to hold and exist only up to coherent natural isomorphism. So strict 2-groups are particularly rigid incarnations of 2-groups.
We may think of any 2-group in terms of its delooping , a 2-groupoid with a single object, with morphisms the objects of and 2-morphisms the morphisms of . If is a strict 2-group, then is a strict 2-groupoid. This is often a useful point of view. In particular, the general strictification result of bicategories implies that any such 2-groupoid is equivalent to a strict one. So, up to the right notion of equivalence, strict 2-groups already exhaust all 2-groups; we just have to take care to allow for homomorphisms of these -groups to be weak. (However, this theorem may not apply to structured -groups, such as Lie 2-groups.)
Strict 2-groups are also equivalently encoded in terms of crossed modules of ordinary groups: is the group of objects of the groupoid and the group of morphisms in whose source is the neutral element in .
In applications it is usually useful to pass back and forth between the 2-groupoid incarnation of strict 2-groups and their incarnation as crossed modules. The first perspective makes transparent many constructions, while the second perspective gives a useful means to do computations with 2-groups. The translation between the two points of view is described in detail below.
A strict 2-group is equivalently:
We examine the first definition in more detail.
Copying and adapting from the entry on general internal categories we have:
a collection of group homomorphisms of the form
such that the composites and are the identity morphisms on , and such that, writing for the pullback,
there is, in addition, a homomorphism
“respecting and ”;
such that the composition is associative and unital with respect to “in the obvious way”.
has a single object ;
the horizontal composition in is given by the group product operation on .
Conversely, every strict 2-groupoid with a single object defines a 2-group this way.
Beware, however, as discussed in detail at crossed module, that (strict) 2-groups and (strict) one-object 2-groupoids, live is somewhat different 2-categories. If one wants to really identify in a way that respects morphisms between these objects, one needs to think of as a pointed object equipped with its unique pointing .
We describe how a crossed module
encodes a strict one-object 2-groupoid , and hence a strict 2-group .
There are four isomorphic but different ways to construct from , which differ by whether the composition of 1-morphisms and of 1-morphisms with 2-morphisms in is taken to correspond to the product in the groups and , respectively, or in their opposites.
In concrete computations it happens that not all of these choices directly yield the expected formulas in terms of classical group theory from a given diagrammatics involving . While all choices will be isomorphic, some will be more convenient. Therefore often it matters which one of the four choices below one takes in order to get a streamlined translation between diagrammatics and formulas. For concrete examples of this phenomenon in practice see nonabelian group cohomology and gerbe.
We now define the one-object strict 2-groupoid from the crossed module with action .
has a single object ;
The set of 1-morphisms of is the group :
For we write for the corresponding 1-morphism in ;
Compositition of 1-morphisms is given by the product operation in . There are two choice for the order in which to form the product.
(convention F) horizontal composition is given by
(convention B) horizontal composition is given by
the source operation is projection on the first factor
the target operation on morphisms starting at the identity morphism is the boundary map of the crossed module combined with the product in
So in diagrams this means that a 2-morphism corresponding to is labelled as
The target of general 2-morphisms labeled by and starting at some is either of , depending on the choice of conventions discussed in the following.
Horizontal composition of 1-morphisms with 2-morphisms (“whiskering”) is determined by the rule
Horizontal composition of 2-morphisms starting at the identity 1-morphism is fixed by the convention chosen for composition of 1-morphisms
in convention F
in convention B
Notice that this is compatible with the source-target maps due to the fact that that is a group homomorphism.
With these choices made, all other compositions are now fixed by use of the exchange law:
Vertical composition of composable 2-morphisms is given, on the labels, by the product in , in the following order
(in convention L B)
By the above, every crossed module gives an example of a 2-group.
But the nature of some strict 2-groups is best understood by genuinely regarding them as 2-categorical structures. This is true notably for the example of the automorphism 2-groups, discussed below. These, too, of course are equivalenly encoded by crossed modules, but that may hide a bit their structural meaning.
objects are 1-isomorphisms in ;
morphisms are 2-isomorphisms between these 1-isomorphisms.
Accordingly, the crossed module corresponding to the 2-group is
where the boundary map is the one that sends each element to the inner automorphism given by conjugation with :
Perhaps the simplest example of such a structure is a congruence relation on a group . If is a congruence relation on , then we form the 2-group by setting and to be the group of pairs with . That this is a group follows from the definition of congruence given in the above reference. The two maps and are defined by , , whilst . The pullback is a subgroup of given by all ‘pairs of pairs’ and the composition homomorphism sends such a pair to . The other properties are easy to check.
Any congruence relation corresponds to a normal subgroup, given by those elements that are congruent to the identity element of , so that . Likewise given a normal subgroup of you get a congruence, with iff (or equivalently, ) belongs to .
See also the references at 2-group.
The equivalence between strict 2-groups and crossed modules is discussed in