nLab
strict 2-group

Contents

Idea

The notion of strict 2-group is a strict vertical categorification of that of group.

A strict 2-group is a group object internal to the category Grpd of groupoids (regarded as an ordinary category, not as a 2-category).

This means that it is a groupoid GG equipped with a product functor :G×GG\cdot : G \times G \to G that behaves like the product in a group, in that it is unital and associative and such that there are inverses under multiplication.

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 GG in terms of its delooping BG\mathbf{B}G, a 2-groupoid with a single object, with morphisms the objects of GG and 2-morphisms the morphisms of GG. If GG is a strict 2-group, then BG\mathbf{B}G 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 22-groups to be weak. (However, this theorem may not apply to structured 22-groups, such as Lie 2-groups.)

Strict 2-groups are also equivalently encoded in terms of crossed modules (G 2G 1)(G_2 \to G_1) of ordinary groups: G 1G_1 is the group of objects of the groupoid GG and G 2G_2 the group of morphisms in GG whose source is the neutral element in G 1G_1.

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.

Definitions

A strict 2-group is equivalently:

Expanding the definition

We examine the first definition in more detail.

Copying and adapting from the entry on general internal categories we have:

A internal category in Grp is

In terms of strict 2-groupoids

Every strict 2-group GG defines a strict 2-groupoid BG\mathbf{B}G – called its delooping – defined by the fact that

Conversely, every strict 2-groupoid with a single object \bullet 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 BG\mathbf{B}G in a way that respects morphisms between these objects, one needs to think of BG\mathbf{B}G as a pointed object equipped with its unique pointing *BG{*} \to \mathbf{B}G.

In terms of crossed modules

We describe how a crossed module

[BG]=(G 2δG 1) [\mathbf{B}G] = (G_2 \stackrel{\delta}{\to} G_1)

with action

α:G 1Aut(G 2) \alpha : G_1 \to Aut(G_2)

encodes a strict one-object 2-groupoid BG\mathbf{B}G, and hence a strict 2-group GG.

There are four isomorphic but different ways to construct BG\mathbf{B}G from [BG][\mathbf{B}G], which differ by whether the composition of 1-morphisms and of 1-morphisms with 2-morphisms in BG\mathbf{B}G is taken to correspond to the product in the groups G 1G_1 and G 2G_2, 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 BG\mathbf{B}G. 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 BG\mathbf{B}G from the crossed module (δ:G 2G 1)(\delta : G_2 \to G_1) with action α:G 1Aut(G 2)\alpha : G_1 \to Aut(G_2).

Examples

From crossed modules

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.

Automorphism 2-groups

For aa any object in a strict 2-category CC, there is the strict automorphism 2-group Aut C(a)Aut_C(a) whose

In particular, for KK a group and BK\mathbf{B}K its delooping groupoid, we have the automorphism 2-group of BK\mathbf{B}K in the 2-category Grpd. This is usually called the automorphism 2-group of the group KK

AUT(K):=Aut Grpd(BK). AUT(K) := Aut_{Grpd}(\mathbf{B}K) \,.

Its objects are the ordinary automorphisms of KK in Grp, while its 2-morphisms go between two automorphisms that differ by an inner automorphism.

Accordingly, the crossed module corresponding to the 2-group AUT(K)AUT(K) is

[AUT(K)]=(K Ad Aut(K) ), [AUT(K)] = \left( \array{ K &\stackrel{Ad}{\to}& Aut(K) \\ } \right) \,,

where the boundary map is the one that sends each element kKk \in K to the inner automorphism given by conjugation with kk:

Ad(k):qkqk 1. Ad(k) : q \mapsto k q k^{-1} \,.

From congruence relations

Perhaps the simplest example of such a structure is a congruence relation on a group GG. If \sim is a congruence relation on GG, then we form the 2-group by setting C 0=GC_0 = G and C 1C_1 to be the group of pairs (a,b)(a,b) with aba\sim b. That this is a group follows from the definition of congruence given in the above reference. The two maps ss and tt are defined by s(a,b)=as(a,b) = a, t(a,b)=bt(a,b) = b, whilst i(a)=(a,a)i(a) = (a,a). The pullback is a subgroup of C 1×C 1C_1\times C_1 given by all ‘pairs of pairs’ ((a,b),(b,c))((a,b),(b,c)) and the composition homomorphism sends such a pair to (a,c)(a,c). The other properties are easy to check.

Any congruence relation corresponds to a normal subgroup, given by those elements aa that are congruent to the identity element of GG, so that eae\sim a. Likewise given a normal subgroup NN of GG you get a congruence, with aba \sim b iff b 1ab^{-1} a (or equivalently, ab 1a b^{-1}) belongs to NN.

References

See also the references at 2-group.

The equivalence between strict 2-groups and crossed modules is discussed in