strict 2-group



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 1G_1 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.


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

  • a collection of group homomorphisms of the form

    C 1s,tC 0iC 1 C_1 \stackrel{s,t}{\to} C_0 \stackrel{i}{\to} C_1

    such that the composites sis\cdot i and tit\cdot i are the identity morphisms on C 0C_0, and such that, writing C 1× t,sC 1C_1 \times_{t,s} C_1 for the pullback,

    C 1× t,sC 1 C 1 t C 1 s C 0 \array{ C_1 \times_{t,s} C_1 &\to& C_1 \\ \downarrow && \downarrow^{t} \\ C_1 &\stackrel{s}{\to}& C_0 }

    there is, in addition, a homomorphism

    C 1× t,sC 1compC 1 C_1 \times_{t,s} C_1 \stackrel{comp}{\to} C_1

    “respecting ss and tt”;

  • such that the composition compcomp is associative and unital with respect to ii “in the obvious way”.

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

  • BG\mathbf{B}G has a single object \bullet;

  • The hom-groupoid BG(,)=G\mathbf{B}G(\bullet,\bullet) = G is the 2-group GG itself, regarded as a groupoid;

  • the horizontal composition in BG\mathbf{B}G is given by the group product operation on GG.

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).

  • BG\mathbf{B}G has a single object \bullet;

  • The set of 1-morphisms of BG\mathbf{B}G is the group G 1G_1:

    1Mor BG(,):=G 1. 1Mor_{\mathbf{B}G}(\bullet, \bullet) := G_1 \,.

    For gG 1g \in G_1 we write g\bullet \stackrel{g}{\to} \bullet for the corresponding 1-morphism in BG\mathbf{B}G;

  • Compositition of 1-morphisms is given by the product operation in G 1G_1. There are two choice for the order in which to form the product.

    • (convention F) horizontal composition is given by

      (g 1g 2)=(g 1g 2) (\bullet \stackrel{g_1}{\to} \bullet \stackrel{g_2}{\to} \bullet) \;\; = \;\; (\bullet \stackrel{g_1 g_2}{\to} \bullet)
    • (convention B) horizontal composition is given by

      (g 1g 2)=(g 2g 1) (\bullet \stackrel{g_1}{\to} \bullet \stackrel{g_2}{\to} \bullet) \;\; = \;\; (\bullet \stackrel{g_2 g_1}{\to} \bullet)
  • The set of 2-morphisms of BG\mathbf{B}G is the cartesian product G 1×G 2G_1 \times G_2 where

    • the source operation is projection on the first factor

      s:=p 1:G 1×G 2G 1 s := p_1 : G_1 \times G_2 \to G_1
    • the target operation on morphisms starting at the identity morphism is the boundary map δ:G 2G 1\delta : G_2 \to G_1 of the crossed module combined with the product in G 1G_1

      t Id×G 2=δ t|_{{Id}\times G_2} = \delta

    So in diagrams this means that a 2-morphism corresponding to (Id,k)G 1×G 2(Id, k) \in G_1 \times G_2 is labelled as

    Id h δ(h). \array{ & \nearrow \searrow^{\mathrlap{Id}} \\ \bullet &\Downarrow^{\mathrlap{h}}& \bullet \\ & \searrow \nearrow_{\mathrlap{\delta(h)}} } \,.

    The target of general 2-morphisms labeled by hh and starting at some gg is either δ(h)g\delta(h)g of gδ(h)g \delta(h), depending on the choice of conventions discussed in the following.

  • Horizontal composition of 1-morphisms with 2-morphisms (“whiskering”) is determined by the rule

    • (convention R)

      Id h g := g h \array{ & \nearrow \searrow^{Id} \\ \bullet &\Downarrow^h& \bullet &\stackrel{g}{\to}& \bullet \\ & \searrow \nearrow } \;\; := \;\; \array{ & \nearrow \searrow^{g} \\ \bullet &\Downarrow^h& \bullet \\ & \searrow \nearrow }
      Id g h := g α(g)(h) \array{ && & \nearrow \searrow^{Id} \\ \bullet &\stackrel{g}{\to}& \bullet &\Downarrow^h& \bullet \\ && & \searrow \nearrow } \;\; := \;\; \array{ & \nearrow \searrow^{g} \\ \bullet &\Downarrow^{\alpha(g)(h)}& \bullet \\ & \searrow \nearrow }
    • (convention L)

      Id g h := g h \array{ && & \nearrow \searrow^{Id} \\ \bullet &\stackrel{g}{\to}& \bullet &\Downarrow^h& \bullet \\ && & \searrow \nearrow } \;\; := \;\; \array{ & \nearrow \searrow^{g} \\ \bullet &\Downarrow^h& \bullet \\ & \searrow \nearrow }
      Id h g := g α(g)(h) \array{ & \nearrow \searrow^{Id} \\ \bullet &\Downarrow^h& \bullet &\stackrel{g}{\to}& \bullet \\ & \searrow \nearrow } \;\; := \;\; \array{ & \nearrow \searrow^{g} \\ \bullet &\Downarrow^{\alpha(g)(h)}& \bullet \\ & \searrow \nearrow }
  • 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

      Id Id h 1 h 2 = Id h 1h 2 \array{ & \nearrow \searrow^{\mathrlap{Id}} & & \nearrow \searrow^{\mathrlap{Id}} \\ \bullet &\Downarrow^{h_1}& \bullet &\Downarrow^{h_2}& \bullet \\ & \searrow \nearrow && \searrow \nearrow } \;\;\; = \;\;\; \array{ & \nearrow \searrow^{\mathrlap{Id}} \\ \bullet &\Downarrow^{h_1 h_2}& \bullet \\ & \searrow \nearrow }
    • in convention B

      Id Id h 1 h 2 = Id h 2h 1 \array{ & \nearrow \searrow^{\mathrlap{Id}} & & \nearrow \searrow^{\mathrlap{Id}} \\ \bullet &\Downarrow^{h_1}& \bullet &\Downarrow^{h_2}& \bullet \\ & \searrow \nearrow && \searrow \nearrow } \;\;\; = \;\;\; \array{ & \nearrow \searrow^{\mathrlap{Id}} \\ \bullet &\Downarrow^{h_2 h_1}& \bullet \\ & \searrow \nearrow }

    Notice that this is compatible with the source-target maps due to the fact that that δ\delta 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 G 2G_2, in the following order

    (in convention L B)

    h 1 h 2 = h 2h 1 . \array{ & \nearrow &\Downarrow^{\mathrlap{h_1}}& \searrow \\ \bullet &&\to&& \bullet \\ & \searrow &\Downarrow^{\mathrlap{h_2}}& \nearrow } \;\;\; = \;\;\; \array{ & \nearrow \searrow \\ \bullet &\Downarrow^{\mathrlap{h_2 h_1}}& \bullet \\ & \searrow \nearrow } \,.


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

  • objects are 1-isomorphisms aaa \to a in CC;

  • morphisms are 2-isomorphisms between these 1-isomorphisms.

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} \,.

Fom 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.


See also the references at 2-group.

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

  • Ronnie Brown and C. Spencer, G-groupoids, crossed modules and the fundamental groupoid of a topological group, Proc. Kon. Ned. Akad. v. Wet, 79, (1976), 296–302.)

Revised on February 18, 2014 17:26:53 by Mark Gomer? (