group theory

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

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

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

• a collection of group homomorphisms of the form

$C_1 \stackrel{s,t}{\to} C_0 \stackrel{i}{\to} C_1$

such that the composites $s\cdot i$ and $t\cdot i$ are the identity morphisms on $C_0$, and such that, writing $C_1 \times_{t,s} C_1$ for the pullback,

$\array{ C_1 \times_{t,s} C_1 &\to& C_1 \\ \downarrow && \downarrow^{t} \\ C_1 &\stackrel{s}{\to}& C_0 }$

there is, in addition, a homomorphism

$C_1 \times_{t,s} C_1 \stackrel{comp}{\to} C_1$

“respecting $s$ and $t$”;

• such that the composition $comp$ is associative and unital with respect to $i$ “in the obvious way”.

### In terms of strict 2-groupoids

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

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

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

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

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 $\mathbf{B}G$ in a way that respects morphisms between these objects, one needs to think of $\mathbf{B}G$ as a pointed object equipped with its unique pointing ${*} \to \mathbf{B}G$.

### In terms of crossed modules

We describe how a crossed module

$[\mathbf{B}G] = (G_2 \stackrel{\delta}{\to} G_1)$

with action

$\alpha : G_1 \to Aut(G_2)$

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

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

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

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

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

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

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

• (convention F) horizontal composition is given by

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

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

• the source operation is projection on the first factor

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

$t|_{{Id}\times G_2} = \delta$

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

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

The target of general 2-morphisms labeled by $h$ and starting at some $g$ is either $\delta(h)g$ of $g \delta(h)$, 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)

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

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

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

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

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

(in convention L B)

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

## 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 $a$ any object in a strict 2-category $C$, there is the strict automorphism 2-group $Aut_C(a)$ whose

• objects are 1-isomorphisms $a \to a$ in $C$;

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

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

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

Its objects are the ordinary automorphisms of $K$ 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 $AUT(K)$ is

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

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

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

### From congruence relations

Perhaps the simplest example of such a structure is a congruence relation on a group $G$. If $\sim$ is a congruence relation on $G$, then we form the 2-group by setting $C_0 = G$ and $C_1$ to be the group of pairs $(a,b)$ with $a\sim b$. That this is a group follows from the definition of congruence given in the above reference. The two maps $s$ and $t$ are defined by $s(a,b) = a$, $t(a,b) = b$, whilst $i(a) = (a,a)$. The pullback is a subgroup of $C_1\times C_1$ given by all ‘pairs of pairs’ $((a,b),(b,c))$ and the composition homomorphism sends such a pair to $(a,c)$. The other properties are easy to check.

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