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\begin{document}

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\section*{strict 2-group}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{higher_group_theory}{}\paragraph*{{Higher group theory}}\label{higher_group_theory}

[[!include group theory - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definitions}{Definitions}\dotfill \pageref*{definitions} \linebreak
\noindent\hyperlink{expanding_the_definition}{Expanding the definition}\dotfill \pageref*{expanding_the_definition} \linebreak
\noindent\hyperlink{in_terms_of_strict_2groupoids}{In terms of strict 2-groupoids}\dotfill \pageref*{in_terms_of_strict_2groupoids} \linebreak
\noindent\hyperlink{InTermsOfCrossedModules}{In terms of crossed modules}\dotfill \pageref*{InTermsOfCrossedModules} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{from_crossed_modules}{From crossed modules}\dotfill \pageref*{from_crossed_modules} \linebreak
\noindent\hyperlink{automorphism_2groups}{Automorphism 2-groups}\dotfill \pageref*{automorphism_2groups} \linebreak
\noindent\hyperlink{from_congruence_relations}{From congruence relations}\dotfill \pageref*{from_congruence_relations} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

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

A strict 2-group is a [[group object]] [[internalization|internal]] to the [[category]] [[Grpd]] of [[groupoid]]s (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-group]]s 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-group]]s.

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 [[k-morphism|2-morphism]]s 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 [[bicategory|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 \emph{homomorphisms} of these $2$-groups to be weak. (However, this theorem may not apply to structured $2$-groups, such as [[Lie 2-group]]s.)

Strict 2-groups are also equivalently encoded in terms of [[crossed module]]s $(G_2 \to G_1)$ of ordinary groups: $G_1$ is the group of [[object]]s of the groupoid $G$ and $G_2$ the group of [[morphism]]s 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.

\hypertarget{definitions}{}\subsection*{{Definitions}}\label{definitions}

A \textbf{strict [[2-group]]} is equivalently:

\begin{itemize}%
\item an [[internal category]] in [[Grp]],
\item an [[internal groupoid]] in [[Grp]],
\item an internal [[group object]] in [[Cat]],
\item an internal [[group object]] in [[Grpd]],
\item a [[strict 2-groupoid]] with one object.

\end{itemize}
\hypertarget{expanding_the_definition}{}\subsection*{{Expanding the definition}}\label{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

\begin{itemize}%
\item a collection of group homomorphisms of the form

\begin{displaymath}
C_1 \stackrel{s,t}{\to} C_0 \stackrel{i}{\to} C_1
\end{displaymath}
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,

\begin{displaymath}
\itexarray{
    C_1 \times_{t,s} C_1 &\to& C_1
    \\
    \downarrow && \downarrow^{t}
    \\
    C_1 &\stackrel{s}{\to}& C_0
  }
\end{displaymath}
there is, in addition, a homomorphism

\begin{displaymath}
C_1 \times_{t,s} C_1 \stackrel{comp}{\to} C_1
\end{displaymath}
``respecting $s$ and $t$'';


\item such that the \emph{composition} $comp$ is associative and unital with respect to $i$ ``in the obvious way''.



\end{itemize}
\hypertarget{in_terms_of_strict_2groupoids}{}\subsubsection*{{In terms of strict 2-groupoids}}\label{in_terms_of_strict_2groupoids}

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

\begin{itemize}%
\item $\mathbf{B}G$ has a single object $\bullet$;


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


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



\end{itemize}
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 \emph{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 ${*} \to \mathbf{B}G$.

\hypertarget{InTermsOfCrossedModules}{}\subsubsection*{{In terms of crossed modules}}\label{InTermsOfCrossedModules}

We describe how a [[crossed module]]

\begin{displaymath}
[\mathbf{B}G] = (G_2 \stackrel{\delta}{\to} G_1)
\end{displaymath}
with action

\begin{displaymath}
\alpha : G_1 \to Aut(G_2)
\end{displaymath}
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 $\alpha : G_1 \to Aut(G_2)$.

\begin{itemize}%
\item $\mathbf{B}G$ has a single [[object]] $\bullet$;


\item The set of [[k-morphism|1-morphisms]] of $\mathbf{B}G$ is the group $G_1$:

\begin{displaymath}
1Mor_{\mathbf{B}G}(\bullet, \bullet) := G_1
  \,.
\end{displaymath}
For $g \in G_1$ we write $\bullet \stackrel{g}{\to} \bullet$ for the corresponding 1-morphism in $\mathbf{B}G$;


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

\begin{itemize}%
\item \textbf{(convention F)} horizontal composition is given by

\begin{displaymath}
(\bullet \stackrel{g_1}{\to} \bullet
  \stackrel{g_2}{\to} \bullet)
  \;\; 
  =
  \;\;
  (\bullet \stackrel{g_1 g_2}{\to} \bullet)
\end{displaymath}

\item \textbf{(convention B)} horizontal composition is given by

\begin{displaymath}
(\bullet \stackrel{g_1}{\to} \bullet
  \stackrel{g_2}{\to} \bullet)
  \;\; 
  =
  \;\;
  (\bullet \stackrel{g_2 g_1}{\to} \bullet)
\end{displaymath}


\end{itemize}

\item The [[set]] of [[k-morphism|2-morphism]]s of $\mathbf{B}G$ is the cartesian [[product]] $G_1 \times G_2$ where

\begin{itemize}%
\item the source operation is projection on the first factor

\begin{displaymath}
s := p_1 : G_1 \times G_2 \to G_1
\end{displaymath}

\item 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 $G_1$

\begin{displaymath}
t|_{{Id}\times G_2} = \delta
\end{displaymath}


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

\begin{displaymath}
\itexarray{
     &  \nearrow \searrow^{\mathrlap{Id}}
     \\
    \bullet &\Downarrow^{\mathrlap{h}}& \bullet
    \\
     &  \searrow \nearrow_{\mathrlap{\delta(h)}}
  }
  \,.
\end{displaymath}
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.


\item Horizontal composition of 1-morphisms with 2-morphisms (``whiskering'') is determined by the rule

\begin{itemize}%
\item \textbf{(convention R)}

\begin{displaymath}
\itexarray{
    & \nearrow \searrow^{Id}  
    \\
    \bullet &\Downarrow^h& \bullet 
    &\stackrel{g}{\to}&
    \bullet
    \\
    & \searrow \nearrow
  }
  \;\;
  :=
  \;\;
  \itexarray{
    & \nearrow \searrow^{g}
    \\
    \bullet &\Downarrow^h& \bullet
    \\
    & \searrow \nearrow
  }
\end{displaymath}
\begin{displaymath}
\itexarray{
   && & \nearrow \searrow^{Id}  
    \\
    \bullet
    &\stackrel{g}{\to}&
    \bullet &\Downarrow^h& \bullet 
    \\
    && & \searrow \nearrow
   }
  \;\;
  :=
  \;\;
  \itexarray{
    & \nearrow \searrow^{g}
    \\
    \bullet &\Downarrow^{\alpha(g)(h)}& \bullet
    \\
    & \searrow \nearrow
  }
\end{displaymath}

\item \textbf{(convention L)}

\begin{displaymath}
\itexarray{
   && & \nearrow \searrow^{Id}  
    \\
    \bullet
    &\stackrel{g}{\to}&
    \bullet &\Downarrow^h& \bullet 
    \\
    && & \searrow \nearrow
   }
  \;\;
  :=
  \;\;
  \itexarray{
    & \nearrow \searrow^{g}
    \\
    \bullet &\Downarrow^h& \bullet
    \\
    & \searrow \nearrow
  }
\end{displaymath}
\begin{displaymath}
\itexarray{
    & \nearrow \searrow^{Id}  
    \\
    \bullet &\Downarrow^h& \bullet 
    &\stackrel{g}{\to}&
    \bullet
    \\
    & \searrow \nearrow
  }
  \;\;
  :=
  \;\;
  \itexarray{
    & \nearrow \searrow^{g}
    \\
    \bullet &\Downarrow^{\alpha(g)(h)}& \bullet
    \\
    & \searrow \nearrow
  }
\end{displaymath}


\end{itemize}

\item Horizontal composition of 2-morphisms starting at the identity 1-morphism is fixed by the convention chosen for composition of 1-morphisms

\begin{itemize}%
\item in \textbf{convention F}

\begin{displaymath}
\itexarray{
    & \nearrow \searrow^{\mathrlap{Id}} & 
    &
  \nearrow \searrow^{\mathrlap{Id}}
    \\
    \bullet &\Downarrow^{h_1}&
    \bullet
    &\Downarrow^{h_2}&
    \bullet
    \\
    & \searrow \nearrow && \searrow \nearrow
  }
  \;\;\;
  =
  \;\;\;
  \itexarray{
    & \nearrow \searrow^{\mathrlap{Id}}
    \\
    \bullet &\Downarrow^{h_1 h_2}&
    \bullet
    \\
    & \searrow \nearrow
  }
\end{displaymath}

\item in \textbf{convention B}

\begin{displaymath}
\itexarray{
    & \nearrow \searrow^{\mathrlap{Id}} & 
    &
   \nearrow \searrow^{\mathrlap{Id}}
    \\
    \bullet &\Downarrow^{h_1}&
    \bullet
    &\Downarrow^{h_2}&
    \bullet
    \\
    & \searrow \nearrow && \searrow \nearrow
  }
  \;\;\;
  =
  \;\;\;
  \itexarray{
    & \nearrow \searrow^{\mathrlap{Id}}
    \\
    \bullet &\Downarrow^{h_2 h_1}&
    \bullet
    \\
    & \searrow \nearrow
  }
\end{displaymath}


\end{itemize}
Notice that this is compatible with the source-target maps due to the fact that that $\delta$ is a group homomorphism.


\item With these choices made, all other compositions are now fixed by use of the exchange law:


\item Vertical composition of composable 2-morphisms is given, on the labels, by the product in $G_2$, in the following order

\textbf{(in convention L B)}

\begin{displaymath}
\itexarray{
     & \nearrow &\Downarrow^{\mathrlap{h_1}}& \searrow
     \\
     \bullet &&\to&& \bullet
     \\
     & \searrow &\Downarrow^{\mathrlap{h_2}}& \nearrow
  }
  \;\;\;
  =
  \;\;\;
  \itexarray{
    & \nearrow \searrow
    \\
    \bullet
    &\Downarrow^{\mathrlap{h_2 h_1}}& \bullet
    \\
    & \searrow \nearrow
  }   
  \,.
\end{displaymath}


\end{itemize}
\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\hypertarget{from_crossed_modules}{}\subsubsection*{{From crossed modules}}\label{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.

\hypertarget{automorphism_2groups}{}\subsubsection*{{Automorphism 2-groups}}\label{automorphism_2groups}

For $a$ any [[object]] in a [[strict 2-category]] $C$, there is the strict \textbf{automorphism 2-group} $Aut_C(a)$ whose

\begin{itemize}%
\item objects are 1-[[isomorphism]]s $a \to a$ in $C$;


\item morphisms are 2-isomorphisms between these 1-isomorphisms.



\end{itemize}
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 \textbf{automorphism 2-group of the group $K$}

\begin{displaymath}
AUT(K) := Aut_{Grpd}(\mathbf{B}K)
  \,.
\end{displaymath}
Its objects are the ordinary [[automorphism]]s 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

\begin{displaymath}
[AUT(K)] =
  \left(
    \itexarray{
      K &\stackrel{Ad}{\to}& Aut(K)
      \\
    }
  \right)
  \,,
\end{displaymath}
where the boundary map is the one that sends each element $k \in K$ to the inner automorphism given by conjugation with $k$:

\begin{displaymath}
Ad(k) : q \mapsto k q k^{-1}
  \,.
\end{displaymath}
\hypertarget{from_congruence_relations}{}\subsubsection*{{From congruence relations}}\label{from_congruence_relations}

Perhaps the \emph{simplest example} of such a structure is a [[congruence|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 $N$.

\hypertarget{references}{}\subsection*{{References}}\label{references}

See also the references at [[2-group]].

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

\begin{itemize}%
\item [[Ronnie Brown]] and C. Spencer, \emph{G-groupoids, crossed modules and the fundamental groupoid of a topological group}, Proc. Kon. Ned. Akad. v. Wet, 79, (1976), 296--302.)

\end{itemize}
[[!redirects strict 2-groups]]



\end{document}