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\section*{2-group}

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

\hypertarget{group_theory}{}\paragraph*{{Group Theory}}\label{group_theory}

[[!include group theory - contents]]

\hypertarget{higher_category_theory}{}\paragraph*{{Higher category theory}}\label{higher_category_theory}

[[!include higher category theory - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{strict_groups}{Strict $2$-groups}\dotfill \pageref*{strict_groups} \linebreak
\noindent\hyperlink{weak_groups}{Weak $2$-groups}\dotfill \pageref*{weak_groups} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{PresentationByCrossedModules}{Presentation by crossed modules}\dotfill \pageref*{PresentationByCrossedModules} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{specific_examples}{Specific examples}\dotfill \pageref*{specific_examples} \linebreak
\noindent\hyperlink{picard_2group}{Picard 2-group}\dotfill \pageref*{picard_2group} \linebreak
\noindent\hyperlink{automorphism_2groups}{Automorphism 2-groups}\dotfill \pageref*{automorphism_2groups} \linebreak
\noindent\hyperlink{inner_automorphism_2groups}{Inner automorphism 2-groups}\dotfill \pageref*{inner_automorphism_2groups} \linebreak
\noindent\hyperlink{string_2group}{String 2-group}\dotfill \pageref*{string_2group} \linebreak
\noindent\hyperlink{platonic_2group}{Platonic 2-group}\dotfill \pageref*{platonic_2group} \linebreak
\noindent\hyperlink{ExamplesForEquivalencesOf2Groups}{Equivalences of 2-groups}\dotfill \pageref*{ExamplesForEquivalencesOf2Groups} \linebreak
\noindent\hyperlink{FromInclusionsOfNormalSubgroups}{From inclusions of normal subgroups}\dotfill \pageref*{FromInclusionsOfNormalSubgroups} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{References}{References}\dotfill \pageref*{References} \linebreak


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

The notion of \emph{2-group} is a [[vertical categorification]] of the notion of \emph{[[group]]}.

It is the special case of an [[n-group]] for $n=2$, equivalently an [[∞-group]] which is [[n-truncated|1-truncated]]. Under the [[looping and delooping]]-equivalence, 2-groups are equivalent to [[pointed object|pointed]] [[n-connected|connected]] [[homotopy n-type|homotopy 2-types]].

Somewhat more precisely, a \emph{$2$-group} is a [[group object]] in the [[(2,1)-category]] of [[groupoids]]. Equivalently, it is a [[monoidal category|monoidal groupoid]] in which the [[tensor product]] with any [[object]] has an [[inverse]] up to [[isomorphism]]. Also equivalently, by the [[looping and delooping]]-equivalence, it is a [[pointed object|pointed]] [[2-groupoid]] with a single [[equivalence class]] of objects.

Like other notions of [[higher category theory]], $2$-groups come in weak and strict forms, depending on how you interpret the above.

\hypertarget{strict_groups}{}\subsubsection*{{Strict $2$-groups}}\label{strict_groups}

The earliest version studied is that of [[strict 2-group]]s.

A \textbf{strict $2$-group} consists of:

\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}
See [[strict 2-group]] for further discussion and examples.

\hypertarget{weak_groups}{}\subsubsection*{{Weak $2$-groups}}\label{weak_groups}

A \textbf{weak $2$-group}, or simply \textbf{$2$-group}, is a (weak) [[monoidal category]] where every morphism is invertible and \emph{such that}:

\begin{itemize}%
\item given any object $x$, there exists an object $x^{-1}$ such that the monoidal products $x \otimes x^{-1}$ and $x^{-1} \otimes x$ are each [[isomorphism|isomorphic]] to the monoidal unit $1$.

\end{itemize}
A \textbf{coherent $2$-group} is a monoidal category where every morphism is invertible and \emph{equipped with}:

\begin{itemize}%
\item for each object $x$ a specific object $x^{-1}$ and specific [[isomorphism]]s from $x \otimes x^{-1}$ and $x^{-1} \otimes x$ to $1$ which form an [[adjoint equivalence]].

\end{itemize}
A theorem in HDA V (see references) shows that every weak $2$-group may be made coherent. For purposes of [[internalization]], one probably wants to use the coherent version.

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

\begin{defn}
\label{}\hypertarget{}{}
The [[(2,1)-category]] $2Grp$ of \textbf{2-groups} is equivalently

\begin{itemize}%
\item the [[full sub-(∞,1)-category|full sub-(2,1)-category]] of that of [[monoidal categories]] and [[strong monoidal functors]] on those that are [[groupoids]] and whose [[tensor product]] has weak [[inverses]] for each object;


\item the [[full sub-(∞,1)-category]] of that of [[∞-groups]] on the [[n-truncated|1-truncated]] objects;


\item the [[full sub-(∞,1)-category]] of that of [[group object in an (∞,1)-category|group objects]] in [[∞Grpd]] on the [[n-truncated|1-truncated]] objects;


\item the [[full sub-(∞,1)-category]]

\begin{displaymath}
\infty Grpd_{1 \leq \bullet \leq 2}^{*/}
  \hookrightarrow
  \infty Grpd^{*/}
\end{displaymath}
of [[∞Grpd]]$^{*/}$ on those objects which are both [[n-connected|connected]] as well as [[n-truncated|2-truncated]].



\end{itemize}
\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
The last equivalent characterization is related to the previous ones by the [[looping and delooping]]-equivalence

\begin{displaymath}
\itexarray{
    Grp(\infty Grpd)
    &\stackrel{\overset{\Omega}{\leftarrow}}{\underset{\mathbf{B}}{\to}}&
    \infty Grpd^{*/}_{\geq 1}
    \\
    \uparrow^{\mathrlap{full\;inc.}}
    &&
    \uparrow^{\mathrlap{full\;inc.}}    
    \\
    2 Grp
    &\stackrel{\overset{\Omega}{\leftarrow}}{\underset{\mathbf{B}}{\to}}&
    2Grpd_{\geq 1}^{*/}
  }
  \,.
\end{displaymath}
Here $(-)^{*/}$ denotes taking [[pointed objects]], hence the [[over-(∞,1)-category|slice under]] the point, and $(-)_{\geq}$ denotes the full [[sub-(∞,1)-category|full inclusion]] on [[n-connected|connected]] objects.

\end{remark}
By replacing in the last of these equivalent characterizations the ambient [[(∞,1)-topos]] [[∞Grpd]] with any other one, to be denoted $\mathbf{H}$, obtains notions of 2-groups with extra structure. For instance for $\mathbf{H} =$ [[Smooth∞Grpd]] the $(\infty,1)$-topos of [[smooth ∞-groupoids]] one obtains:

\begin{defn}
\label{}\hypertarget{}{}
The [[(2,1)-category]] $Smooth2Grp$ of \textbf{smooth 2-groups} is

\begin{displaymath}
\itexarray{
    Grp(Smooth \infty Grpd)
    &\stackrel{\overset{\Omega}{\leftarrow}}{\underset{\mathbf{B}}{\to}}&
    Smooth\infty Grpd^{*/}_{\geq 1}
    \\
    \uparrow^{\mathrlap{full\;inc.}}
    &&
    \uparrow^{\mathrlap{full\;inc.}}    
    \\
    Smooth 2 Grp
    &\stackrel{\overset{\Omega}{\leftarrow}}{\underset{\mathbf{B}}{\to}}&
    Smooth2Grpd_{\geq 1}^{*/}
  }
  \,.
\end{displaymath}
\end{defn}
Below in \hyperlink{PresentationByCrossedModules}{presentation by crossed modules} are discussed more explict presentations of $2Grp$ and $Smooth2Grpd$ etc. by explicit algebraic data.

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{PresentationByCrossedModules}{}\subsubsection*{{Presentation by crossed modules}}\label{PresentationByCrossedModules}

By the discussion there, every \emph{[[∞-group]]} has a \emph{presentation} by a [[simplicial group]]. More precisely, the [[(∞,1)-category]], $\infty Grp$, is [[presentable (∞,1)-category|presented]] by the [[model structure on simplicial groups]] (for instance under [[simplicial localization]])

\begin{displaymath}
\infty Grpd \simeq L_W Grp^{\Delta^{op}}
  \,.
\end{displaymath}
Moreover, if $G \in Grp^{\Delta^{op}}$ is an [[n-group]], then it is equivalent to a [[coskeleton|n-coskeletal]] simplicial group. For $n = 2$ one finds that these are naturally identified with \emph{[[crossed modules]]} of groups (see there for more details).

In conclusion, this means that

\begin{prop}
\label{PresentationOf2GrpByCrossedModules}\hypertarget{PresentationOf2GrpByCrossedModules}{}
The [[(2,1)-category]] $2Grp$ of 2-groups is [[equivalence of (infinity,1)-categories|equivalent]] to the [[simplicial localization]] of the [[category with weak equivalences]] whose

\begin{itemize}%
\item objects are [[crossed modules]]


\item morphisms are [[homomorphisms]] of crossed modules;


\item weak equivalences are those morphisms of crossed modules which correspond to [[weak homotopy equivalences]] of the corresponding simplicial groups.



\end{itemize}
\end{prop}
A straightforward analysis shows that

\begin{prop}
\label{HomotopyGroupsOfCrossedModule}\hypertarget{HomotopyGroupsOfCrossedModule}{}
For $(G_1 \stackrel{\delta}{\to} G_0, G_0 \stackrel{\alpha}{\to} Aut(G_1))$ a [[crossed module]], the [[homotopy groups]] of the corresponding [[2-group]]/[[simplicial group]] are

\begin{itemize}%
\item $\pi_0 = G_0 / im(\delta)$ (the [[quotient]] of $G_0$ by the [[image]] of $\delta$, which is necessarily a [[normal subgroup]] of $G_0$);


\item $\pi_1 = ker(\delta)$ (the [[kernel]] of $\delta$).



\end{itemize}
Accordingly, a weak equivalence of crossed modules $f : G \to H$ is a morphism of crossed modules which induces an [[isomorphism]] of kernel and cokernel of $\delta_G$ with that of $\delta_H$.

\end{prop}
Similar statements hold for 2-groups with extra structure. For instance the $(2,1)$-category $Smooth2Grp$ of smooth 2-groups is equivalent to the [[simplicial localization]] of the category whose

\begin{itemize}%
\item objects are [[sheaves]] of [[crossed modules]] on [[CartSp]]${}_{smooth}$;


\item weak equivalences are those morphisms of sheaves of crossed modules which on every [[stalk]] induce weak equivalences of crossed modules as above.



\end{itemize}
(See the discussion at [[Smooth∞Grpd]] for more on this.)

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

\hypertarget{specific_examples}{}\subsubsection*{{Specific examples}}\label{specific_examples}

\hypertarget{picard_2group}{}\paragraph*{{Picard 2-group}}\label{picard_2group}

\begin{itemize}%
\item [[Picard 2-group]]

\end{itemize}
\hypertarget{automorphism_2groups}{}\paragraph*{{Automorphism 2-groups}}\label{automorphism_2groups}

For $C$ any [[2-category]] and $c \in C$ any object of it, the category $Aut_C(c) \subset Hom_C(c,c)$ of auto-equivalences of $c$ and invertible 2-morphisms between these is naturally a 2-group, whose group product comes from the horizontal composition in $C$.

If $C$ is a [[strict 2-category]] there is the notion of strict [[automorphism 2-group]]. See there for more details on that case.

For instance if $C = Grp_2 \subset Grpd$ is the 2-category of [[group]] obtained by regarding groups as one-object [[groupoid]]s, then for $H \in Grp$ a group, its automorphism 2-group obtained this way is the strict 2-group

\begin{displaymath}
AUT(H) := Aut_{Grp_2}(H)
\end{displaymath}
corresponding to the [[crossed module]] $(H \stackrel{Ad}{\to} Aut(H))$, where $Aut(H)$ is the ordinary [[automorphism group]] of $H$.

\hypertarget{inner_automorphism_2groups}{}\paragraph*{{Inner automorphism 2-groups}}\label{inner_automorphism_2groups}

See [[inner automorphism 2-group]].

\hypertarget{string_2group}{}\paragraph*{{String 2-group}}\label{string_2group}

See [[string 2-group]].

\hypertarget{platonic_2group}{}\paragraph*{{Platonic 2-group}}\label{platonic_2group}

See \emph{[[Platonic 2-group]]}

\hypertarget{ExamplesForEquivalencesOf2Groups}{}\subsubsection*{{Equivalences of 2-groups}}\label{ExamplesForEquivalencesOf2Groups}

We discuss some weak equivalences in the [[category with weak equivalences]] of [[crossed modules]] and crossed module homomorphisms, which presents $2Grp$ by the discussion \hyperlink{PresentationByCrossedModules}{above}.

\hypertarget{FromInclusionsOfNormalSubgroups}{}\paragraph*{{From inclusions of normal subgroups}}\label{FromInclusionsOfNormalSubgroups}

Let $G$ be a [[group]] and $N \hookrightarrow G$ the inclusion of a [[normal subgroup]]. Equipped with the canonical [[action]] of $G$ on $N$ by [[conjugation]], this inclusion constitutes a [[crossed module]]. There is a canonical morphism of crossed modules from $(N \hookrightarrow G)$ to $(1 \to G/N)$, hence to the ordinary [[quotient]] group, regarded as a crossed module.

\begin{prop}
\label{}\hypertarget{}{}
The morphism $(N \hookrightarrow G) \to G/N$ is a weak equivalence of crossed modules, prop. \ref{PresentationOf2GrpByCrossedModules}. Accordingly, it presents an [[equivalence in an (infinity,1)-category|equivalence]] of 2-groups.

\end{prop}
\begin{proof}
The canonical morphism in question is given by the commuting diagram of groups

\begin{displaymath}
\itexarray{
    N &\stackrel{f_1}{\to}& 1
    \\
    \downarrow && \downarrow
    \\
    G &\stackrel{f_0}{\to}& G/N
  }
  \,.
\end{displaymath}
By prop. \ref{HomotopyGroupsOfCrossedModule} we need to check that this induces an isomorphism on the [[kernel]] and [[cokernel]] of the vertical morphisms.

The kernel of the left vertical morphism is the trivial group, because $N \hookrightarrow G$ is an inclusion, by definition. Clearly also the kernel of the right vertical morphisms is the trivial group. Hence $f_1$ restricted to the kernels is the unique morphism from the trivial group to itself, hence is an isomrphism.

Moreover, the cokernel of the left vertical morphism is of course the quotient $G/N$ and $f_0$, being the quotient map, is manifestly an isomorphism on cokernels.

\end{proof}
This class of weak equivalence plays an important role as constituting [[∞-anafunctor|2-anafunctors]] that exhibit long [[fiber sequence]] extensions of [[short exact sequences]] of [[central extensions]] of groups.

\begin{prop}
\label{}\hypertarget{}{}
Let $A \to \hat G$ be the inclusion of a [[center|central]] subgroup, exhibiting a [[central extension]] $A \to \hat G \to G$ with $G := \hat G/A$. Then this [[short exact sequence]] of groups extends to a long [[fiber sequence]] of 2-groups

\begin{displaymath}
A \to \hat G \to G \to \mathbf{B}A \to \mathbf{B}\hat G \to \mathbf{B}G \to \mathbf{B}^2 A
  \,,
\end{displaymath}
where $\mathbf{B}A$ denotes the 2-group given by the [[crossed module]] $(A \to 1)$, and similarly for the other cases.

Here the [[connecting homomorphism]] $G \to \mathbf{B}A$ is presented in the category of crossed modules by a zig-zag / [[anafunctor]] whose left leg is the above weak equivalence:

\begin{displaymath}
(1 \to G) \stackrel{\simeq}{\leftarrow}
  (A \to \hat G)
   \to 
  (A \to 1)
  \,.
\end{displaymath}
\end{prop}
\begin{example}
\label{}\hypertarget{}{}
For smooth 2-groups, useful examples of the above are smooth refinements of various [[universal characteristic classes]]:

\begin{itemize}%
\item the second [[Stiefel-Whitney class]]

\begin{displaymath}
w_2 : \mathbf{B}SO \to \mathbf{B}^2\mathbb{Z}_2
\end{displaymath}
is induced this way from the central extension $\mathbb{Z}_2 \to Spin \to SO$ of the [[special orthogonal group]] by the [[spin group]];


\item the first [[Chern class]]

\begin{displaymath}
c_1 : \mathbf{B}U(1) \to \mathbf{B}^2 \mathbb{Z}
\end{displaymath}
induced from the central extension $\mathbb{Z} \to \mathbb{R} \to U(1)$.



\end{itemize}
\end{example}
\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[group]]


\item \textbf{2-group}, [[crossed module]], [[differential crossed module]]

\begin{itemize}%
\item [[braided 2-group]], [[symmetric 2-group]]


\item [[quantum 2-group]]



\end{itemize}

\item [[3-group]], [[2-crossed module]] / [[crossed square]], [[differential 2-crossed module]]


\item [[n-group]]


\item [[∞-group]], [[simplicial group]], [[crossed complex]], [[hypercrossed complex]]


\item [[group stack]]

[[smooth 2-group]]



\end{itemize}
[[!include homotopy n-types - table]]

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

2-groups were introduced, under the name \emph{gr-categories}, in

\begin{itemize}%
\item [[Hoàng Xuân Sính]], \emph{Gr}-cat\'e{}gories, PhD thesis (1973), (\href{http://w5.mathematik.uni-stuttgart.de/fachbereich/Kuenzer/Kuenzer/sinh.html}{web})

\end{itemize}
supervised by [[Grothendieck]].

Exposition and discussion of 2-groups as special [[monoidal categories]] ([[Picard 2-group]]) is in

\begin{itemize}%
\item [[John Baez]], [[Aaron Lauda]], \emph{HDA V: 2-Groups}, Theory and Applications of Categories 12 (2004), 423-491. (\href{http://arxiv.org/abs/math.QA/0307200}{arXiv:math.QA/0307200}).

\end{itemize}
Computational enumeration of geometrically [[discrete group|discrete]] 2-groups using the computer program \href{http://pages.bangor.ac.uk/~mas023/chda/xmod/xmod244.html}{XMod} is reported on in

\begin{itemize}%
\item Murat Alp, Christopher Wensley, \emph{Enumeration of $Cat^1$-groups of low order}, Int. J. Algebra Comput. 10, 407 (2000) (\href{http://www.worldscientific.com/doi/abs/10.1142/S0218196700000170}{publisher})


\item Graham Ellis, Le van Luyen, \emph{Homotopy 2-types of Low order} (\href{http://hamilton.nuigalway.ie/preprints/2t.pdf}{pdf})



\end{itemize}
Discussion of structured 2-groups (e.g. smooth 2-groups) is in sections 2.6.5 and 3.4.2 of

\begin{itemize}%
\item [[Urs Schreiber]], \emph{[[schreiber:differential cohomology in a cohesive topos]]}

\end{itemize}
For more on this see the references at [[string 2-group]].

[[!redirects 2-groups]]

[[!redirects weak 2-group]] [[!redirects weak 2-groups]]



\end{document}