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\section*{crossed module}

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

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

[[!include higher category theory - contents]]

\hypertarget{homological_algebra}{}\paragraph*{{Homological algebra}}\label{homological_algebra}

[[!include homological algebra - contents]]

\hypertarget{content}{}\section*{{Content}}\label{content}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{diagrammatic_definition}{Diagrammatic definition}\dotfill \pageref*{diagrammatic_definition} \linebreak
\noindent\hyperlink{definition_in_terms_of_equations}{Definition in terms of equations}\dotfill \pageref*{definition_in_terms_of_equations} \linebreak
\noindent\hyperlink{Morphisms}{Morphisms}\dotfill \pageref*{Morphisms} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \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}

A \emph{crossed module} (of groups) is:

\begin{itemize}%
\item from the [[nPOV]]: a convenient way to encode a [[strict 2-group]] $G$ in terms of a morphism of two ordinary [[group]]s $\partial : G_2 \to G_1$.

\end{itemize}
From other points of view it is:

\begin{itemize}%
\item like the inclusion of a [[normal subgroup]], but isn't an inclusion in general;
\item like a [[module]] with a twisted `multiplication';
\item like the action of automorphisms on a group;
\item a [[crossed complex]] concentrated in degrees $1$ and $2$;
\item a nonabelian [[chain complex|chain-complex]];
\item a [[Moore complex]] of certain [[simplicial group]]s.

\end{itemize}
Historically they were the first example of [[higher dimensional algebra]] to be studied.

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

\hypertarget{diagrammatic_definition}{}\subsubsection*{{Diagrammatic definition}}\label{diagrammatic_definition}

A \textbf{crossed module} is

\begin{itemize}%
\item a pair of [[group]]s $G_2, G_1$,


\item morphisms of groups

\begin{displaymath}
G_2 \stackrel{\delta }{\to}{G_1}
\end{displaymath}
and

\begin{displaymath}
G_1 \stackrel{\alpha}{\to} Aut(G_2)
\end{displaymath}
(which below we will conceive of as a map $\alpha : G_1 \times G_2 \to G_2$ analogously to the adjoint action $Ad : G \times G \to G$ of a group on itself)


\item such that

\begin{displaymath}
\itexarray{
  G_2 \times G_2
  &&\stackrel{\delta \times Id}{\to}&&
  G_1 \times G_2
  \\
  & {}_{Ad}\searrow && \swarrow_\alpha
  \\
  &&
  G_2
}
\end{displaymath}
and

\begin{displaymath}
\itexarray{
  G_1 \times G_2 &\stackrel{\alpha}{\to}& G_2
  \\
  \downarrow^{Id \times \delta} && \downarrow^{\delta}
  \\
  G_1 \times G_1 &\stackrel{Ad}{\to}& G_1    
}
\end{displaymath}
commute.



\end{itemize}
We may use the notation $(G_2,G_1,\delta)$, for this if the action is fairly obvious, including an explicit [[action]], $(G_2,G_1,\delta,\alpha)$, if there is a risk of confusion.

If one unwraps the definitions in terms of automorphism groups to use merely finite products (for instance, writing the actions as $G_1 \times G_2 \to G_2$, together with commuting diagrams encoding the necessary properties), then crossed modules can be defined internal to any [[cartesian monoidal category]] $C$, namely as a structure involving internal groups in $C$. For instance, one might consider Lie crossed modules, which are crossed modules of [[Lie groups]]. These are relevant for certain models of the [[String group]].

Alternatively, one can take another tack, and define crossed module objects in categories that support enough structure without using internal groups, the most general case of which, in practice, are [[semiabelian categories]]. There one considers the objects to behave `like groups' in the sense that the category they form looks very much like the category of groups. Janelidze (\hyperlink{Janelidze_03}{Janelidze 2003}) defined the notion of internal crossed module in a semiabelian category (so that in the prototypical example of the category of groups, they reduce to the above notion).

A key result, also due to (\hyperlink{Janelidze_03}{Janelidze 2003}) and generalising the Brown-Spencer theorem from the case of ordinary crossed modules, is the following:

\begin{theorem}
\label{}\hypertarget{}{}
\textbf{(Janelidze's Brown-Spencer theorem).} Let $C$ be a semiabelian category. Then the category $XMod(C)$ of crossed modules in $C$ is equivalent to the category $Gpd(C)$ of internal groupoids in $C$.

\end{theorem}
Here the notion of [[internal groupoid]] is the usual diagrammatic notion.

\hypertarget{definition_in_terms_of_equations}{}\subsubsection*{{Definition in terms of equations}}\label{definition_in_terms_of_equations}

The two [[diagram]]s can be translated into equations, which may often be helpful.

\begin{itemize}%
\item If we write the effect of acting with $g_1\in G_1$ on $g_2\in G_2$ as ${}^{g_1}g_2$, then the second diagram translates as the equation:

\begin{displaymath}
\delta({}^{g_1}g_2) = g_1\delta(g_2)g_1^{-1}.
\end{displaymath}
In other words, $\delta$ is equivariant for the action of $G_1$.


\item The first diagram is slightly more subtle. The group $G_2$ can act on itself in two different ways, (i) by the usual conjugation action, ${}^{g_2}g^\prime_2=g_2g^\prime_2g_2^{-1}$ and (ii) by first mapping $g_2$ down to $G_1$ and then using the action of that group back on $G_2$. The first diagram says that the two actions coincide. Equationally this gives:

\begin{displaymath}
{}^{\delta(g_2)}g^\prime_2 = g_2g^\prime_2g_2^{-1}.
\end{displaymath}
This equation is known as the \textbf{Peiffer rule} in the literature. Another way to interpret it is to rewrite it slightly:

\begin{displaymath}
{}^{\delta(g_2)}g^\prime_2 g_2 = g_2g^\prime_2
\end{displaymath}
The Peiffer rule can thus be seen as a `twisted commutativity law' for $G_2$.



\end{itemize}
\hypertarget{Morphisms}{}\subsubsection*{{Morphisms}}\label{Morphisms}

For $G$ and $H$ two [[strict 2-group]]s coming from crossed modules $[G]$ and $[H]$, a morphism of strict 2-groups $f : G \to H$, and hence a morphism of crossed modules $[f] : [G] \to [H]$ is a [[2-functor]]

\begin{displaymath}
\mathbf{B}f : \mathbf{B}G \to \mathbf{B}H
\end{displaymath}
between the corresponding [[delooping|delooped]] [[2-groupoid]]s. Expressing this in terms of a diagram of the ordinary groups appearing in $[G]$ and $[H]$ yields a diagram called a [[butterfly]]. See there for more details.

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

\begin{itemize}%
\item For $H$ any group, its [[automorphism]] crossed module is

\begin{displaymath}
AUT(H) := (G_2 = H, G_1 = Aut(H), \delta = Ad, \alpha = Id)
 \,.
\end{displaymath}
Under the equivalence of crossed modules with [[strict 2-group]]s this corresponds to the [[automorphism 2-group]]

\begin{displaymath}
Aut_{Grpd}(\mathbf{B}H)
\end{displaymath}
of [[automorphism]]s in the category [[Grpd]] of [[groupoid]]s on the one-object [[delooping]] [[groupoid]] $\mathbf{B}H$ of $H$.


\item Almost the canonical example of a crossed module is given by a group $G$ and a normal subgroup $N$ of $G$. We take $G_2 = N$, and $G_1 = G$ with the action given by conjugation, whilst $\delta$ is the inclusion, $inc : N \to G$. This is `almost canonical', since if we replace the groups by simplicial groups $G_.$ and $N_.$, then $(\pi_0(G_.),\pi_0(N_.),\pi_0(inc))$ is a crossed module, and given any crossed module, $(C,P,\delta)$, there is a simplicial group $G_.$ and a normal subgroup $N_.$, such that the construction above gives the given crossed module up to isomorphism.


\item Another standard example of a crossed module is $M \to ^0 P$ where $P$ is a group and $M$ is a $P$-module. Thus the category of modules over groups embeds in the category of crossed modules.


\item If $\mu: M \to P$ is a crossed module with cokernel $G$, and $M$ is abelian, then the operation of $P$ on $M$ factors through $G$. In fact such crossed modules in which both $M$ and $P$ are abelian should not be sneezed at! A good example is $\mu: C_2 \times C_2 \to C_4$ where $C_n$ denotes the cyclic group of order $n$, $\mu$ is injective on each factor, and $C_4$ acts on the product by the twist. This crossed module has a [[classifying space]] $X$ with fundamental and second homotopy groups $C_2$ and non trivial $k$-invariant in $H^3(C_2, C_2)$, so $X$ is not a product of [[Eilenberg-MacLane space]]s. However the crossed module is an algebraic model and so one one can do algebraic constructions with it. It gives in some ways a better feel for the space than the $k$-invariant. The [[higher homotopy van Kampen theorem]] implies that the above $X$ gives the 2-type of the [[mapping cone]] of the map of [[classifying space]]s $BC_2 \to BC_4$.


\item Suppose $F\stackrel{i}{\to}E\stackrel{p}{\to}B$ is a [[fibration sequence]]

of [[pointed object|pointed spaces]], thus $p$ is a [[fibration]] in the [[model structure on topological spaces|topological sense]] (lifting of paths and homotopies of paths will suffice), $F = p^{-1}(b_0)$, where $b_0$ is the basepoint of $B$. The fibre $F$ is pointed at $f_0$, say, and $f_0$ is taken as the basepoint of $E$ as well.

There is an induced map on [[homotopy group]]s

\begin{displaymath}
\pi_1(F) \stackrel{\pi_1(i)}{\longrightarrow} \pi_1(E)
\end{displaymath}
and if $a$ is a loop in $E$ based at $f_0$, and $b$ a loop in $F$ based at $f_0$, then the composite path corresponding to $a b a^{-1}$ is [[homotopy|homotopic]] to one wholly within $F$. To see this, note that $p(a b a^{-1})$ is [[null homotopic loop|null homotopic]]. Pick a [[homotopy]] in $B$ between it and the constant map, then lift that homotopy back up to $E$ to one starting at $a b a^{-1}$. This homotopy is the required one and its other end gives a well defined element ${}^a b \in \pi_1(F)$ (abusing notation by confusing paths and their homotopy classes). With this action $(\pi_1(F), \pi(E), \pi_1(i))$ is a crossed module. This will not be proved here, but is not that difficult. (Of course, secretly, this example is `really' the same as the previous one since a fibration of [[simplicial group]]s is just morphism that is an [[epimorphism]] in each degree, and the [[fibration sequence|fibre]] is thus just a [[simplicial group|normal simplicial subgroup]]. What is fun is that this generalises to `higher dimensions'.)


\item A particular case of this last example can be obtained from the inclusion of a subspace $A\to X$ into a pointed space $(X,x_0)$, (where we assume $x_0\in A$). We can replace this inclusion by a homotopic fibration, $\overline{A}\to X$ in `the standard way', and then find that the fundamental group of its fibre is $\pi_2(X,A,x_0)$.



\end{itemize}
A deep theorem of J.H.C. Whitehead is that the crossed module

\begin{displaymath}
\delta: \pi_2(A \cup \{e^2_\lambda\}_{\lambda \in \Lambda},A,x) \to \pi_1(A,x)
\end{displaymath}
is the [[free crossed module]] on the characteristic maps of the $2$-cells. One utility of this is that it enables the expression of nonabelian chains and boundaries ideas in dimensions $1$ and $2$: thus for the standard picture of a Klein Bottle formed by identifications from a square $\sigma$ the formula

\begin{displaymath}
\delta \sigma = a+b-a +b
\end{displaymath}
makes sense with $\sigma$ a generator of a free crossed module; in the usual abelian chain theory we can write only $\partial \sigma =2b$, thus losing information.

Whitehead's proof of this theorem used knot theory and transversality. The theorem is also a consequence of the $2$-dimensional Seifert-van Kampen Theorem, proved by Brown and Higgins, which states that the functor

$\Pi_2$: (pairs of pointed spaces) $\to$ (crossed modules)

preserves certain colimits (see reference below).

This last example was one of the first investigated by Whitehead and his proof appears also in a little book by [[Hilton]]; see also [[Nonabelian algebraic topology]], however the more general result of Brown and Higgins determines also the group $\pi_2(X \cup CA,X,x)$ as a crossed $\pi_1(X,x)$ module, and then Whitehead's result is the case with $A$ is a wedge of circles.

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

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


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


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


\item [[n-group]]


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



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

The concept was introduced in

\begin{itemize}%
\item [[J. H. C. Whitehead]], \emph{Combinatorial Homotopy II}, Bull. Amer. Math. Soc., 55 (1949), 453--496.

\end{itemize}
see also

\begin{itemize}%
\item [[Peter J. Hilton]], \emph{An Introduction to Homotopy Theory}, Cambridge University Press 1953

\end{itemize}
Further discussion is in

\begin{itemize}%
\item [[Ronnie Brown]], [[Philip Higgins]], \emph{On the connections between the second relative homotopy groups of some related spaces}, \emph{Proc. London Math. Soc.} (3) 36 (1978) 193-212.


\item [[Ronnie Brown]], \emph{Groupoids and crossed objects in algebraic topology}, \emph{Homology, Homotopy and Applications}, 1 (1999) 1-78.


\item [[George Janelidze]], \emph{Internal crossed modules}, Georgian Mathematical Journal \textbf{10} (2003) pp 99-114. (\href{https://eudml.org/doc/51553}{EuDML})


\item [[Ronnie Brown]], [[Philip Higgins]], and R. Sivera, \emph{[[Nonabelian Algebraic Topology]]: Filtered spaces, Crossed Complexes, Cubical Homotopy Groupoids}, EMS Tracts in Mathematics, Vol. 15, (Autumn 2010).



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[[!redirects crossed modules]]



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