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\begin{document}

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\section*{2-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{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{remarks}{Remarks}\dotfill \pageref*{remarks} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{from_simplicial_groups_to_2crossed_modules}{From simplicial groups to 2-crossed modules}\dotfill \pageref*{from_simplicial_groups_to_2crossed_modules} \linebreak
\noindent\hyperlink{from_crossed_squares_to_2crossed_modules}{From crossed squares to 2-crossed modules}\dotfill \pageref*{from_crossed_squares_to_2crossed_modules} \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{2-crossed module} encodes a semistrict 3-group -- a [[Gray-group]] -- in generalization of how a [[crossed module]] encodes a [[strict 2-group]].

A [[simplicial group]] whose [[Moore complex]] has length $1$ (that is, at most stuff in dimensions $0$ and $1$) will be the internal [[nerve]] of a strict $2$-[[2-group|group]] and the Moore complex will be the corresponding [[crossed module]]. What if we have a simplicial group whose Moore complex has at most stuff in dimensions $0$, $1$, and $2$; can we describe its structure in some similar way? Yes, and Conduch\'e{} provided a neat description of the structure involved. From the structure one can rebuild a simplicial group, a type of internal $2$-[[nerve]] construction.

In other words, a $2$-crossed module \emph{is} the Moore complex of a $2$-[[truncated]] simplicial group.

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

A \textbf{$2$-crossed module} is a [[normal complex of groups]]

\begin{displaymath}
L\stackrel{\partial_2}{\to} M \stackrel{\partial_1}{\to}N,
\end{displaymath}
together with an [[action]] of $N$ on all three groups and a mapping

\begin{displaymath}
\{ - ,- \} : M\times M \to L
\end{displaymath}
such that

\begin{enumerate}%
\item the action of $N$ on itself is by conjugation, and $\partial_2$ and $\partial_1$ are $N$-equivariant;


\item for all $m_0,m_1 \in M$,

\begin{displaymath}
\partial_2\{m_0,m_1\} = \,^{\partial_1 m_0}m_1 . m_0m_1^{-1}m_0^{-1};
\end{displaymath}

\item if $\ell_0,\ell_0 \in L$, then

\begin{displaymath}
\{\partial_2\ell_0,\partial_2\ell\} = [\ell_1,\ell_0];
\end{displaymath}

\item if $\ell \in L$ and $m\in M$, then

\begin{displaymath}
\{m,\partial \ell\}\{\partial \ell,m\} = \,^{\partial m}\ell.\ell^{-1};
\end{displaymath}

\item for all $m_0,m_1,m_2 \in M$,

\begin{itemize}%
\item $\{m_0,m_1m_2\} = \{m_0,m_1\}\{ \partial \{m_0,m_2\},(m_0m_1m_0^{-1})\}\{m_0,m_2\}$;
\item $\{m_0m_1,m_2\} = \,^{\partial m_0}\{m_1,m_2\}\{m_0,m_1m_2m_1^{-1}\}$;

\end{itemize}

\item if $n\in N$ and $m_0,m_1 \in M$, then

\begin{displaymath}
\,^{n} \{m_0,m_1\} = \{ \,^{n}m_0, \,^{n}m_1\}.
\end{displaymath}


\end{enumerate}
The pairing $\{ - ,- \} : M\times M \to L$ is often called the \textbf{Peiffer lifting} of the $2$-crossed module.

\hypertarget{remarks}{}\subsection*{{Remarks}}\label{remarks}

\begin{itemize}%
\item In a $2$-crossed module as above the structure $\partial_2: L \to M$ is a [[crossed module]], but $\partial_1: M\to N$ may not be one, as the Peiffer identity need not hold. The \emph{[[Peiffer commutator]]}, which measures the failure of that identity, may not be trivial, but it will be a boundary element and the Peiffer lifting gives a structured way of getting an element in $L$ that maps down to it.


\item It is sometimes useful to consider a [[crossed module]] as being a [[crossed complex]] of length 1 (i.e. on possibly non-trivial morphism only). Likewise one can consider a 2-crossed module as a special case of a [[2-crossed complex]]. Such a gadget is intuitively a 2-crossed module with a `tail', which is a chain complex of modules over the $\pi_0$ of the base 2-crossed module, much as a crossed complex is a crossed module together with a `tail'.


\item A quadratic module, as developed by [[H.-J. Baues]], is a special case of a 2-crossed module, satisfying nilpotency conditions at the level of the underlying pre-crossed module (which is a close to being a crossed module as possible.) The fundamental quadratic module of a CW-complex yields an equivalence of categories between the category of pointed 3-types and the category of quadratic modules.


\item A functorial fundamental 2-crossed module of a CW-complex can also be defined, by using Graham Ellis fundamental crossed square of a CW-complex; this is explained in the article of Jo\~a{}o Faria Martins, below. We can also define this fundamental 2-crossed module of a CW-complex, by using Kan's fundamental simplicial group of a CW-complex, and by applying the usual reflection from simplicial groups to simplicial groups of Moore complex of lenght two, known to be equivalent to 2-crossed modules.


\item The homotopy theory of 2-crossed modules can be addressed by noting that 2-crossed modules, inducing a reflective subcategory of the category of simplicial groups, inherit a natural Quillen model structure, as explored in the article of Cabello and Garzon below. A version very close to the usual homotopy theory of crossed complexes was developed in the article of Joao Faria Martins below in a parallel way to the homotopy theory of [[quadratic module]]s and [[quadratic complex]]es as introduced by [[H. J. Baues]].



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

Any [[crossed module]], $G_2 \stackrel{\delta }{\to}{G_1}$ gives a 2-crossed module, $L\stackrel{\partial_2}{\to} M \stackrel{\partial_1}{\to}N,$ by setting $L = 1$, the trivial group, and, of course, $M = G_2$, $N = G_1$. Conversely any 2-crossed module having trivial top dimensional group ($L=1$) `is' a crossed module. This gives an inclusion of the category of crossed modules into that of 2-crossed modules, as a [[reflective subcategory]].

The reflection is given by noting that, if

\begin{displaymath}
L\stackrel{\partial_2}{\longrightarrow} M \stackrel{\partial_1}{\longrightarrow}N
\end{displaymath}
is a 2-crossed module, then $Im\, \partial_2$ is a normal subgroup of $M$, and then there is an obvious induced crossed module structure on

\begin{displaymath}
\partial_1 : \frac{M}{Im\, \partial_2} \to N.
\end{displaymath}
But we can do better than this. More generally, let

\begin{displaymath}
\ldots \to 1 \to 1 \to C_3\stackrel{\partial_3}{\longrightarrow} C_2 \stackrel{\partial_2}{\longrightarrow}C_1,
\end{displaymath}
be a truncated [[crossed complex]] (of groups) in which all higher dimensional terms are trivial, then taking $L = C_3$, $M = C_2$ and $N = C_1$, with trivial Peiffer lifting, gives one a 2-crossed complex. Conversely suppose we have a 2-crossed module with trivial Peiffer lifting: $\{m_1,m_2\} = 1$ for all $m_1$, $m_2 \in M$, axiom 3 then shows that $L$ is an Abelian group, and similarly the other axioms can be analysed to show that the result is a truncated crossed complex.

This gives:

\begin{uprop}
The category $Crs_{2]}$ of crossed complexes of length 2 is equivalent to the full subcategory of $2-CMod$ given by those 2-crossed modules with trivial Peiffer lifting.

\end{uprop}
Of course, the resulting `inclusion' has a left adjoint, which is quite fun to check out! (You kill off the subgroup of $L$ generated by the Peiffer lifting, \ldots{}. is that all?)

\hypertarget{from_simplicial_groups_to_2crossed_modules}{}\subsubsection*{{From simplicial groups to 2-crossed modules}}\label{from_simplicial_groups_to_2crossed_modules}

If $G$ is a simplicial group then

\begin{displaymath}
\frac{\mathcal{N}G_2}{d_0(\mathcal{N}G_3)} \to \mathcal{N}G_1\to \mathcal{N}G_0,
\end{displaymath}
is a 2-crossed module. (You are invited to find the Peiffer lifting!)

\hypertarget{from_crossed_squares_to_2crossed_modules}{}\subsubsection*{{From crossed squares to 2-crossed modules}}\label{from_crossed_squares_to_2crossed_modules}

Both [[crossed square|crossed squares]] and 2-crossed modules model all connected homotopy 3-types so one naturally asks how to pass from one description to the other. Going from crossed squares to 2-crossed modules is easy, so will be given here (going back is harder).

Let

\begin{displaymath}
\itexarray{& L & {\to}^\lambda & M & \\
         \lambda^\prime & \downarrow &&\downarrow & \mu\\
          &N & {\to}_{\nu}& P & \\
}
\end{displaymath}
be a crossed square then $N$ acts on $M$ via $P$, so ${}^n m := {}^{\nu(n)}m$, and so we can form $M\rtimes N$ and the sequence

\begin{displaymath}
L\stackrel{((\lambda')^{-1},\lambda)}{\longrightarrow}M\rtimes N\stackrel{\mu\nu}{\longrightarrow}P
\end{displaymath}
is then a 2-crossed complex.

(And, yes, these are actually group homomorphisms: $(\mu,\nu)(m,n) = \mu(m)\nu(n)$, the product of the two elements! Try it!)

The full result and an explanation of what is going on here is given in

\begin{itemize}%
\item D. Conduch\'e{}, \emph{Simplicial Crossed Modules and Mapping Cones}, Georgian Math. J., 10, (2003), 623--636

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

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


\item [[2-group]], [[crossed module]], [[differential crossed module]]


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


\item [[n-group]]


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



\end{itemize}
[[!redirects 2-crossed modules]]

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

\begin{itemize}%
\item [[H. J. Baues]]: \emph{Combinatorial homotopy and $4$-dimensional complexes.} With a preface by Ronald Brown. de Gruyter Expositions in Mathematics, 2. Walter de Gruyter $\backslash$\& Co., Berlin, 1991.


\item [[Julia G. Cabello]], [[Antonio R. Garzón]]: \emph{Quillen's theory for algebraic models of $n$-types}. Extracta Math. 9 (1994), no. 1, 42--47. ((EuDML](https://eudml.org/doc/38395))


\item [[P. Carrasco]] and [[T. Porter]], \emph{Coproduct of 2-crossed modules. Applications to a definition of a tensor product for 2-crossed complexes}, Collectanea Mathematica, (DOI) 10.1007/s13348-015-0156-9.


\item [[Daniel Conduché]], \emph{Modules crois\'e{}s g\'e{}n\'e{}ralis\'e{}s de longueur $2$}, in: Proceedings of the Luminy conference on algebraic $K$-theory (Luminy, 1983). J. Pure Appl. Algebra 34 (1984), no. 2-3, 155--178.


\item [[Joao Faria Martins]], \emph{Homotopy of 2-crossed complexes and the homotopy category of pointed 3-types}, (wed \href{http://ferrari.dmat.fct.unl.pt/~jnm/Hom2XComplexes.pdf}{pdf})


\item [[Graham Ellis]], \emph{Crossed squares and combinatorial homotopy}, Math. Z. 214 (1993), no. 1, 93--110.



\end{itemize}


\end{document}