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\begin{document}

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\section*{2-crossed complex}

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{examples}{Examples:}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{crossed_complexes_and_2crossed_complexes}{Crossed complexes and 2-crossed complexes.}\dotfill \pageref*{crossed_complexes_and_2crossed_complexes} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

[[crossed complex|Crossed complexes]] are a useful extension of [[crossed module]]s allowing not only the encoding of an algebraic model for the [[homotopy 2-type]], but also information on the `[[complex of chains on the universal cover]]'. The category of crossed complexes is a [[monoidal closed category]] equivalent to various types of [[strict infinity-groupoid]].

To model the [[homotopy 3-type]] of a space, we can use either a [[2-crossed module]] or a [[crossed square]] (or various other algebraic models to be added some time in the future). A [[crossed complex]] is a `hybrid', part [[crossed module]] but with a `tail' which is a [[chain complex]]. What would be the `hybrid' between a 2-crossed module and a chain complex? Are there examples that are easily constructed? What sort of information do they encode? Are they easy to analyse, understand, \ldots{} and useful?

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

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

\begin{displaymath}
\ldots \to C_n \stackrel{\partial_n}{\longrightarrow} C_{n-1} \longrightarrow \ldots \longrightarrow C_0,
\end{displaymath}
together with a 2-crossed module structure given on $C_2\to C_1\to C_0$ by a Peiffer lifting function $\{ -,-\} : C_1\times C_1 \to C_2$, such that, on writing $\pi = Coker(C_1\to C_0)$,

\begin{enumerate}%
\item each $C_n$, $n\geq 3$ and $Ker \,\partial_2$ are $\pi$-modules and the $\partial_n$ for $n\geq 4$, together with the codomain restriction of $\partial_3$, are $\pi$-module homomorphisms;


\item the $\pi$-module structure on $Ker \partial_2$ is the action induced from the $C_0$-action on $C_2$ for which the action of $\partial_1 C_1$ is trivial.



\end{enumerate}
A 2-crossed complex morphism is defined in the obvious way, being compatible with all the actions, the pairings and Peiffer liftings. We will denote by $2 Crs$, the corresponding category.

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

\begin{itemize}%
\item Any 2-crossed module clearly gives a 2-crossed complex (with trivial `tail').


\item \emph{From simplicial groups to 2-crossed complexes}. If $G$ is a simplicial group, then



\end{itemize}
\begin{displaymath}
\ldots \to C(G)_3 \to  \frac{\mathcal{N}G_2}{d_0(\mathcal{N}G_3\cap D_3)} \to \mathcal{N}G_1\to \mathcal{N}G_0,
\end{displaymath}
has the structure of a 2-crossed complex, where $\mathcal{N}G$ is the [[Moore complex]] of $G$, $D_n$ is the subgroup of $G_n$ generated by the degenerate elements, and, for $n\gt2$,

\begin{displaymath}
{C}(G)_{n} = \frac{\mathcal{N}G_n}{(\mathcal{N}G_n\cap D_n)d_0(\mathcal{N}G_{n+1}\cap D_{n+1})},
\end{displaymath}
is the $n$-dimensional term of the crossed complex, $C(G)$, associated to the simplicial group $G$ as in the entry [[crossed complex]] (in the section \textbf{From simplicial group(oid)s to crossed complexes}.)

(There is an obvious extension of the group based definition above to a groupoid based one, and of this construction to one which takes as input a simplicially enriched groupoid.)

The Moore complex of a simplicial group $G$ has the structure of a 2-crossed complex if and only if for each $n\gt 2$, $\mathcal{N}G_n\cap D_n$ is trivial. This means that the axioms of a [[group T-complex]] are almost satisfied, but not necessarily in dimension 2.

\begin{itemize}%
\item A quadratic chain complex as defined by H.J. Baues is a special case of a 2-crossed complex, satisfying additional (pre-crossed module) nilpotency condition at the level of the underlying pre-crossed module. (In fact the category of quadratic chain complexes is a reflective subcategory of the category of 2-crossed complexes.) In Baues' book referenced below, there is the construction of the fundamental quadratic chain complex of a pointed CW-complex. The reflection (or cotruncation) of this to the category of quadratic modules (i.e. 3-truncated quadratic chain complexes) faithfully represents the homotopy 3-type of a CW-space (at the level of spaces and maps between them).


\item Graham Ellis defined the fundamental squared complex of a CW-complex from triad homotopy groups and generalised Whitehead products, and showed how Baues fundamental quadratic chain complex of a CW-complex can be obtained from it. A homotopy 2-crossed complex of a CW-complex can also be defined is the same way, see the work of Jo\~a{}o Faria Martins below.



\end{itemize}
\hypertarget{crossed_complexes_and_2crossed_complexes}{}\subsection*{{Crossed complexes and 2-crossed complexes.}}\label{crossed_complexes_and_2crossed_complexes}

Any [[crossed complex]] can be given the structure of a 2-crossed complex simply by defining a trivial Peiffer lifting, $\{-,-\}$. As the Peiffer lifting covers the Peiffer commutators in $C_1$, and these are trivial (since the bottom of the crossed complex is a crossed module), this trivial Peiffer lifting works and gives a 2-crossed complex structure. This defines a functor from the category of crossed complexes to that of 2-crossed complexes.

Any 2-crossed complex which has a Peiffer lifting that is trivial $\{x,y\} = 1$, for all $x,y \in C_1$) is isomorphic to a crossed complex in this sense.

This functor, from $Crs$ to $2-Crs$, has a left adjoint which is the identity on the subcategory of $2-Crs$ with trivial Peiffer liftings, so $Crs$ is equivalent to a reflective subcategory of $2-Crs$

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

\begin{itemize}%
\item See the [[Crossed Menagerie]], chapter 5.


\item [[H.-J. Baues]], Combinatorial Homotopy and 4-Dimensional Complexes , de Gruyter Expositions in Mathematics 2, Walter de Gruyter, (1991).


\item [[Graham Ellis]], Crossed squares and combinatorial homotopy. Mathematische Zeitschrift Volume 214, Number 1, 93-110, DOI: 10.1007/BF02572393


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



\end{itemize}


\end{document}