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\section*{crossed square}

\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{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{homotopical_example}{Homotopical example}\dotfill \pageref*{homotopical_example} \linebreak
\noindent\hyperlink{algebraic_example}{Algebraic example}\dotfill \pageref*{algebraic_example} \linebreak
\noindent\hyperlink{simplicial_group_example}{Simplicial group example}\dotfill \pageref*{simplicial_group_example} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{relation_to_groups}{Relation to $cat^2$-groups}\dotfill \pageref*{relation_to_groups} \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 [[crossed module]] is a bit like a [[normal subgroup]] \ldots{} without being a [[subgroup]]. In fact if a crossed module has a boundary map which is a monomorphism then it is isomorphic to the inclusion crossed module of a normal subgroup.

Crossed modules model all connected [[homotopy 2-type]]s (which by the [[looping and delooping]]-theorem means: all [[2-group]]s). \emph{Crossed squares} model all connected [[homotopy 3-type]]s (hence all [[3-group]]s) and correspond in much the same way to pairs of normal subgroups.

Suppose $G$ is a group and $M$ and $N$ are normal subgroups of $G$; then of course, so is $M \cap N$. Put these groups in a square, with the inclusion maps between them. Finally note that if $m \in M$ and $n \in N$, then $[m,n]$ is in the intersection $M \cap N$. This gives you a crossed square with $h$-map $h(m,n) = [m,n]$. Removing the condition that the inclusions are inclusions (!) gives the general form.

(The definition that follows is that given by Guin-Valery and Loday in their paper (see references). Another definition can be given that is just the case $n = 2$ of that of [[crossed n-cube]], for which see that entry.

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

A \textbf{crossed square}

\begin{displaymath}
\itexarray{& L & {\to}^\lambda & M & \\
         \lambda^\prime & \downarrow &&\downarrow & \mu\\
          &N & {\to}_{\nu}& P & \\
}
\end{displaymath}
consists of four morphisms of groups $\lambda: L \to M$, $\lambda': L \to N$, $\mu: M \to P$, $\nu: N \to P$, such that $\nu \lambda'= \mu \lambda$ together with [[action]]s of the group $P$ on $L, M, N$ on the left, conventionally, (and hence actions of $M$ on $L$ and $N$ via $\mu$ and of $N$ on $L$ and $M$ via $\mu$) and a function $h: M \times N \to L$.

This structure shall satisfy the following axioms:

\begin{enumerate}%
\item the maps $\lambda, \lambda '$ preserve the actions of $P$; further, with the given actions, the maps $\lambda, \lambda',\mu, \nu$ and $\kappa = \mu\lambda = \mu
'\lambda '$ are crossed modules;
\item $\lambda h(m,n)=m^n m^{-1},\lambda 'h(m,n)= {^m}n\,n^{-1}$
\item $h(\lambda l, n)= l^n l^{-1}, h(m,\lambda 'l)= {^m}l\, l^{-1}$
\item $h(mm',n) = {^m}h(m',n) h(m,n), h(m,nn') = h(m,n) ^n h(m,n')$;
\item $h(^p m, {^p n})= {^p}h(m,n)$;

\end{enumerate}
for all $l\in L, \,m, m'\in M,\, n,n'\in N$ and $p\in P$.

The similarity of these axioms to [[commutator]] identities is no accident (see below).

This should be thought of as a \emph{crossed module of crossed modules} (in either direction!). For instance horizontally:

\begin{displaymath}
\itexarray{& L & & M & \\
         \lambda^\prime & \downarrow &\to &\downarrow & \mu\\
          &N & & P & \\
}
\end{displaymath}
The image of this morphism is a normal sub-crossed module of $(M,P,\mu)$, so we can form a quotient

\begin{displaymath}
\overline{\mu} : M/\lambda L \to P/\nu N,
\end{displaymath}
and this is a crossed module, as is the kernal crossed module of this (horizontal) morphism.

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

\hypertarget{homotopical_example}{}\subsubsection*{{Homotopical example}}\label{homotopical_example}

The classical homotopical example $\Pi(X;A,B)$ is determined by a pointed [[triad]] $(X; A,B)$ where $A,B \subseteq X$, and $P = \pi_1(A \cap B)$, $M = \pi_2(A, A \cap B), N = \pi_2(B, A \cap B)$ and $L=\pi_3(X; A,B)$. The operations of $P$ are the standard ones and $h$ is the generalised [[Whitehead product]]. (The conventions may be slightly different from the standard ones in homotopy theory.) This can be generalised to a functor $\Pi$ from squares of pointed spaces to crossed squares.

Ellis uses this construction in

\begin{itemize}%
\item G.J. Ellis, Crossed squares and combinatorial homotopy, Math. Z., 461 (1993) 93--110,

\end{itemize}
where the fact that that the crossed square associated to a triad is defined directly in terms of certain homotopy classes is important.

The fact that there is a [[van Kampen theorem|van Kampen type theorem]] for $\Pi$ implies that one calculates some nonabelian groups. It also implies that one is calculating some (pointed) homotopy 3-types.

\hypertarget{algebraic_example}{}\subsubsection*{{Algebraic example}}\label{algebraic_example}

The example hinted at above has $P$ a group, $M$ and $N$ normal subgroups, and $L = M \cap N$,

\begin{displaymath}
\itexarray{& M\cap N & {\to}^\lambda & M & \\
         \lambda^\prime & \downarrow &&\downarrow & \mu\\
          &N & {\to}_{\nu}& P & \\
}
\end{displaymath}
with all maps the evident inclusions, all actions by conjugation, and $h : M \times N \to M\cap N$ given by $h(m,n) = [m,n]$.

\hypertarget{simplicial_group_example}{}\subsubsection*{{Simplicial group example}}\label{simplicial_group_example}

If we replace each group in the algebraic example by a simplicial group, we would have a simplicial crossed square, now apply the connected component functor to that and you get a crossed square, and in fact any crossed square can be constructed up to isomorphism in this way.

If we start with a simplicial group, $G$, using the [[decalage]] functor we can construct a simplicial group and two normal subgroups and thus get to the previous situation. The result can be interpreted in terms of the [[Moore complex]] as follows:

\begin{displaymath}
\itexarray{& \frac{NG_2}{d_0(NG_3)}& {\to} & Ker d_1 & \\
          & \downarrow &&\downarrow & \\
          &Ker d_2 & {\to}& G_1 & \\
}
\end{displaymath}
The two morphisms with codomain $G_1$ are inclusions, the other two are induced by $d_0$. The $h$-map can be explicitly given. It can be found in

T. Porter, \emph{n-types of simplicial groups and crossed n-cubes}, Topology, 32, (1993), 5 -- 24,

which also contains the discussion of the generalisation to [[crossed n-cube|crossed n-cubes]]

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

\hypertarget{relation_to_groups}{}\subsubsection*{{Relation to $cat^2$-groups}}\label{relation_to_groups}

A crossed module $\mu: M \to P$ determines a $cat^1$-structure on the [[semidirect product]] group $M \rtimes P$. Thus to say that the above crossed square is a \emph{crossed module of crossed modules} suggests that we should ask for $L \rtimes N \to M \rtimes P$ to be a crossed module, so that there is an action which allows the \emph{big group} $G = (L \rtimes N) \rtimes (M \rtimes P)$ to be a $cat^1$-group. Then $G$ becomes a \emph{$cat^2$-[[cat-2-group|group]]}. The $h$-map of the crossed square derives from a commutator in $G$.

This equivalence between crossed squares and $cat^2$-groups confirms the completeness of the axioms for crossed squares. Notice also that to prove a diagram of crossed squares is a [[colimit]] diagram, it looks as if you have to make appallingly detailed verifications of axioms. It is much easier to prove the corresponding diagram of $cat^2$-groups is a colimit!

This theme of using two equivalent categories, one for conjecture and proof, the other for calculation and application to traditional invariants, runs through the story of [[higher homotopy van Kampen theorems]].

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

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


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


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


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



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

\begin{itemize}%
\item D. Guin-Walery and [[J.-L. Loday]], 1981, \emph{Obstructions \`a{} l'excision en K-th\'e{}orie alg\`e{}brique}, in Evanston Conference on Algebraic K-theory, 1980, volume 854 of Lecture Notes in Maths., 179 -- 216, Springer.

\end{itemize}
[[!redirects crossed squares]]



\end{document}