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\section*{cat-n-group}

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

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

[[!include higher category theory - contents]]

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

[[!include group theory - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{abstract_definition}{Abstract definition}\dotfill \pageref*{abstract_definition} \linebreak
\noindent\hyperlink{algebraic_definition}{Algebraic definition}\dotfill \pageref*{algebraic_definition} \linebreak
\noindent\hyperlink{special_cases}{Special cases}\dotfill \pageref*{special_cases} \linebreak
\noindent\hyperlink{homotopical_example}{Homotopical example}\dotfill \pageref*{homotopical_example} \linebreak
\noindent\hyperlink{remarks}{Remarks}\dotfill \pageref*{remarks} \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}

Just as [[cat-1-group]]s (i) give models for connected [[homotopy 2-type]]s, (ii) are equivalent to [[crossed modules]], or [[2-group]]s,and are an algebraic encoding of [[internal category|internal categories]] within the category [[Grp]] of groups, so it is not surprising that higher dimensional analogues encode higher order homotopy information. This gives one the abstract definition:

\hypertarget{abstract_definition}{}\subsection*{{Abstract definition}}\label{abstract_definition}

A \emph{$cat^n$-group} is a strict [[n-fold category]] [[internal category|internal]] to [[Grp]].

Regarding a [[group]] as a [[groupoid]] with a single object, this is the same as an [[n-fold category|(n+1)-fold groupoid]] in which in one direction all morphisms are [[endomorphism]]s and there is corresponding notion of cat$^n$-groupoid.

As with the cases $n=1$ and 2, there is a neat purely group theoretic definition of these objects.

\hypertarget{algebraic_definition}{}\subsection*{{Algebraic definition}}\label{algebraic_definition}

A \emph{cat$^n$-group} is a [[group]] $G$ together with $2n$ [[endomorphism]]s $s_i, t_i, (1 \le i \le n)$ such that

\begin{displaymath}
s_i t_i = t_i, and  t_i s_i = s_i  for all  i,
\end{displaymath}
\begin{displaymath}
s_i s_j = s_j s_i,   t_i t_j = t_j t_i, s_i t_j = t_j s_i  for i\neq  j
\end{displaymath}
and, for all $i$,

\begin{displaymath}
[ Ker\, s_i, Ker\, t_i] = 1.
\end{displaymath}
Morphisms of cat$^n$-groups are the obvious things, morphisms of the groups compatible with the endomorphisms.

A cat$^{n}$-group is thus a group with $n$ independent cat$^{1}$-group structures on it.

\hypertarget{special_cases}{}\subsection*{{Special cases}}\label{special_cases}

\begin{itemize}%
\item A $cat^0$-group is a group.


\item A [[cat-1-group]] is a strict [[2-group]], viewed in a slightly different way.



\end{itemize}
\hypertarget{homotopical_example}{}\subsection*{{Homotopical example}}\label{homotopical_example}

For simplicity, we describe $\Pi X_{*}$ in a special case, namely when the $n$-cube of spaces $X_{*}$ arises from a pointed $(n +
1)$-ad $(X;X_1,\ldots ,X_n)$ by the rule: $X_{
\langle n \rangle}  = X$ and for $A$ properly contained in $\langle n \rangle$, $X_A = \bigcap _{i \not\in A} X_i$, with maps the inclusions. Let $\Phi$ be the space of maps $I^n
\to   X$ which take the faces of $I^n$ in the $i$th direction into $X_i$. Notice that $\Phi$ has the structure of $n$ compositions derived from the gluing of cubes in each direction. Let $\bullet  \in  \Phi$ be the constant map at the base point. Then $G = \pi_1(\Phi ,\bullet )$ is certainly a group. Gilbert, 1988, identifies $G$ with Loday's $\Pi
X_{*}$, so that Loday's results, obtained by methods of simplicial spaces, show that $G$ becomes a cat$^n$-group. It may also be shown that the extra groupoid structures are inherited from the compositions on $\Phi$. It is this cat$^n$-group which is written $\Pi X_*$ and is called the \emph{fundamental cat$^n$-group of the $(n + 1)$-ad}.

See also [[crossed n-cube]] for an alternative description.

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

\begin{itemize}%
\item Even though $cat^n$-groups are examples of strict [[n-fold category|(n+1)-fold categories]], Loday has shown that the [[homotopy category]] of $cat^{n-1}$-groups is equivalent to that of spaces which are pointed connected [[homotopy n-type]]s. Hence for $cat^n$-groups (thought of as $(n+1)$-fold groupoids) the [[homotopy hypothesis]] is true in this sense. See there for more details.

\end{itemize}
[[Tim Porter|Tim]] Is the first statement above correct? $Cat^n$-groups are examples of strict (n+1)-fold categories, not strict n=categories or am I missing something? (28-09-2010$<$- corrected)

[[Ronnie Brown|Ronnie]] Agreed, and I have corrected that. This is important since an [[n-category]] internal to [[Grp]] is equivalent to a single vertex [[crossed complex]] of length $n+1$.

It is not so clear how to construct a homotopical functor from $n$-cubes of \emph{non pointed spaces}, and what should be the receiving category.

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

\begin{itemize}%
\item [[crossed n-cube]]

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

The original proof of Loday's result is to be found in

\begin{itemize}%
\item [[J.-L. Loday]], \emph{Spaces with finitely many nontrivial homotopy groups}, J. Pure Appl. Alg., 24, (1982), 179--202.

\end{itemize}
This paper also uses the term n-cat-group, but later use favours the term cat$^n$-group to make it clearer that these were an [[n-fold category]] [[internalization|internal]] to [[Grp]]. There are one or two gaps in that proof and various patches and complete proofs were then given. The main one is in

\begin{itemize}%
\item [[R. Steiner]], Resolutions of spaces by cubes of fibrations. J. London Math. Soc. (2) 34 (1986), 169--176.

\end{itemize}
A proof using $cat^n$-groups and a neat detailed analysis of multisimplicial groups and related topics was given in

\begin{itemize}%
\item [[Manuel Bullejos|M. Bullejos]], [[Antonio Cegarra|A. M. Cegarra]], and [[John Duskin|J. Duskin]], \emph{On $cat^n$ -groups and homotopy types}, J. Pure Appl. Alg., 86, (1993), 135--154.

\end{itemize}
Porter (1993) gave a simple proof in terms of [[crossed n-cube|crossed n-cubes]] using as little high-powered simplicial techniques as possible in

\begin{itemize}%
\item [[Tim Porter|T. Porter]], \emph{n-types of simplicial groups and crossed n-cubes}, Topology, 32, (1993), 5--24.

\end{itemize}
[[!redirects cat-n-groups]]



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