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\begin{document}

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\section*{normal complex of groups}

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{example}{Example}\dotfill \pageref*{example} \linebreak
\noindent\hyperlink{remark}{Remark}\dotfill \pageref*{remark} \linebreak


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

[[chain complex|Chain complexes]] are most often encountered in an Abelian context, but the definition makes sense even when the [[group]]s involved need not be [[abelian group|abelian]]. What does not work well is the formation of the [[homology]] of such a chain complex of groups, except if it is `normal' in the following sense:

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

A \emph{chain complex} of groups is a sequence (of any length, finite or infinite) of groups and homomorphisms, for instance,

\begin{displaymath}
\ldots \to C_n \stackrel{\partial_n}{\longrightarrow} C_{n-1} \stackrel{\partial_{n-1}}{\longrightarrow}C_{n-2}\longrightarrow \ldots ,
\end{displaymath}
in which each composite $\partial_{n-1} \circ \partial_n$ is the trivial homomorphism.

The chain complex is \textbf{normal} if each image $\partial_n C_n$ is a [[normal subgroup]] of the next group $C_{n-1}$.

\hypertarget{example}{}\subsection*{{Example}}\label{example}

If $G$ is a [[simplicial group]] then its [[Moore complex]] is a normal complex of groups. See there.

\hypertarget{remark}{}\subsection*{{Remark}}\label{remark}

There is an obvious generalisation to normal complex of groupoids, provided one keeps to working with groupoids all having the same set of objects and morphisms which are the identity on objects.

Likewise, one can talk about normal complexes in a [[semi-abelian category]].

[[!redirects normal complexes of groups]]



\end{document}