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\begin{document}

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\section*{nonabelian homological algebra}

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

\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{terminology_and_relations_to_other_fields}{Terminology and relations to other fields}\dotfill \pageref*{terminology_and_relations_to_other_fields} \linebreak
\noindent\hyperlink{contributors}{Contributors}\dotfill \pageref*{contributors} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

\textbf{Nonabelian homological algebra} studies the generalizations of elements of [[homological algebra]] to ambient [[categories]] which are not [[abelian categories]].

For example, the [[5-lemma]], [[9-lemma]], the [[snake lemma]] are some of the ``elements'' which can be generalized to a wide class of categories. Typical classes of categories suitable for some elements of homological algebra are [[protomodular categories]], homological categories, and [[semi-abelian categories]].

\hypertarget{terminology_and_relations_to_other_fields}{}\subsection*{{Terminology and relations to other fields}}\label{terminology_and_relations_to_other_fields}

Though they may be shown to be related the subject of [[nonabelian cohomology]] is a bit different from nonabelian homological algebra.

The nonabelian groups were historical motivation for much of the subject. While abelian groups are the template of an object in an abelian category, large part of work was directed toward understanding of what is the right generalization of concept of a group, what lead to a definition of a [[semi-abelian category]].

\hypertarget{contributors}{}\subsection*{{Contributors}}\label{contributors}

Main contributors are [[Dominique Bourn]], [[George Janelidze]], [[Francis Borceux]]. Their direction of work is largely influenced by motivations from universal algebra.

A recent independent development is the work of [[Alexander Rosenberg]] listed below in references. It features also the nonadditive triangulated categories and a new version of algebraic K-theory as byproducts. In unfinished part of the work further generalization of the homological algebra in the fibered categories is considered.

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

\begin{itemize}%
\item [[Francis Borceux]], [[Dominique Bourn]], [[Borceux-Bourn|Mal'cev, protomodular, homological and semi-abelian categories]], Mathematics and Its Applications \textbf{566}, Kluwer 2004


\item [[George Janelidze|G. Z. Janelidze]], \emph{On [[satellites]] in arbitrary categories}, Bull. Georgian Acad. Sci. \textbf{82} (1976), no. 3, 529-532, in Russian, with a reprint translated in English at \href{http://front.math.ucdavis.edu/0809.1504}{arXiv:0809.1504}.


\item [[Alexander Rosenberg]], \emph{\href{http://www.mpim-bonn.mpg.de/preprints/send?bid=3623}{Homological algebra of noncommutative `spaces' I}} , 199 pages, preprint Max Planck, Bonn: MPIM2008-91.



\end{itemize}


\end{document}