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\begin{document}

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\section*{semi-abelian category}

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

\hypertarget{homological_algebra}{}\paragraph*{{Homological algebra}}\label{homological_algebra}

[[!include homological algebra - contents]]

\hypertarget{semiabelian_categories}{}\section*{{Semiabelian categories}}\label{semiabelian_categories}

\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{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak
\noindent\hyperlink{doldkan_correspondence}{Dold--Kan correspondence}\dotfill \pageref*{doldkan_correspondence} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

The axioms of a \emph{semi-abelian} category are supposed to capture the properties of the categories of [[groups]], [[rng|rings without unit]], [[associative algebras]] without unit, [[Lie algebras]], as nicely as the axioms of an [[abelian category]] captures the properties of the categories of [[abelian groups]] and of [[modules]].

[[Mike Shulman|Mike]]: Why only rings without units (that is, rngs)? Intuitively, what important properties do the above listed examples share that are not shared by rings with units?

[[Zoran Skoda]]: I want to know the answer as well. It might be something in the self-dual axioms. For unital rings artinian implies noetherian but not other way around; though the definitions of the two notions are dual.

\emph{Toby}: The category of unital rings and unitary ring homomorphisms has no zero object.

[[Mike Shulman|Mike]]: Ah, right. Is it protomodular? I think I will understand this definition better from some non-examples that violate each clause individually.

[[arsmath|walt]]: It is protomodular. This follows from the main theorem of \emph{Characterization of Protomodular Varieties of Universal Algebra} by Bourn and Janelidze. By that theorem any variety that contains a group will be protomodular. Unital rings only fail to be semiabelian for the trivial reason that ideals aren't subrings.

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

A [[category]] $C$ is \textbf{semi-abelian} if it

\begin{itemize}%
\item is [[exact category|Barr-exact]] (hence [[regular category|regular]] and in particular has finite [[limit]]s);


\item has a [[zero object]];


\item has finite [[coproduct]]s; and


\item is [[protomodular category|protomodular]].



\end{itemize}
In other words, it is a [[homological category]] which is [[Barr-exact]] and has finite [[coproducts]].

Equivalently, $C$ is semi-abelian if:

\begin{itemize}%
\item it has finite [[product]]s and [[coproduct]]s and a [[zero object]];


\item it has [[pullback]]s of [[monomorphism]]s (or even only of [[split monomorphism]]s);


\item it has [[coequalizer]]s of [[kernel pair]]s;


\item [[regular epimorphism]]s are stable under [[pullback]];


\item [[congruence|equivalence relations]] are effective; and


\item the \emph{Split Short [[five lemma|Five Lemma]]} holds:



\end{itemize}
\begin{udefn}
\textbf{(split short five lemma)}

Given a [[commutative diagram]]

\begin{displaymath}
\itexarray{
    L & \overset{l}{\to} & F & \overset{q}{\to} & C
   \\
   {}^{\mathllap{u}}\downarrow 
     && \downarrow^{\mathrlap{w}} && \downarrow^{\mathrlap{v}} 
   \\
    K & \underset{k}{\to} & E& \underset{p}{\to} & B
   }
\end{displaymath}
where

\begin{itemize}%
\item $p$ and $q$ are [[split epimorphism]]s


\item and $l$ and $k$ are their [[kernel]]s,



\end{itemize}
then if $u$ and $v$ are [[isomorphism]]s so is $w$.

\end{udefn}
To see that the second list of axioms implies the existence of finite limits, observe that the pullback

\begin{displaymath}
\itexarray{P & \to & A\\
  \downarrow && \downarrow^f\\
  B& \underset{g}{\to} & C}
\end{displaymath}
can be computed as the pullback

\begin{displaymath}
\itexarray{P & \to & A\times B\\
  \downarrow && \downarrow^{(1,1,f)}\\
  A\times B& \underset{(1,1,g)}{\to} & A\times B\times C}
\end{displaymath}
in which both legs are split monics. Filling in one of the equivalent definitions of Barr-exactness, the equivalence of the two lists of axioms reduces to showing that in a Barr-exact category with coproducts and a zero object, protomodularity is equivalent to the Split Short Five Lemma; see the paper referenced below for a proof.

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

\begin{itemize}%
\item Every [[abelian category]] is semi-abelian. Conversely, a semi-abelian category is abelian if and only if it is [[additive category|additive]] (since any exact additive category is abelian), and if and only if its opposite is semi-abelian.


\item The category [[Grp]] of not-necessarily-abelian [[group]]s is semi-abelian but not [[abelian category|abelian]]. So are the categories of rings without units, algebras without units, Lie algebras, and many other sorts of algebras. (The category of rings with unit is not semi-abelian since it lacks a zero object.)


\item More generally, the category of internal [[group object]]s in any exact category is semi-abelian as soon as it has finite coproducts. For instance, this applies to internal groups in any [[topos]] with a [[natural numbers object|NNO]].


\item The opposite of any [[topos]], such as $Set^{op}$, is Barr-exact and protomodular, but obviously lacks a zero object.


\item The category of Heyting semilattices


\item The category of (ordinary) Lie algebras


\item The category $Set_*$ of [[pointed set]]s is Barr-exact with finite coproducts and a zero object, but is not semi-abelian: protomodularity and the Split Short Five Lemma fail to hold.


\item If $C$ is exact and protomodular with finite colimits, then for any $x\in C$ the [[over category|over]]-[[under category|under]] category $(x/C/x)$ is semi-abelian. For example, the opposite of the category of [[pointed object|pointed objects]] in a [[topos]] is semi-abelian, and in particular, $Set_*^{op}$ is semi-abelian.



\end{itemize}
[[Urs Schreiber|Urs]]: how can I understand that this (has to?) involve the opposite category?

[[Mike Shulman|Mike]]: Well, as the previous example shows, $Set_*$ itself is not semi-abelian. The way I'm thinking of it is that a surjection of pointed sets is not determined by its kernel, but an injection of pointed sets is determined by its cokernel.

\begin{itemize}%
\item The categories of [[crossed module|crossed modules]], [[crossed complex]]es, and their friends are semi-abelian; see example 4.2.6 of the Van der Linden paper referenced below.

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

\hypertarget{general}{}\subsubsection*{{General}}\label{general}

(\ldots{})

\hypertarget{doldkan_correspondence}{}\subsubsection*{{Dold--Kan correspondence}}\label{doldkan_correspondence}

\begin{itemize}%
\item The analogue of a [[Dold–Kan correspondence]] holds for [[simplicial object]]s in semi-abelian categories.

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

\begin{itemize}%
\item [[George Janelidze]], L\'a{}szl\'o{} M\'a{}rki, [[Walter Tholen]], \emph{Semi-abelian categories}, J. Pure Appl. Alg. \textbf{168}, 2-3 (2002) 367-386, 


\item [[Dominique Bourn]], [[Francis Borceux]], [[Mal'cev, protomodular, homological and semi-abelian categories]], Kluwer 2004.


\item Dominique Bourn, [[Maria Manuel Clementino]], \emph{Categorical and topological aspects of semi-abelian theories} , lecture notes Haute Bodeux 2007. (\href{http://www.math.yorku.ca/~tholen/HB07BournClementino.pdf}{pdf})


\item [[Tim Van der Linden]], \emph{Homology and homotopy in semi-abelian categories}, \href{http://arxiv.org/abs/math/0607100}{math/0607100}.



\end{itemize}
[[!redirects semi-abelian category]] [[!redirects semi-abelian categories]] [[!redirects semiabelian category]] [[!redirects semiabelian categories]]



\end{document}