abelian categories, examples ofExamplesOfAbelianCategory2007-02-20 03:50:022008-10-28 13:24:52Exampleabelian categoryAbelian groupsGrothendieck categoriescommutative groupoidsreversible automataquantum automataringsmodules% this is the default PlanetMath preamble. as your knowledge
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The axiomatization of abelian categories was intended to capture some of the useful properties of categories in homological algebra. This entry gives some examples of abelian categories.
\begin{example}
The category $\Ab$ of abelian groups is an abelian category. [\PMlinkname{Proof}{ProofThatAbelianGroupsFormAnAbelianCategory}]
\end{example}
Since abelian groups are a special case of $R$-modules, one might ask whether categories of $R$-modules are also abelian categories. This is the case.
\begin{example}
For any ring $R$, the category $\Mod{R}$ of left $R$-modules is an abelian category.
\end{example}
\begin{example}
For any ring $R$, the category $\mathcal{C}(R)$ of \PMlinkname{complexes}{ChainComplex} of left $R$-modules is an abelian category.
\end{example}
\begin{example}
For any topological space $X$, the category of sheaves of abelian groups over $X$ is an abelian category.
\end{example}
\begin{example}
Every Grothendieck category that satisfies the $\mathcal{A}b6$ axiom is Abelian.
\end{example}
\begin{example}
For any topological groupoid $\mathcal{G}$, the 2--category of sheaves of commutative groupoids over the groupoid space $X_G$ is an abelian 2-category.
\end{example}
\begin{example}
The 2--category of commutative groupoids $\mathcal{G_C}$ is an Abelian 2--category.
\end{example}
\begin{example}
For any reversible sequential machine, or automaton, $R_S$, (with all state transitions reversible), the category $\mathcal{A}(R_S)$ of such reversible automata is an Abelian category.
\end{example}
\textbf{Counter-example}
For general quantum automata $Q_A$s, the category $\mathcal{Q_A}$ of such quantum automata is in general, non-Abelian
(or nonabelian).