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%%% Owner: bci1
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\begin{document}
\section{Abelian $C_3$-category}
\begin{definition}
Let $\mathcal{A}$ be an \emph{Abelian cocomplete \htmladdnormallink{category}{http://planetphysics.us/encyclopedia/Cod.html}}, defined as the dual of an Abelian complete category.
A \emph{$C_3$-category} is defined as a \htmladdnormallink{cocomplete Abelian category}{http://planetphysics.us/encyclopedia/CocompleteAbelianCategory.html} $\mathcal{A}$
such that the following distributivity \htmladdnormallink{relation}{http://planetphysics.us/encyclopedia/Bijective.html} holds for any direct family $\left\{A_i\right\}$ and any subobject $B$:
$$(\bigcup A_i) \bigcap B = \bigcup (A_i \bigcap B),$$
(\cite{BM266})
\end{definition}
\begin{remark}
A \emph{$C_3$-category} is also called an $\mathcal{A}b5$-category.
\end{remark}
\begin{example}
The dual of the Cartesian closed category of finite Abelian \htmladdnormallink{quantum groups}{http://planetphysics.us/encyclopedia/QuantumGroup4.html} with exponential
elements (including \htmladdnormallink{Lie groups}{http://planetphysics.us/encyclopedia/BilinearMap.html}) and quantum group \htmladdnormallink{homomorphisms}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} is a $C_3$-category.
\end{example}
\begin{thebibliography}{9}
\bibitem{BM266}
See p.82 and eq. (1) in ref. $[266]$ in the
\htmladdnormallink{Bibliography for categories and algebraic topology}{http://planetphysics.us/encyclopedia/BibliographyForCategoryTheoryAndAlgebraicTopologyApplicationsInTheoreticalPhysics.html}
\bibitem{NP288}
Ref. $[288]$ in the \htmladdnormallink{Bibliography for categories and algebraic topology}{http://planetphysics.us/encyclopedia/BibliographyForCategoryTheoryAndAlgebraicTopologyApplicationsInTheoreticalPhysics.html}
\end{thebibliography}
\end{document}
