%%% This file is part of PlanetPhysics snapshot of 2011-09-01
%%% Primary Title: $C_3$-category theorem
%%% Primary Category Code: 00.
%%% Filename: C_3CategoryTheorem.tex
%%% Version: 2
%%% Owner: bci1
%%% Author(s): bci1
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\begin{document}
\begin{theorem} (\htmladdnormallink{proposition}{http://planetphysics.us/encyclopedia/Predicate.html} 1.2. in ref. \cite{BM266}.)
A cocomplete Abelian category is $C_3$ if and only if the direct limit of every direct family
of subobjects $\left\{A_i\right\}$ of an \htmladdnormallink{object}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} $A$ is equal to $\bigcup A_i$.
\end{theorem}
\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}
\end{thebibliography}
\end{document}
