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%%% Primary Title: Grothendieck category lemma
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\begin{document}
\subsection{Introduction: proper generator}
\begin{definition}
Let us recall that a \emph{\htmladdnormallink{generator}{http://planetphysics.us/encyclopedia/Generator.html}} of a \htmladdnormallink{Grothendieck category}{http://planetphysics.us/encyclopedia/GrothendieckCategory.html} $\mathcal{G}$ is called \emph{proper} if $U$ has the property that a \htmladdnormallink{monomorphism}{http://planetphysics.us/encyclopedia/InjectiveMap.html} $i: U' \to U$ induces an \htmladdnormallink{isomorphism}{http://planetphysics.us/encyclopedia/IsomorphicObjectsUnderAnIsomorphism.html} $$Hom_{\mathcal{G}}(U,U) \cong Hom_{\mathcal{G}}(U',U)$$ if and only if $i$ is an isomorphism (viz. p. 251 in ref.
\cite{NP65}).
\end{definition}
\subsection{Grothendieck category lemma}
\begin{lemma}
Any Grothendieck category $\mathcal{G}$ has a proper generator.
\end{lemma}
\begin{thebibliography}{9}
\bibitem[AG4]{sga}
Alexander Grothendieck et al. \emph{S\'eminaires en G\'eometrie Alg\`ebrique- 4}, Tome 1, Expos\'e 1
(or the Appendix to Expos\'ee 1, by `N. Bourbaki' for more detail and a large number of results.),
AG4 is \htmladdnormallink{freely available}{http://modular.fas.harvard.edu/sga/sga/pdf/index.html} in French;
also available here is an extensive
\htmladdnormallink{Abstract in English}{http://planetmath.org/?op=getobj&from=books&id=158}.
\bibitem{NP65}
Nicolae Popescu. {\em Abelian Categories with Applications to Rings and Modules.},
Academic Press: New York and London, 1973 and 1976 edns., ({\em English translation by I. C. Baianu}.)
\bibitem{LS94}
Leila Schneps. 1994.
\htmladdnormallink{The Grothendieck Theory of Dessins d'Enfants}{http://planetmath.org/?op=getobj&from=books&id=163}.
(London Mathematical Society Lecture Note Series), Cambridge University Press, 376 pp.
\bibitem{DHSL2k}
David Harbater and Leila Schneps. 2000.
\htmladdnormallink{Fundamental groups of moduli and the Grothendieck-Teichm\"uller group}{http://www.ams.org/tran/2000-352-07/S0002-9947-00-02347-3/home.html}, \emph{Trans. Amer. Math. Soc}. 352 (2000), 3117-3148.
MSC: Primary 11R32, 14E20, 14H10; Secondary 20F29, 20F34, 32G15.
\end{thebibliography}
\end{document}
