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\begin{document}
\subsection{Generator, Generator Family and Cogenerator}
Let $\mathcal{C}$ be a \htmladdnormallink{category}{http://planetphysics.us/encyclopedia/Cod.html}. Moreover, let $\left\{U\right\}= \left\{U_i\right\}_{i \in I}$ be a family of \htmladdnormallink{objects}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} of $\mathcal{C}$. The \emph{family} $\left\{U\right\}$ is said to be a \emph{family of \htmladdnormallink{generators}{http://planetphysics.us/encyclopedia/Generator.html}} of the category $\mathcal{C}$ if for any object $A$ of $\mathcal{C}$ and any subobject $B$ of $A$, distinct from $A$, there is at least an index $i \in I$, and a \htmladdnormallink{morphism}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html}, $u : U_i \to A$, that cannot be factorized through the canonical \htmladdnormallink{injection}{http://planetphysics.us/encyclopedia/InjectiveMap.html} $i : B \to A$. Then, an object $U$ of $\mathcal{C}$ is said to be a \emph{generator} of the category $\mathcal{C}$ provided that $U$ belongs to the family of generators $\left\{U_i\right\}_{i \in I}$ of $\mathcal{C}$ (\cite{NP65}).
By \htmladdnormallink{duality}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html}, that is, by simply reversing all arrows in the above definition one obtains the notion of a
{\em family of cogenerators} $\left\{U^*\right\}$ of the same category $\mathcal{C}$, and also the notion of {\em cogenerator} $U^*$ of $\mathcal{C}$, if all of the required, reverse arrows exist. Notably, in a groupoid-- regarded as a \htmladdnormallink{small category}{http://planetphysics.us/encyclopedia/Cod.html} with all its morphisms invertible-- this is always possible, and thus a \htmladdnormallink{groupoid}{http://planetphysics.us/encyclopedia/QuantumOperatorAlgebra5.html} can always be cogenerated {\em via} duality. Moreover, any generator in the dual category $\mathcal{C}^{op}$ is a cogenerator of $\mathcal{C}$.
\subsection{Ab-conditions: Ab3 and Ab5 conditions}
\begin{enumerate}
\item \emph{(Ab3)}. Let us recall that an \emph{Abelian} category $\mathcal{A}b$ is \emph{cocomplete}
(or an $\mathcal{A}b3$-category) if it has arbitrary direct sums.
\item \emph{(Ab5).} A \emph{\htmladdnormallink{cocomplete Abelian category}{http://planetphysics.us/encyclopedia/CocompleteAbelianCategory.html}} $\mathcal{A}b$ is said to be an $\mathcal{A}b5$-category if for any directed family $\left\{A_i\right\}_{i \in I}$ of subobjects of $\mathcal{A}$, and for any subobject $B$ of
$\mathcal{A}$, the following equation holds
$(\sum_{i \in I}A_i) \bigcap B = \sum_{i \in I} (A_i \bigcap B).$
\end{enumerate}
\subsubsection{Remarks}
\begin{itemize}
\item One notes that the condition \emph{Ab3} is \emph{equivalent to the existence of arbitrary direct limits}.
\item Furthermore, \emph{Ab5} is equivalent to the following \htmladdnormallink{proposition}{http://planetphysics.us/encyclopedia/Predicate.html}:
\emph{there exist inductive limits and the inductive limits over directed families of indices are exact},
that is, if $I$ is a directed set and $0 \to A_i \to B_i \to C_i \to 0$ is an exact
sequence for any $i \in I$, then
$$0 \to \limdir{(A_i)} \to \limdir{(B_i)} \to \limdir{(C_i)} \to 0$$
is also an exact sequence.
\item By duality, one readily obtains conditions \emph{Ab3*} and \emph{Ab5*} simply
by reversing the arrows in the above conditions defining \emph{Ab3} and \emph{Ab5}.
\end{itemize}
\subsection{Grothendieck and co-Grothendieck Categories}
\begin{definition} A {\em Grothendieck category} is an $\mathcal{\A}b5$ category
with a generator.
\end{definition}
As an example consider the category $\mathcal{\A}b$ of \htmladdnormallink{Abelian groups}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html}
such that if $\left\{X_i \right\}_{i \in I}$ is a family of abelian groups, then
a {\em direct product} $\Pi$ is defined by the Cartesian product $\Pi _i (X_i)$
with addition defined by the rule: $(x_i) + (y_i) = (x_i + y_i)$.
One then defines a projection $\rho : \Pi _i (X_i) \rightarrow X_i$ given by
$p_i ((x_i)) = x_i$. A {\em direct sum} is obtained by taking the appropriate subgroup
consisting of all elements $(x_i)$ such that $x_i = 0$ for all but a finite number of indices
$i$. Then one also defines a {\em structural injection} , and it is straightforward
to prove that $\mathcal{\A}b$ is an $\mathcal{\A}b6$ and $\mathcal{\A}b4^*$
category. (\emph{viz}. p 61 in ref. \cite{NP65}).
\begin{definition} A \emph{co-Grothendieck category} is an $\mathcal{A}b5^*$ category that has a set of cogenerators,
i.e., a category whose dual is a Grothendieck category.
\end{definition}
\subsubsection{Remarks}
\begin{enumerate}
\item Let $\mathcal{\A}$ be an \htmladdnormallink{abelian category}{http://planetphysics.us/encyclopedia/AbelianCategory2.html} and $\mathcal{C}$ a small category.
One defines then a \htmladdnormallink{functor}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} $k_c: \mathcal{\A} \rightarrow [\mathcal{C},\mathcal{\A}]$
as follows: for any $X \in Ob \mathcal{\A}$, $k_{\mathcal{C}}(X) : \mathcal{C} \rightarrow \mathcal{\A}$ is the
{\em constant functor} which is associated to $X$. Then $\mathcal{\A}$ is an {\em $\mathcal{\A}b5$ category} (respectively, $\mathcal{\A}b5^*$), if and only if for any directed set $I$, as above, the functor $k_I$ has an exact left (or respectively, right) adjoint.
\item With $\mathcal{\A}b4$, $\mathcal{\A}b5$, $\mathcal{\A}b4^*$, and $\mathcal{\A}b6$
one can construct categories of (pre) additive functors.
\item A \emph{preabelian category} is {\em an \htmladdnormallink{additive category}{http://planetphysics.us/encyclopedia/DenseSubcategory.html} with the additional ($\mathcal{\A}b1$) condition} that for any morphism $f$ in the category there exist also \emph{both} $ker f$ and $coker f$;
\item An \emph{Abelian category} can be then also defined as a \em{preabelian category} in which for any morphism $f:X \to Y$, the morphism $ \overline{f} : coim f \to im f$ is an \htmladdnormallink{isomorphism}{http://planetphysics.us/encyclopedia/IsomorphicObjectsUnderAnIsomorphism.html} (the $\mathcal{\A}b2$ condition).
\end{enumerate}
\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{Alex84}
Alexander Grothendieck, 1984. ``Esquisse d'un Programme'', (1984 manuscript), {\em finally published in ``Geometric Galois Actions''}, L. Schneps, P. Lochak, eds.,
{\em London Math. Soc. Lecture Notes} {\bf 242}, Cambridge University Press, 1997, pp.5-48;
English transl., ibid., pp. 243-283. MR 99c:14034 .
\bibitem{Alex81}
Alexander Grothendieck, ``La longue marche in \'a travers la th\'eorie de Galois''
\emph{= ``The Long March Towards/Across the Theory of Galois''}, 1981 manuscript, University of Montpellier preprint series 1996, edited by J. Malgoire.
\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}
