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Grothendieck category
0.1 Introduction: generator, generator family and cogenerator definitions
Let $\mathcal{C}$ be a category. Moreover, let $\left\{U\right\}=\left\{U_{i}\right\}_{{i\in I}}$ be a family of objects of $\mathcal{C}$. The family $\left\{U\right\}$ is said to be a family of generators 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 morphism, $u:U_{i}\to A$, that cannot be factorized through the canonical injection $i:B\to A$. Then, an object $U$ of $\mathcal{C}$ is said to be a generator of the category $\mathcal{C}$ provided that $U$ belongs to the family of generators $\left\{U_{i}\right\}_{{i\in I}}$ of $\mathcal{C}$ ([4]).
By duality, that is, by simply reversing all arrows in the above definition one obtains the notion of a family of cogenerators $\left\{U^{*}\right\}$ of the same category $\mathcal{C}$, and also the notion of cogenerator $U^{*}$ of $\mathcal{C}$, if all of the required, reverse arrows exist. Notably, in a groupoidβ regarded as a small category with all its morphisms invertibleβ this is always possible, and thus a groupoid can always be cogenerated via duality. Moreover, any generator in the dual category $\mathcal{C}^{{op}}$ is a cogenerator of $\mathcal{C}$.
0.2 Abconditions: Ab3 and Ab5 conditions
1. (Ab3). Let us recall that an Abelian category $\mathcal{A}b$ is cocomplete (or an $\mathcal{A}b3$category) if it has arbitrary direct sums.
2. (Ab5). A cocomplete Abelian category $\mathcal{A}b$ 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
$(\sum_{{i\in I}}A_{i})\bigcap B=\sum_{{i\in I}}(A_{i}\bigcap B).$
0.2.1 Remarks

One notes that the condition Ab3 is equivalent to the existence of arbitrary direct limits.

Furthermore, Ab5 is equivalent to the following proposition: 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.

By duality, one readily obtains conditions Ab3* and Ab5* simply by reversing the arrows in the above conditions defining Ab3 and Ab5.
0.3 Grothendieck and coGrothendieck categories
Definition 0.1.
A Grothendieck category is an $\mathcal{\mathcal{A}}b5$ category with a generator.
As an example consider the category $\mathcal{\mathcal{A}}b$ of Abelian groups such that if $\left\{X_{i}\right\}_{{i\in I}}$ is a family of abelian groups, then a 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 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 structural injection , and it is straightforward to prove that $\mathcal{\mathcal{A}}b$ is an $\mathcal{\mathcal{A}}b6$ and $\mathcal{\mathcal{A}}b4^{*}$ category. (viz. p 61 in ref. [4]).
Definition 0.2.
A coGrothendieck category is an $\mathcal{A}b5^{*}$ category that has a set of cogenerators, i.e., a category whose dual is a Grothendieck category.
0.3.1 Remarks
1. Let $\mathcal{\mathcal{A}}$ be an Abelian category and $\mathcal{C}$ a small category. One defines then a functor $k_{c}:\mathcal{\mathcal{A}}\rightarrow[\mathcal{C},\mathcal{\mathcal{A}}]$ as follows: for any $X\in Ob\mathcal{\mathcal{A}}$, $k_{{\mathcal{C}}}(X):\mathcal{C}\rightarrow\mathcal{\mathcal{A}}$ is the constant functor which is associated to $X$. Then $\mathcal{\mathcal{A}}$ is an $\mathcal{\mathcal{A}}b5$ category (respectively, $\mathcal{\mathcal{A}}b5^{*}$), if and only if for any directed set $I$, as above, the functor $k_{I}$ has an exact left (or respectively, right) adjoint.
2. With $\mathcal{\mathcal{A}}b4$, $\mathcal{\mathcal{A}}b5$, $\mathcal{\mathcal{A}}b4^{*}$, and $\mathcal{\mathcal{A}}b6$ one can construct categories of (pre) additive functors.
3. A preabelian category is an additive category with the additional ($\mathcal{\mathcal{A}}b1$) condition that for any morphism $f$ in the category there exist also both $kerf$ and $cokerf$;
4. An Abelian category can be then also defined as a preabelian category in which for any morphism $f:X\to Y$, the morphism $\overline{f}:coimf\to imf$ is an isomorphism (the $\mathcal{\mathcal{A}}b2$ condition).
References
 1 Alexander Grothendieck et al. SΓ©minaires en GΓ©ometrie AlgΓ¨brique 4, Tome 1, ExposΓ© 1 (or the Appendix to ExposΓ©e 1, by βN. Bourbakiβ for more detail and a large number of results.), AG4 is freely available in French; also available here is an extensive Abstract in English.
 2 Alexander Grothendieck, 1984. βEsquisse dβun Programmeβ, (1984 manuscript), finally published in βGeometric Galois Actionsβ, L. Schneps, P. Lochak, eds., London Math. Soc. Lecture Notes 242, Cambridge University Press, 1997, pp.548; English transl., ibid., pp. 243283. MR 99c:14034 .
 3 Alexander Grothendieck, βLa longue marche in Γ‘ travers la thΓ©orie de Galoisβ = βThe Long March Towards/Across the Theory of Galoisβ, 1981 manuscript, University of Montpellier preprint series 1996, edited by J. Malgoire.
 4 Nicolae Popescu. Abelian Categories with Applications to Rings and Modules., Academic Press: New York and London, 1973 and 1976 edns., (English translation by I. C. Baianu.)
 5 Leila Schneps. 1994. The Grothendieck Theory of Dessins dβEnfants. (London Mathematical Society Lecture Note Series), Cambridge University Press, 376 pp.
 6 David Harbater and Leila Schneps. 2000. Fundamental groups of moduli and the GrothendieckTeichmΓΌller group, Trans. Amer. Math. Soc. 352 (2000), 31173148. MSC: Primary 11R32, 14E20, 14H10; Secondary 20F29, 20F34, 32G15.
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