Preliminary Data: Ab3 and Ab5 conditions, generator and generator family definitions.

Let be a category. Moreover, let be a family of objects of . The family is said to be a family of generators of the category if for any object of and any subobject of , distinct from , there is at least an index , and a morphism, , that cannot be factorized through the canonical injection . Then, an object of is said to be a generator of the category provided that the family is a family of generators [3] of the category .

Ab-conditions

1. (Ab3). Let us recall that an Abelian category is cocomplete (or an -category) if it has arbitrary direct sums.
2. (Ab5). A cocomplete Abelian category is said to be an -category if for any directed family of subobjects of , and for any subobject of , the following equation holds

   

Note that the condition Ab3 is equivalent to the existence of arbitrary direct limits. Please note also that Ab5 is equivalent to the fact that there exist inductive limits and the inductive limits over directed families of indices are exact, that is, if is a directed set and

is an exact sequence for any , then

is also an exact sequence.

Definition 0.1   A Grothendieck category is an category with a generator.

As an example consider the category of Abelian groups such that if is a family of abelian groups, then a direct product is defined by the Cartesian product with addition defined by the rule: . One then defines a projection given by . A direct sum is obtained by taking the appropriate subgroup consisting of all elements such that for all but a finite number of indices . Then one also defines a structural injection , and it is straightforward to prove that is an and category. (viz. p 61 in ref. [3]).

Remarks

1. Let be an Abelian category and a small category. One defines then a functor as follows: for any , is the constant functor which is associated to . Then is an category (respectively, ), if and only if for any directed set , as above, the functor has an exact left (or respectively, right) adjoint.
2. With , , , and one can construct categories of (pre) additive functors.
3. A preabelian category is an additive category with the additional ( ) condition that for any morphism in the category there exist also both and ;
4. An Abelian category can be then also defined as a preabelian category in which for any morphism , the morphism is an isomorphism (the condition).

Bibliography

AG4

Alexander Grothendieck et al. Séminaires en Géometrie Algèbrique- 4, Tome 1, Exposé 1 (or the Appendix to Exposée 1, by `N. Bourbaki' for more detail and a large number of results.), AG4 is freely available in French; also available here is an extensive Abstract in English.

1

Alexander Grothendieck, 1984. ``Esquisse d'un Programme'', (1984 manuscript), finally published in ``Geometric Galois Actions'', L. Schneps, P. Lochak, eds., London Math. Soc. Lecture Notes 242, Cambridge University Press, 1997, pp.5-48; English transl., ibid., pp. 243-283. MR 99c:14034 .

2

Alexander Grothendieck, ``La longue marche in à travers la théorie de Galois'' = ``The Long March Towards/Across the Theory of Galois'', 1981 manuscript, University of Montpellier preprint series 1996, edited by J. Malgoire.

3

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.)

4

Leila Schneps. 1994. The Grothendieck Theory of Dessins d'Enfants. (London Mathematical Society Lecture Note Series), Cambridge University Press, 376 pp.

5

David Harbater and Leila Schneps. 2000. Fundamental groups of moduli and the Grothendieck-Teichmüller group, Trans. Amer. Math. Soc. 352 (2000), 3117-3148. MSC: Primary 11R32, 14E20, 14H10; Secondary 20F29, 20F34, 32G15.