Grothendieck category

Generator, Generator Family and Cogenerator

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 groupoid- regarded as a small category with all its morphisms invertible- 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}$ .

Ab-conditions: 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).$

Remarks

Grothendieck and co-Grothendieck 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 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.

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 $ ker f$ and $ coker f$ ;
  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} : coim f \to im f$ is an isomorphism (the $ \mathcal{\mathcal A}b2$ condition).

Bibliography

1
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.

2
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 .

3
Alexander Grothendieck, ``La longue marche in á 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.

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 d'Enfants. (London Mathematical Society Lecture Note Series), Cambridge University Press, 376 pp.

6
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.



Contributors to this entry (in most recent order):

As of this snapshot date, this entry was owned by bci1.