Grothendieck category

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 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

• 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.

Grothendieck and co-Grothendieck categories

As an example consider the category $\mathcal{\A}b$ 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 $\mathcal{\A}b$ is an $\mathcal{\A}b6$ and $\mathcal{\A}b4^*$ category. (viz. p 61 in ref. [4]).

Remarks

1. Let $\mathcal{\A}$ be an Abelian category and $\mathcal{C}$ a small category. One defines then a functor $k_c: \mathcal{\A} \rightarrow [\mathcal{C},\mathcal{\A}]$ as follows: for any $X \in Ob \mathcal{\A}$ , $k_{\mathcal{C}}(X) : \mathcal{C} \rightarrow \mathcal{\A}$ is the constant functor which is associated to $X$ . Then $\mathcal{\A}$ is an $\mathcal{\A}b5$ 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.
2. With $\mathcal{\A}b4$ , $\mathcal{\A}b5$ , $\mathcal{\A}b4^*$ , and $\mathcal{\A}b6$ one can construct categories of (pre) additive functors.
3. A preabelian category is an additive category with the additional ($\mathcal{\A}b1$ ) 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 $\mathcal{\A}b2$ 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.