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Grothendieck category (Definition)

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

  • 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

    $\displaystyle 0 \to \limdir {(A_i)} \to \limdir {(B_i)} \to \limdir {(C_i)} \to 0$
    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

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.



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See Also: proper generator theorem, $C_2$-category, $C_1$-category, $C_3$-category

Other names:  Ab5-category with a generator
Keywords:  Grothendieck category

Cross-references: isomorphism, additive category, functor, Abelian category, Abelian groups, proposition, cocomplete Abelian category, groupoid, small category, duality, injection, morphism, generators, objects, category
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This is version 4 of Grothendieck category, born on 2009-02-16, modified 2009-02-16.
Object id is 525, canonical name is GrothendieckCategory.
Accessed 1344 times total.

Classification:
Physics Classification: 00. (GENERAL)
 02. (Mathematical methods in physics)
 03. (Quantum mechanics, field theories, and special relativity )
 03.65.Fd (Algebraic methods )
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