PlanetPhysics/Grothendieck Category

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Generator, Generator Family and CogeneratorEdit

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 belongs to the family of generators of ([1]).

By duality, that is, by simply reversing all arrows in the above definition one obtains the notion of a family of cogenerators of the same category , and also the notion of cogenerator of , 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 is a cogenerator of .

Ab-conditions: Ab3 and Ab5 conditionsEdit

  1. (Ab3) . Let us recall that an Abelian category is cocomplete

(or an -category) if it has arbitrary direct sums.

  1. (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


  • 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 is a directed set and is an exact sequence for any , then Failed to parse (unknown function "\limdir"): {\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 CategoriesEdit

A Grothendieck category  is an Failed to parse (unknown function "\A"): {\displaystyle \mathcal{\A}b5}

with a generator.

As an example consider the category Failed to parse (unknown function "\A"): {\displaystyle \mathcal{\A}b} 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 Failed to parse (unknown function "\A"): {\displaystyle \mathcal{\A}b} is an Failed to parse (unknown function "\A"): {\displaystyle \mathcal{\A}b6} and Failed to parse (unknown function "\A"): {\displaystyle \mathcal{\A}b4^*} category. (viz . p 61 in ref. [1]).

A co-Grothendieck category  is an  category that has a set of cogenerators,

i.e., a category whose dual is a Grothendieck category.


  1. Let Failed to parse (unknown function "\A"): {\displaystyle \mathcal{\A}} be an abelian category and a small category.

One defines then a functor Failed to parse (unknown function "\A"): {\displaystyle k_c: \mathcal{\A} \rightarrow [\mathcal{C},\mathcal{\A}]} as follows: for any Failed to parse (unknown function "\A"): {\displaystyle X \in Ob \mathcal{\A}} , Failed to parse (unknown function "\A"): {\displaystyle k_{\mathcal{C}}(X) : \mathcal{C} \rightarrow \mathcal{\A}} is the constant functor which is associated to . Then Failed to parse (unknown function "\A"): {\displaystyle \mathcal{\A}} is an Failed to parse (unknown function "\A"): {\displaystyle \mathcal{\A'' b5} category} (respectively, Failed to parse (unknown function "\A"): {\displaystyle \mathcal{\A}b5^*} ), if and only if for any directed set , as above, the functor has an exact left (or respectively, right) adjoint.

  1. With Failed to parse (unknown function "\A"): {\displaystyle \mathcal{\A}b4} , Failed to parse (unknown function "\A"): {\displaystyle \mathcal{\A}b5} , Failed to parse (unknown function "\A"): {\displaystyle \mathcal{\A}b4^*} , and Failed to parse (unknown function "\A"): {\displaystyle \mathcal{\A}b6}

one can construct categories of (pre) additive functors.

  1. A preabelian category is an \htmladdnormallink{additive category {} with the additional (Failed to parse (unknown function "\A"): {\displaystyle \mathcal{\A}b1} ) condition} that for any morphism in the category there exist also both and ;
  2. An Abelian category can be then also defined as a \em{preabelian category} in which for any morphism , the morphism is an isomorphism (the Failed to parse (unknown function "\A"): {\displaystyle \mathcal{\A}b2} condition).

All SourcesEdit



  1. 1.0 1.1 1.2 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 .)
  2. Alexander Grothendieck et al. S\'eminaires en G\'eometrie Alg\`ebrique- 4 , Tome 1, Expos\'e 1 (or the Appendix to Expos\'ee 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.
  3. 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 {\mathbf 242}, Cambridge University Press, 1997, pp.5-48; English transl., ibid., pp. 243-283. MR 99c:14034 .
  4. Alexander Grothendieck, "La longue marche in \'a travers la th\'eorie de Galois" = "The Long March Towards/Across the Theory of Galois" , 1981 manuscript, University of Montpellier preprint series 1996, edited by J. Malgoire.
  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-Teichm\"uller group, Trans. Amer. Math. Soc . 352 (2000), 3117-3148. MSC: Primary 11R32, 14E20, 14H10; Secondary 20F29, 20F34, 32G15.