PlanetPhysics/Grothendieck Category
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Generator, Generator Family and Cogenerator
editLet 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 conditions
edit- (Ab3) . Let us recall that an Abelian category is cocomplete
(or an -category) if it has arbitrary direct sums.
- (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
Remarks
edit- 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 (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikiversity.org/v1/":): {\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
editA Grothendieck category is an Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikiversity.org/v1/":): {\displaystyle \mathcal{\A}b5} category
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 (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikiversity.org/v1/":): {\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.
Remarks
edit- 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 (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikiversity.org/v1/":): {\displaystyle k_c: \mathcal{\A} \rightarrow [\mathcal{C},\mathcal{\A}]} as follows: for any Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikiversity.org/v1/":): {\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 (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikiversity.org/v1/":): {\displaystyle \mathcal{\A}} is an Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikiversity.org/v1/":): {\displaystyle \mathcal{\A'' b5} category} (respectively, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikiversity.org/v1/":): {\displaystyle \mathcal{\A}b5^*} ), if and only if for any directed set , as above, the functor has an exact left (or respectively, right) adjoint.
- With Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikiversity.org/v1/":): {\displaystyle \mathcal{\A}b4} , Failed to parse (unknown function "\A"): {\displaystyle \mathcal{\A}b5} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikiversity.org/v1/":): {\displaystyle \mathcal{\A}b4^*} , and Failed to parse (unknown function "\A"): {\displaystyle \mathcal{\A}b6}
one can construct categories of (pre) additive functors.
- A preabelian category is an \htmladdnormallink{additive category {http://planetphysics.us/encyclopedia/DenseSubcategory.html} 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 ;
- 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 (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikiversity.org/v1/":): {\displaystyle \mathcal{\A}b2} condition).
All Sources
editReferences
edit- ↑ 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 .)
- ↑ 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.
- ↑ 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 .
- ↑ 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.
- ↑ Leila Schneps. 1994. The Grothendieck Theory of Dessins d'Enfants. (London Mathematical Society Lecture Note Series), Cambridge University Press, 376 pp.
- ↑ 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.