# 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

*(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

#### RemarksEdit

- 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

AGrothendieck categoryis anFailed to parse (unknown function "\A"): {\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 (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]}).

Aco-Grothendieck categoryis an category that has a set of cogenerators,

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

#### RemarksEdit

- 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.

- 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.

- 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 (unknown function "\A"): {\displaystyle \mathcal{\A}b2}**condition).

## All SourcesEdit

^{[2]}^{[3]}^{[4]}^{[1]}^{[5]}^{[6]}

## ReferencesEdit

- ↑
^{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.