# PlanetPhysics/Grothendieck Category

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### Generator, Generator Family and Cogenerator

Let ${\displaystyle {\mathcal {C}}}$ be a category. Moreover, let ${\displaystyle \left\{U\right\}=\left\{U_{i}\right\}_{i\in I}}$ be a family of objects of ${\displaystyle {\mathcal {C}}}$. The family ${\displaystyle \left\{U\right\}}$ is said to be a family of generators of the category ${\displaystyle {\mathcal {C}}}$ if for any object ${\displaystyle A}$ of ${\displaystyle {\mathcal {C}}}$ and any subobject ${\displaystyle B}$ of ${\displaystyle A}$, distinct from ${\displaystyle A}$, there is at least an index ${\displaystyle i\in I}$, and a morphism, ${\displaystyle u:U_{i}\to A}$, that cannot be factorized through the canonical injection ${\displaystyle i:B\to A}$. Then, an object ${\displaystyle U}$ of ${\displaystyle {\mathcal {C}}}$ is said to be a generator of the category ${\displaystyle {\mathcal {C}}}$ provided that ${\displaystyle U}$ belongs to the family of generators ${\displaystyle \left\{U_{i}\right\}_{i\in I}}$ of ${\displaystyle {\mathcal {C}}}$ ([1]).

By duality, that is, by simply reversing all arrows in the above definition one obtains the notion of a family of cogenerators ${\displaystyle \left\{U^{*}\right\}}$ of the same category ${\displaystyle {\mathcal {C}}}$, and also the notion of cogenerator ${\displaystyle U^{*}}$ of ${\displaystyle {\mathcal {C}}}$, 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 ${\displaystyle {\mathcal {C}}^{op}}$ is a cogenerator of ${\displaystyle {\mathcal {C}}}$.

### Ab-conditions: Ab3 and Ab5 conditions

1. (Ab3) . Let us recall that an Abelian category ${\displaystyle {\mathcal {A}}b}$ is cocomplete

(or an ${\displaystyle {\mathcal {A}}b3}$-category) if it has arbitrary direct sums.

1. (Ab5). A cocomplete Abelian category ${\displaystyle {\mathcal {A}}b}$ is said to be an ${\displaystyle {\mathcal {A}}b5}$-category if for any directed family ${\displaystyle \left\{A_{i}\right\}_{i\in I}}$ of subobjects of ${\displaystyle {\mathcal {A}}}$, and for any subobject ${\displaystyle B}$ of

${\displaystyle {\mathcal {A}}}$, the following equation holds

${\displaystyle (\sum _{i\in I}A_{i})\bigcap B=\sum _{i\in I}(A_{i}\bigcap B).}$

#### 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 ${\displaystyle I}$ is a directed set and ${\displaystyle 0\to A_{i}\to B_{i}\to C_{i}\to 0}$ is an exact sequence for any ${\displaystyle i\in I}$, 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

A Grothendieck category  is an $\displaystyle \mathcal{\A}b5$
category


with a generator.

As an example consider the category $\displaystyle \mathcal{\A}b$ of Abelian groups such that if ${\displaystyle \left\{X_{i}\right\}_{i\in I}}$ is a family of abelian groups, then a direct product ${\displaystyle \Pi }$ is defined by the Cartesian product ${\displaystyle \Pi _{i}(X_{i})}$ with addition defined by the rule: ${\displaystyle (x_{i})+(y_{i})=(x_{i}+y_{i})}$. One then defines a projection ${\displaystyle \rho :\Pi _{i}(X_{i})\rightarrow X_{i}}$ given by ${\displaystyle p_{i}((x_{i}))=x_{i}}$. A direct sum is obtained by taking the appropriate subgroup consisting of all elements ${\displaystyle (x_{i})}$ such that ${\displaystyle x_{i}=0}$ for all but a finite number of indices ${\displaystyle i}$. Then one also defines a structural injection , and it is straightforward to prove that $\displaystyle \mathcal{\A}b$ is an $\displaystyle \mathcal{\A}b6$ and $\displaystyle \mathcal{\A}b4^*$ category. (viz . p 61 in ref. [1]).

A co-Grothendieck category  is an ${\displaystyle {\mathcal {A}}b5^{*}}$ category that has a set of cogenerators,


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

#### Remarks

1. Let $\displaystyle \mathcal{\A}$ be an abelian category and ${\displaystyle {\mathcal {C}}}$ a small category.

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

1. With $\displaystyle \mathcal{\A}b4$ , $\displaystyle \mathcal{\A}b5$ , $\displaystyle \mathcal{\A}b4^*$ , and $\displaystyle \mathcal{\A}b6$

one can construct categories of (pre) additive functors.

1. A preabelian category is an \htmladdnormallink{additive category {http://planetphysics.us/encyclopedia/DenseSubcategory.html} with the additional ($\displaystyle \mathcal{\A}b1$ ) condition} that for any morphism ${\displaystyle f}$ in the category there exist also both ${\displaystyle kerf}$ and ${\displaystyle cokerf}$;
2. An Abelian category can be then also defined as a \em{preabelian category} in which for any morphism ${\displaystyle f:X\to Y}$, the morphism ${\displaystyle {\overline {f}}:coimf\to imf}$ is an isomorphism (the $\displaystyle \mathcal{\A}b2$ condition).

## References

1. 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.