PlanetPhysics/Grothendieck Category

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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 $$\mathcal{C}$$.

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
(or an $$\mathcal{A}b3$$-category) if it has arbitrary direct sums.
 * 1) (Ab3) . Let us recall that an Abelian category $$\mathcal{A}b$$ is cocomplete

$$\mathcal{A}$$, the following equation holds
 * 1) (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

$$(\sum_{i \in I}A_i) \bigcap B = \sum_{i \in I} (A_i \bigcap B).$$

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 $$0 \to \limdir{(A_i)} \to \limdir{(B_i)} \to \limdir{(C_i)} \to 0$$ 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 $$\mathcal{\A}b5$$ category with a generator.

As an example consider the category $$\mathcal{\A}b$$ 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 $$\mathcal{\A}b$$ is an $$\mathcal{\A}b6$$ and $$\mathcal{\A}b4^*$$ category. (viz . p 61 in ref. ).

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
One defines then a functor $$k_c: \mathcal{\A} \rightarrow [\mathcal{C},\mathcal{\A}]$$ as follows: for any $$X \in Ob \mathcal{\A}$$, $$k_{\mathcal{C}}(X) : \mathcal{C} \rightarrow \mathcal{\A}$$ is the constant functor which is associated to $$X$$. Then $$\mathcal{\A}$$ is an $$\mathcal{\A'' b5$$ category} (respectively, $$\mathcal{\A}b5^*$$), if and only if for any directed set $$I$$, as above, the functor $$k_I$$ has an exact left (or respectively, right) adjoint. one can construct categories of (pre) additive functors.
 * 1) Let $$\mathcal{\A}$$ be an abelian category and $$\mathcal{C}$$ a small category.
 * 1) With $$\mathcal{\A}b4$$, $$\mathcal{\A}b5$$, $$\mathcal{\A}b4^*$$, and $$\mathcal{\A}b6$$
 * 1) A preabelian category is an \htmladdnormallink{additive category {http://planetphysics.us/encyclopedia/DenseSubcategory.html} with the additional ($$\mathcal{\A}b1$$) condition} that for any morphism $$f$$ in the category there exist also both  $$ker f$$ and $$coker f$$;
 * 2) An Abelian category can be then also defined as a \em{preabelian category} in which for any morphism $$f:X \to Y$$, the morphism $$ \overline{f} : coim f \to im f$$ is an isomorphism (the $$\mathcal{\A}b2$$ condition).