# Grothendieck category

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An Abelian category with a set of generators (cf. Generator of a category) and satisfying the following axiom: There exist coproducts (sums) of arbitrary families of objects, and for each directed family of subobjects $U_i$, $i \in I$, of an object $A$, and any subobject $V$, the following equality holds: $$\left({ \bigcup_{i \in I} U_i }\right) \cap V = \bigcup_{i \in I} \left({ U_i \cap V }\right)$$

The category of modules (left or right) over an arbitrary associative ring $R$ with an identity element and the category of sheaves of $R$-modules over an arbitrary topological space (cf. Sheaf theory) are Grothendieck categories. A full subcategory $\mathfrak{S}$ of the category ${}_R \mathfrak{M}$ of left $R$-modules is known as a localizing subcategory if it is closed with respect to colimits and if, in an exact sequence $$0 \rightarrow A' \rightarrow A \rightarrow A''' \rightarrow 0$$ the object $A$ belongs to $\mathfrak{S}$ if and only if both $A'$ and $A''$ belong to $\mathfrak{S}$. Each localizing subcategory makes it possible to construct the quotient category ${}_R \mathfrak{M} / \mathfrak{S}$. An Abelian category is a Grothendieck category if and only if it is equivalent to some quotient category of the type ${}_R \mathfrak{M} / \mathfrak{S}$.

In a Grothendieck category each object has an injective envelope, and for this reason Grothendieck categories are well suited for use in homological applications.

#### References

 [1] A. Grothendieck, "Sur quelques points d'algèbre homologique" Tôhoku Math. J. (2) , 9 (1957) pp. 119–221 [2] I. Bucur, A. Deleanu, "Introduction to the theory of categories and functors" , Wiley (1968) [3] N. [N. Popescu] Popesco, P. Gabriel, "Charactérisation des catégories abéliennes avec générateurs et limites inductives exactes" C.R. Acad. Sci. , 258 (1964) pp. 4188–4190