# Grothendieck category

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 , , of an object , and any subobject , the following equality is valid:

The category of left (right) -modules over an arbitrary associative ring with an identity element and the category of sheaves of -modules over an arbitrary topological space are Grothendieck categories. A full subcategory of the category of left -modules is known as a localizing subcategory if it is closed with respect to colimits and if, in an exact sequence

the object belongs to if and only if both and belong to . Each localizing subcategory makes it possible to construct the quotient category . An Abelian category is a Grothendieck category if and only if it is equivalent to some quotient category of the type .

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 |

#### Comments

#### References

[a1] | N. Popescu, "Abelian categories with applications to rings and modules" , Acad. Press (1973) |

**How to Cite This Entry:**

Grothendieck category.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Grothendieck_category&oldid=14714