Grothendieck category [SEP] named after
Grothendieck category
In mathematics, a Grothendieck category is a certain kind of abelian category, introduced in Alexander Grothendieck's Tôhoku paper of 1957 in order to develop the machinery of homological algebra for modules and for sheaves in a unified manner. The theory of these categories was further developed in Peter Gabriel's seminal thèse in 1962.
To every algebraic variety formula_1 one can associate a Grothendieck category formula_2, consisting of the quasi-coherent sheaves on formula_1. This category encodes all the relevant geometric information about formula_1, and formula_1 can
Giraud subcategory
In mathematics, Giraud subcategories form an important class of subcategories of Grothendieck categories. They are named after Jean Giraud.
Definition.
Let formula_1 be a Grothendieck category. A full subcategory formula_2 is called "reflective", if the inclusion functor formula_3 has a left adjoint. If this left adjoint of formula_4 also preserves 
kernels, then formula_2 is called a "Giraud subcategory".
Properties.
Let formula_2 be Giraud in the Grothendieck category formula_1 and formula_3 the inclusion functor.
- formula_2 is