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Mathematical subcategories of Grothendieck categories
In mathematics, Giraud subcategories form an important class of subcategories of Grothendieck categories. They are named after Jean Giraud.
Let A {\displaystyle {\mathcal {A}}} be a Grothendieck category. A full subcategory B {\displaystyle {\mathcal {B}}} is called reflective, if the inclusion functor i : B → A {\displaystyle i\colon {\mathcal {B}}\rightarrow {\mathcal {A}}} has a left adjoint. If this left adjoint of i {\displaystyle i} also preserves kernels, then B {\displaystyle {\mathcal {B}}} is called a Giraud subcategory.
Let B {\displaystyle {\mathcal {B}}} be Giraud in the Grothendieck category A {\displaystyle {\mathcal {A}}} and i : B → A {\displaystyle i\colon {\mathcal {B}}\rightarrow {\mathcal {A}}} the inclusion functor.
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