Giraud subcategory
In mathematics, Giraud subcategories form an important class of subcategories of Grothendieck categories. They are named after Jean Giraud.
Definition
Let be a Grothendieck category. A full subcategory is called reflective, if the inclusion functor has a left adjoint. If this left adjoint of also preserves kernels, then is called a Giraud subcategory.
Properties
Let be Giraud in the Grothendieck category and the inclusion functor.
- is again a Grothendieck category.
- An object in is injective if and only if is injective in .
- The left adjoint of is exact.
- Let be a localizing subcategory of and be the associated quotient category. The section functor is fully faithful and induces an equivalence between and the Giraud subcategory given by the -closed objects in .
See also
References
- Bo Stenström; 1975; Rings of quotients. Springer.
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