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