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.
Propertiesedit
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 .
giraud, subcategory, mathematics, giraud, subcategories, form, important, class, subcategories, grothendieck, categories, they, named, after, jean, giraud, contents, definition, properties, also, referencesdefinition, editlet, displaystyle, mathcal, nbsp, grot. In mathematics Giraud subcategories form an important class of subcategories of Grothendieck categories They are named after Jean Giraud Contents 1 Definition 2 Properties 3 See also 4 ReferencesDefinition editLet A displaystyle mathcal A nbsp be a Grothendieck category A full subcategory B displaystyle mathcal B nbsp is called reflective if the inclusion functor i B A displaystyle i colon mathcal B rightarrow mathcal A nbsp has a left adjoint If this left adjoint of i displaystyle i nbsp also preserves kernels then B displaystyle mathcal B nbsp is called a Giraud subcategory Properties editLet B displaystyle mathcal B nbsp be Giraud in the Grothendieck category A displaystyle mathcal A nbsp and i B A displaystyle i colon mathcal B rightarrow mathcal A nbsp the inclusion functor B displaystyle mathcal B nbsp is again a Grothendieck category An object X displaystyle X nbsp in B displaystyle mathcal B nbsp is injective if and only if i X displaystyle i X nbsp is injective in A displaystyle mathcal A nbsp The left adjoint a A B displaystyle a colon mathcal A rightarrow mathcal B nbsp of i displaystyle i nbsp is exact Let C displaystyle mathcal C nbsp be a localizing subcategory of A displaystyle mathcal A nbsp and A C displaystyle mathcal A mathcal C nbsp be the associated quotient category The section functor S A C A displaystyle S colon mathcal A mathcal C rightarrow mathcal A nbsp is fully faithful and induces an equivalence between A C displaystyle mathcal A mathcal C nbsp and the Giraud subcategory B displaystyle mathcal B nbsp given by the C displaystyle mathcal C nbsp closed objects in A displaystyle mathcal A nbsp See also editLocalizing subcategoryReferences editBo Stenstrom 1975 Rings of quotients Springer Retrieved from https en wikipedia org w index php title Giraud subcategory amp oldid 1172602120, wikipedia, wiki, book, books, library,