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
additionally 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.