Giraud subcategory explained
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.
in
is
injective if and only if
is injective in
.
of
is
exact.
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.