In mathematics, Giraud subcategories form an important class of subcategories of Grothendieck categories. They are named after Jean Giraud.
Let
A
be a Grothendieck category. A full subcategory
B
is called reflective, if the inclusion functor
i
:
B
→
A
has a left adjoint. If this left adjoint of
i
also preserves kernels, then
B
is called a Giraud subcategory.
Let
B
be Giraud in the Grothendieck category
A
and
i
:
B
→
A
the inclusion functor.
B
is again a Grothendieck category.
An object
X
in
B
is injective if and only if
i
(
X
)
is injective in
A
.
The left adjoint
a
:
A
→
B
of
i
is exact.
Let
C
be a localizing subcategory of
A
and
A
/
C
be the associated quotient category. The section functor
S
:
A
/
C
→
A
is fully faithful and induces an equivalence between
A
/
C
and the Giraud subcategory
B
given by the
C
-closed objects in
A
.
