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