Limit preservation properties of functor.op
and related constructions #
We formulate conditions about F
which imply that F.op
, F.unop
, F.left_op
and F.right_op
preserve certain (co)limits.
Future work #
- Dually, it is possible to formulate conditions about
F.op
ec. forF
to preserve certain (co)limits.
If F : C ⥤ D
preserves colimits of K.left_op : Jᵒᵖ ⥤ C
, then F.op : Cᵒᵖ ⥤ Dᵒᵖ
preserves
limits of K : J ⥤ Cᵒᵖ
.
Instances For
If F : C ⥤ Dᵒᵖ
preserves colimits of K.left_op : Jᵒᵖ ⥤ C
, then F.left_op : Cᵒᵖ ⥤ D
preserves limits of K : J ⥤ Cᵒᵖ
.
Instances For
If F : Cᵒᵖ ⥤ D
preserves colimits of K.op : Jᵒᵖ ⥤ Cᵒᵖ
, then F.right_op : C ⥤ Dᵒᵖ
preserves
limits of K : J ⥤ C
.
Instances For
If F : Cᵒᵖ ⥤ Dᵒᵖ
preserves colimits of K.op : Jᵒᵖ ⥤ Cᵒᵖ
, then F.unop : C ⥤ D
preserves
limits of K : J ⥤ C
.
Instances For
If F : C ⥤ D
preserves limits of K.left_op : Jᵒᵖ ⥤ C
, then F.op : Cᵒᵖ ⥤ Dᵒᵖ
preserves
colimits of K : J ⥤ Cᵒᵖ
.
Instances For
If F : C ⥤ Dᵒᵖ
preserves limits of K.left_op : Jᵒᵖ ⥤ C
, then F.left_op : Cᵒᵖ ⥤ D
preserves
colimits of K : J ⥤ Cᵒᵖ
.
Instances For
If F : Cᵒᵖ ⥤ D
preserves limits of K.op : Jᵒᵖ ⥤ Cᵒᵖ
, then F.right_op : C ⥤ Dᵒᵖ
preserves
colimits of K : J ⥤ C
.
Instances For
If F : Cᵒᵖ ⥤ Dᵒᵖ
preserves limits of K.op : Jᵒᵖ ⥤ Cᵒᵖ
, then F.unop : C ⥤ D
preserves
colimits of K : J ⥤ C
.
Instances For
If F : C ⥤ D
preserves colimits of shape Jᵒᵖ
, then F.op : Cᵒᵖ ⥤ Dᵒᵖ
preserves limits of
shape J
.
Instances For
If F : C ⥤ Dᵒᵖ
preserves colimits of shape Jᵒᵖ
, then F.left_op : Cᵒᵖ ⥤ D
preserves limits
of shape J
.
Instances For
If F : Cᵒᵖ ⥤ D
preserves colimits of shape Jᵒᵖ
, then F.right_op : C ⥤ Dᵒᵖ
preserves limits
of shape J
.
Instances For
If F : Cᵒᵖ ⥤ Dᵒᵖ
preserves colimits of shape Jᵒᵖ
, then F.unop : C ⥤ D
preserves limits of
shape J
.
Instances For
If F : C ⥤ D
preserves limits of shape Jᵒᵖ
, then F.op : Cᵒᵖ ⥤ Dᵒᵖ
preserves colimits of
shape J
.
Instances For
If F : C ⥤ Dᵒᵖ
preserves limits of shape Jᵒᵖ
, then F.left_op : Cᵒᵖ ⥤ D
preserves colimits
of shape J
.
Instances For
If F : Cᵒᵖ ⥤ D
preserves limits of shape Jᵒᵖ
, then F.right_op : C ⥤ Dᵒᵖ
preserves colimits
of shape J
.
Instances For
If F : Cᵒᵖ ⥤ Dᵒᵖ
preserves limits of shape Jᵒᵖ
, then F.unop : C ⥤ D
preserves colimits
of shape J
.
Instances For
If F : C ⥤ D
preserves colimits, then F.op : Cᵒᵖ ⥤ Dᵒᵖ
preserves limits.
Instances For
If F : C ⥤ Dᵒᵖ
preserves colimits, then F.left_op : Cᵒᵖ ⥤ D
preserves limits.
Instances For
If F : Cᵒᵖ ⥤ D
preserves colimits, then F.right_op : C ⥤ Dᵒᵖ
preserves limits.
Instances For
If F : Cᵒᵖ ⥤ Dᵒᵖ
preserves colimits, then F.unop : C ⥤ D
preserves limits.
Instances For
If F : C ⥤ D
preserves limits, then F.op : Cᵒᵖ ⥤ Dᵒᵖ
preserves colimits.
Instances For
If F : C ⥤ Dᵒᵖ
preserves limits, then F.left_op : Cᵒᵖ ⥤ D
preserves colimits.
Instances For
If F : Cᵒᵖ ⥤ D
preserves limits, then F.right_op : C ⥤ Dᵒᵖ
preserves colimits.
Instances For
If F : Cᵒᵖ ⥤ Dᵒᵖ
preserves limits, then F.unop : C ⥤ D
preserves colimits.
Instances For
If F : C ⥤ D
preserves finite colimits, then F.op : Cᵒᵖ ⥤ Dᵒᵖ
preserves finite
limits.
Instances For
If F : C ⥤ Dᵒᵖ
preserves finite colimits, then F.left_op : Cᵒᵖ ⥤ D
preserves finite
limits.
Instances For
If F : Cᵒᵖ ⥤ D
preserves finite colimits, then F.right_op : C ⥤ Dᵒᵖ
preserves finite
limits.
Instances For
If F : Cᵒᵖ ⥤ Dᵒᵖ
preserves finite colimits, then F.unop : C ⥤ D
preserves finite
limits.
Instances For
If F : C ⥤ D
preserves finite limits, then F.op : Cᵒᵖ ⥤ Dᵒᵖ
preserves finite
colimits.
Instances For
If F : C ⥤ Dᵒᵖ
preserves finite limits, then F.left_op : Cᵒᵖ ⥤ D
preserves finite
colimits.
Instances For
If F : Cᵒᵖ ⥤ D
preserves finite limits, then F.right_op : C ⥤ Dᵒᵖ
preserves finite
colimits.
Instances For
If F : Cᵒᵖ ⥤ Dᵒᵖ
preserves finite limits, then F.unop : C ⥤ D
preserves finite
colimits.