Exponentiable morphisms in a category #
We define cartesian exponentiable morphisms,
and prove that if a morphism f : X ⟶ Y
is cartesian exponentiable
then f
is exponentiable in the slice category Over Y
.
In particular, for a morphism g : I ⟶ Y
,
the exponential f^* g
is the functor composition (baseChange g) ⋙ (Over.map g)
.
Notation #
We provide the following notations:
Π_ f
is the functorOver X ⥤ Over Y
. As such, for an objectX : Over X
, we haveΠ_ f X : Over Y
.
class
CartesianExponentiable
{C : Type u_1}
[CategoryTheory.Category.{u_2, u_1} C]
[CategoryTheory.Limits.HasPullbacks C]
{X Y : C}
(f : X ⟶ Y)
:
Type (max u_1 u_2)
A morphism f : X ⟶ Y
in a category C
is cartesian exponentiable
if there is a right adjoint to the base-change functor along f
.
- functor : CategoryTheory.Functor (CategoryTheory.Over X) (CategoryTheory.Over Y)
A functor
C/X ⥤ C/Y
right adjoint tof*
.
Instances
A functor C/X ⥤ C/Y
right adjoint to f*
.
Equations
- «termΠ__» = Lean.ParserDescr.node `«termΠ__» 90 (Lean.ParserDescr.binary `andthen (Lean.ParserDescr.symbol "Π_ ") (Lean.ParserDescr.cat `term 90))
Instances For
instance
CartesianExponentiable.id
{C : Type u_1}
[CategoryTheory.Category.{u_2, u_1} C]
[CategoryTheory.Limits.HasPullbacks C]
{I : C}
:
The identity morphisms 𝟙
are cartesian exponentiable.
Equations
- CartesianExponentiable.id = { functor := CategoryTheory.Functor.id (CategoryTheory.Over I), adj := CategoryTheory.Adjunction.id.ofNatIsoLeft (CategoryTheory.baseChange.id I).symm }
instance
CartesianExponentiable.comp
{C : Type u_1}
[CategoryTheory.Category.{u_2, u_1} C]
[CategoryTheory.Limits.HasPullbacks C]
{X Y Z : C}
(f : X ⟶ Y)
(g : Y ⟶ Z)
[fexp : CartesianExponentiable f]
[gexp : CartesianExponentiable g]
:
Equations
- One or more equations did not get rendered due to their size.
def
CartesianExponentiable.pushforwardCompIso
{C : Type u_1}
[CategoryTheory.Category.{u_2, u_1} C]
[CategoryTheory.Limits.HasPullbacks C]
{X Y Z : C}
(f : X ⟶ Y)
(g : Y ⟶ Z)
[fexp : CartesianExponentiable f]
[gexp : CartesianExponentiable g]
:
The conjugate isomorphism between pushforward functors.
Equations
- One or more equations did not get rendered due to their size.
Instances For
instance
CartesianExponentiable.exponentiableOverMk
{C : Type u_1}
[CategoryTheory.Category.{u_2, u_1} C]
[CategoryTheory.Limits.HasPullbacks C]
{X : C}
[CategoryTheory.Limits.HasFiniteWidePullbacks C]
{I : C}
(f : X ⟶ I)
[CartesianExponentiable f]
:
An arrow with a pushforward is exponentiable in the slice category.
Equations
- One or more equations did not get rendered due to their size.