Adjunctions related to the over category #
In a category with pullbacks, for any morphism f : X ⟶ Y
, the functor
Over.map f : Over X ⥤ Over Y
has a right adjoint Over.pullback f
.
In a category with binary products, for any object X
the functor
Over.forget X : Over X ⥤ C
has a right adjoint Over.star X
.
Main declarations #
Over.pullback f : Over Y ⥤ Over X
is the functor induced by a morphismf : X ⟶ Y
.Over.mapPullbackAdj
is the adjunctionOver.map f ⊣ Over.pullback f
.star : C ⥤ Over X
is the functor induced by an objectX
.forgetAdjStar
is the adjunctionforget X ⊣ star X
.
TODO #
Show star X
itself has a right adjoint provided C
is Cartesian closed and has pullbacks.
In a category with pullbacks, a morphism f : X ⟶ Y
induces a functor Over Y ⥤ Over X
,
by pulling back a morphism along f
.
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Over.map f
is left adjoint to Over.pullback f
.
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The pullback along an epi that's preserved under pullbacks is faithful.
This "preserved under pullbacks" condition is automatically satisfied in abelian categories:
example [Abelian C] [Epi f] : (pullback f).Faithful := inferInstance
pullback (𝟙 X) : Over X ⥤ Over X is the identity functor.
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pullback commutes with composition (up to natural isomorphism).
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If F
is a left adjoint and its source category has pullbacks, then so is
post F : Over Y ⥤ Over (G Y)
.
If the right adjoint of F
is G
, then the right adjoint of post F
is given by
(Y ⟶ F X) ↦ (G Y ⟶ X ×_{G F X} G Y ⟶ X)
.
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The category over any object X
factors through the category over the terminal object T
.
Equations
- CategoryTheory.Over.forgetMapTerminal X hT = CategoryTheory.NatIso.ofComponents (fun (X_1 : CategoryTheory.Over X) => CategoryTheory.Iso.refl ((CategoryTheory.Over.forget X).obj X_1)) ⋯
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The functor from C
to Over X
which sends Y : C
to π₁ : X ⨯ Y ⟶ X
, sometimes denoted X*
.
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The functor Over.forget X : Over X ⥤ C
has a right adjoint given by star X
.
Note that the binary products assumption is necessary: the existence of a right adjoint to
Over.forget X
is equivalent to the existence of each binary product X ⨯ -
.
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Note that the binary products assumption is necessary: the existence of a right adjoint to
Over.forget X
is equivalent to the existence of each binary product X ⨯ -
.
When C
has pushouts, a morphism f : X ⟶ Y
induces a functor Under X ⥤ Under Y
,
by pushing a morphism forward along f
.
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Under.pushout f
is left adjoint to Under.map f
.
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The pushout along a mono that's preserved under pushouts is faithful.
This "preserved under pushouts" condition is automatically satisfied in abelian categories:
example [Abelian C] [Mono f] : (pushout f).Faithful := inferInstance
pushout (𝟙 X) : Under X ⥤ Under X is the identity functor.
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pushout commutes with composition (up to natural isomorphism).
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Alias of CategoryTheory.Under.pushoutComp
.
pushout commutes with composition (up to natural isomorphism).
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If G
is a right adjoint and its source category has pushouts, then so is
post G : Under Y ⥤ Under (G Y)
.
If the left adjoint of G
is F
, then the left adjoint of post G
is given by
(G Y ⟶ X) ↦ (Y ⟶ Y ⨿_{F G Y} F X ⟶ F X)
.
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The category under any object X
factors through the category under the initial object I
.
Equations
- CategoryTheory.Under.forgetMapInitial X hI = CategoryTheory.NatIso.ofComponents (fun (X_1 : CategoryTheory.Under X) => CategoryTheory.Iso.refl ((CategoryTheory.Under.forget X).obj X_1)) ⋯
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The functor from C
to Under X
which sends Y : C
to in₁ : X ⟶ X ⨿ Y
.
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The functor Under.forget X : Under X ⥤ C
has a left adjoint given by costar X
.
Note that the binary coproducts assumption is necessary: the existence of a left adjoint to
Under.forget X
is equivalent to the existence of each binary coproduct X ⨿ -
.
Equations
Instances For
Note that the binary coproducts assumption is necessary: the existence of a left adjoint to
Under.forget X
is equivalent to the existence of each binary coproduct X ⨿ -
.