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Mathlib.CategoryTheory.Adjunction.Over

Adjunctions related to the over category #

Construct the left adjoint star X to Over.forget X : Over X ⥤ C.

TODO #

Show star X itself has a left adjoint provided C is locally cartesian closed.

@[simp]
theorem CategoryTheory.Over.baseChange_obj_hom {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasPullbacks C] {X : C} {Y : C} (f : X Y) (g : CategoryTheory.Over Y) :
((CategoryTheory.Over.baseChange f).obj g).hom = CategoryTheory.Limits.pullback.snd

Given a morphism f : X ⟶ Y, the functor baseChange f takes morphisms over Y to morphisms over X via pullbacks.

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    @[deprecated CategoryTheory.Over.baseChange]

    Alias of CategoryTheory.Over.baseChange.


    Given a morphism f : X ⟶ Y, the functor baseChange f takes morphisms over Y to morphisms over X via pullbacks.

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      @[simp]
      theorem CategoryTheory.Over.mapAdjunction_counit_app {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasPullbacks C] {X : C} {Y : C} (f : X Y✝) (Y : CategoryTheory.Over Y✝) :
      (CategoryTheory.Over.mapAdjunction f).counit.app Y = CategoryTheory.Over.homMk CategoryTheory.Limits.pullback.fst

      The adjunction Over.map ⊣ baseChange

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        @[simp]
        theorem CategoryTheory.Over.star_obj_hom {C : Type u} [CategoryTheory.Category.{v, u} C] (X : C) [CategoryTheory.Limits.HasBinaryProducts C] (X : C) :
        ((CategoryTheory.Over.star X✝).obj X).hom = CategoryTheory.Limits.prod.fst

        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 ⨯ -.

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            @[deprecated CategoryTheory.Over.star]

            Alias of CategoryTheory.Over.star.


            The functor from C to Over X which sends Y : C to π₁ : X ⨯ Y ⟶ X, sometimes denoted X*.

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              @[deprecated CategoryTheory.Over.forgetAdjStar]

              Alias of CategoryTheory.Over.forgetAdjStar.


              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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