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Mathlib.CategoryTheory.Adjunction.Opposites

Opposite adjunctions #

This file contains constructions to relate adjunctions of functors to adjunctions of their opposites.

Tags #

adjunction, opposite, uniqueness

If G.op is adjoint to F.op then F is adjoint to G.

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    If G is adjoint to F.op then F is adjoint to G.unop.

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      If G.op is adjoint to F then F.unop is adjoint to G.

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        If G is adjoint to F then F.unop is adjoint to G.unop.

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          If G is adjoint to F then F.op is adjoint to G.op.

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            If G is adjoint to F.unop then F is adjoint to G.op.

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              If G.unop is adjoint to F then F.op is adjoint to G.

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                If G.unop is adjoint to F.unop then F is adjoint to G.

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                  def CategoryTheory.Adjunction.leftAdjointsCoyonedaEquiv {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {F : CategoryTheory.Functor C D} {F' : CategoryTheory.Functor C D} {G : CategoryTheory.Functor D C} (adj1 : F G) (adj2 : F' G) :
                  F.op.comp CategoryTheory.coyoneda F'.op.comp CategoryTheory.coyoneda

                  If F and F' are both adjoint to G, there is a natural isomorphism F.op ⋙ coyoneda ≅ F'.op ⋙ coyoneda. We use this in combination with fullyFaithfulCancelRight to show left adjoints are unique.

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                    Given two adjunctions, if the right adjoints are naturally isomorphic, then so are the left adjoints.

                    Note: it is generally better to use Adjunction.natIsoEquiv, see the file Adjunction.Unique. The reason this definition still exists is that apparently CategoryTheory.extendAlongYonedaYoneda uses its definitional properties (TODO: figure out a way to avoid this).

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                      Given two adjunctions, if the left adjoints are naturally isomorphic, then so are the right adjoints.

                      Note: it is generally better to use Adjunction.natIsoEquiv, see the file Adjunction.Unique.

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