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Mathlib.CategoryTheory.Adjunction.FullyFaithful

Adjoints of fully faithful functors #

A left adjoint is

A right adjoint is

This is Lemma 4.5.13 in Riehl's Category Theory in Context [riehl2017]. See also https://stacks.math.columbia.edu/tag/07RB for the statements about fully faithful functors.

In the file Mathlib.CategoryTheory.Monad.Adjunction, we prove that in fact, if there exists an isomorphism L ⋙ R ≅ 𝟭 C, then the unit is an isomorphism, and similarly for the counit. See CategoryTheory.Adjunction.isIso_unit_of_iso and CategoryTheory.Adjunction.isIso_counit_of_iso.

If the left adjoint is faithful, then each component of the unit is an monomorphism.

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If the left adjoint is full, then each component of the unit is a split epimorphism.

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  • h.unitSplitEpiOfLFull X = { section_ := L.preimage (h.counit.app (L.obj X)), id := }
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    If the right adjoint is full, then each component of the counit is a split monomorphism.

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    If the left adjoint is fully faithful, then the unit is an isomorphism.

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    If the right adjoint is faithful, then each component of the counit is an epimorphism.

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    If the right adjoint is full, then each component of the counit is a split monomorphism.

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    • h.counitSplitMonoOfRFull X = { retraction := R.preimage (h.unit.app (R.obj X)), id := }
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      If the right adjoint is full, then each component of the counit is a split monomorphism.

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      If the right adjoint is fully faithful, then the counit is an isomorphism.

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      @[simp]
      theorem CategoryTheory.Adjunction.inv_map_unit {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {L : CategoryTheory.Functor C D} {R : CategoryTheory.Functor D C} (h : L R) {X : C} [CategoryTheory.IsIso (h.unit.app X)] :
      CategoryTheory.inv (L.map (h.unit.app X)) = h.counit.app (L.obj X)

      If the unit of an adjunction is an isomorphism, then its inverse on the image of L is given by L whiskered with the counit.

      @[simp]
      theorem CategoryTheory.Adjunction.whiskerLeftLCounitIsoOfIsIsoUnit_hom_app {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {L : CategoryTheory.Functor C D} {R : CategoryTheory.Functor D C} (h : L R) [CategoryTheory.IsIso h.unit] (X : C) :
      h.whiskerLeftLCounitIsoOfIsIsoUnit.hom.app X = h.counit.app (L.obj X)
      @[simp]
      theorem CategoryTheory.Adjunction.whiskerLeftLCounitIsoOfIsIsoUnit_inv_app {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {L : CategoryTheory.Functor C D} {R : CategoryTheory.Functor D C} (h : L R) [CategoryTheory.IsIso h.unit] (X : C) :
      h.whiskerLeftLCounitIsoOfIsIsoUnit.inv.app X = L.map (h.unit.app X)

      If the unit is an isomorphism, bundle one has an isomorphism L ⋙ R ⋙ L ≅ L.

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        @[simp]
        theorem CategoryTheory.Adjunction.inv_counit_map {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {L : CategoryTheory.Functor C D} {R : CategoryTheory.Functor D C} (h : L R) {X : D} [CategoryTheory.IsIso (h.counit.app X)] :
        CategoryTheory.inv (R.map (h.counit.app X)) = h.unit.app (R.obj X)

        If the counit of an adjunction is an isomorphism, then its inverse on the image of R is given by R whiskered with the unit.

        @[simp]
        theorem CategoryTheory.Adjunction.whiskerLeftRUnitIsoOfIsIsoCounit_hom_app {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {L : CategoryTheory.Functor C D} {R : CategoryTheory.Functor D C} (h : L R) [CategoryTheory.IsIso h.counit] (X : D) :
        h.whiskerLeftRUnitIsoOfIsIsoCounit.hom.app X = R.map (h.counit.app X)
        @[simp]
        theorem CategoryTheory.Adjunction.whiskerLeftRUnitIsoOfIsIsoCounit_inv_app {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {L : CategoryTheory.Functor C D} {R : CategoryTheory.Functor D C} (h : L R) [CategoryTheory.IsIso h.counit] (X : D) :
        h.whiskerLeftRUnitIsoOfIsIsoCounit.inv.app X = h.unit.app (R.obj X)

        If the counit of an is an isomorphism, one has an isomorphism (R ⋙ L ⋙ R) ≅ R.

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          If each component of the unit is a monomorphism, then the left adjoint is faithful.

          If each component of the unit is a split epimorphism, then the left adjoint is full.

          If the unit is an isomorphism, then the left adjoint is fully faithful.

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          • One or more equations did not get rendered due to their size.
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            If each component of the counit is an epimorphism, then the right adjoint is faithful.

            If each component of the counit is a split monomorphism, then the right adjoint is full.

            If the counit is an isomorphism, then the right adjoint is fully faithful.

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              theorem CategoryTheory.Adjunction.isIso_counit_app_of_iso {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {L : CategoryTheory.Functor C D} {R : CategoryTheory.Functor D C} (h : L R) [L.Faithful] [L.Full] {X : D} {Y : C} (e : X L.obj Y) :
              CategoryTheory.IsIso (h.counit.app X)
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              If η_A is an isomorphism, then A is in the essential image of i.

              @[deprecated CategoryTheory.Adjunction.mem_essImage_of_unit_isIso]

              Alias of CategoryTheory.Adjunction.mem_essImage_of_unit_isIso.


              If η_A is an isomorphism, then A is in the essential image of i.

              theorem CategoryTheory.Adjunction.isIso_unit_app_of_iso {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {L : CategoryTheory.Functor C D} {R : CategoryTheory.Functor D C} (h : L R) [R.Faithful] [R.Full] {X : D} {Y : C} (e : Y R.obj X) :
              CategoryTheory.IsIso (h.unit.app Y)