Documentation

Mathlib.CategoryTheory.Adjunction.Whiskering

Whiskering of adjunctions #

Given categories C D E, functors F : D ⥤ E and G : E ⥤ D with an adjunction F ⊣ G, we provide the induced adjunction between the functor categories C ⥤ D and C ⥤ E, and the functor categories E ⥤ C and D ⥤ C.

source
def CategoryTheory.Adjunction.whiskerRight (C : Type u_1) {D : Type u_2} {E : Type u_3} [Category.{v_1, u_1} C] [Category.{v_2, u_2} D] [Category.{v_3, u_3} E] {F : Functor D E} {G : Functor E D} (adj : F ⊣ G) :
(Functor.whiskeringRight C D E).obj F ⊣ (Functor.whiskeringRight C E D).obj G

Given an adjunction F ⊣ G, this provides the natural adjunction (whiskeringRight C _ _).obj F ⊣ (whiskeringRight C _ _).obj G.

Equations
  • One or more equations did not get rendered due to their size.
Instances For
    source
    @[simp]
    theorem CategoryTheory.Adjunction.whiskerRight_unit_app_app (C : Type u_1) {D : Type u_2} {E : Type u_3} [Category.{v_1, u_1} C] [Category.{v_2, u_2} D] [Category.{v_3, u_3} E] {F : Functor D E} {G : Functor E D} (adj : F ⊣ G) (X : Functor C D) (X✝ : C) :
    ((Adjunction.whiskerRight C adj).unit.app X).app X✝ = adj.unit.app (X.obj X✝)
    source
    @[simp]
    theorem CategoryTheory.Adjunction.whiskerRight_counit_app_app (C : Type u_1) {D : Type u_2} {E : Type u_3} [Category.{v_1, u_1} C] [Category.{v_2, u_2} D] [Category.{v_3, u_3} E] {F : Functor D E} {G : Functor E D} (adj : F ⊣ G) (X : Functor C E) (X✝ : C) :
    ((Adjunction.whiskerRight C adj).counit.app X).app X✝ = adj.counit.app (X.obj X✝)
    source
    def CategoryTheory.Adjunction.whiskerLeft (C : Type u_1) {D : Type u_2} {E : Type u_3} [Category.{v_1, u_1} C] [Category.{v_2, u_2} D] [Category.{v_3, u_3} E] {F : Functor D E} {G : Functor E D} (adj : F ⊣ G) :
    (Functor.whiskeringLeft E D C).obj G ⊣ (Functor.whiskeringLeft D E C).obj F

    Given an adjunction F ⊣ G, this provides the natural adjunction (whiskeringLeft _ _ C).obj G ⊣ (whiskeringLeft _ _ C).obj F.

    Equations
    • One or more equations did not get rendered due to their size.
    Instances For
      source
      @[simp]
      theorem CategoryTheory.Adjunction.whiskerLeft_unit_app_app (C : Type u_1) {D : Type u_2} {E : Type u_3} [Category.{v_1, u_1} C] [Category.{v_2, u_2} D] [Category.{v_3, u_3} E] {F : Functor D E} {G : Functor E D} (adj : F ⊣ G) (X : Functor D C) (X✝ : D) :
      ((Adjunction.whiskerLeft C adj).unit.app X).app X✝ = X.map (adj.unit.app X✝)
      source
      @[simp]
      theorem CategoryTheory.Adjunction.whiskerLeft_counit_app_app (C : Type u_1) {D : Type u_2} {E : Type u_3} [Category.{v_1, u_1} C] [Category.{v_2, u_2} D] [Category.{v_3, u_3} E] {F : Functor D E} {G : Functor E D} (adj : F ⊣ G) (X : Functor E C) (X✝ : E) :
      ((Adjunction.whiskerLeft C adj).counit.app X).app X✝ = X.map (adj.counit.app X✝)