Documentation

Mathlib.CategoryTheory.Sites.CompatiblePlus

In this file, we prove that the plus functor is compatible with functors which preserve the correct limits and colimits.

See CategoryTheory/Sites/CompatibleSheafification for the compatibility of sheafification, which follows easily from the content in this file.

def CategoryTheory.GrothendieckTopology.diagramCompIso {C : Type u} [Category.{v, u} C] (J : GrothendieckTopology C) {D : Type w₁} [Category.{max v u, w₁} D] {E : Type w₂} [Category.{max v u, w₂} E] (F : Functor D E) [∀ (α β : Type (max v u)) (fst snd : βα), Limits.HasLimitsOfShape (Limits.WalkingMulticospan fst snd) D] [∀ (α β : Type (max v u)) (fst snd : βα), Limits.HasLimitsOfShape (Limits.WalkingMulticospan fst snd) E] [∀ (X : C) (W : J.Cover X) (P : Functor Cᵒᵖ D), Limits.PreservesLimit (W.index P).multicospan F] (P : Functor Cᵒᵖ D) (X : C) :
(J.diagram P X).comp F J.diagram (P.comp F) X

The diagram used to define P⁺, composed with F, is isomorphic to the diagram used to define P ⋙ F.

Equations
  • One or more equations did not get rendered due to their size.
Instances For
    @[simp]
    theorem CategoryTheory.GrothendieckTopology.diagramCompIso_hom_ι {C : Type u} [Category.{v, u} C] (J : GrothendieckTopology C) {D : Type w₁} [Category.{max v u, w₁} D] {E : Type w₂} [Category.{max v u, w₂} E] (F : Functor D E) [∀ (α β : Type (max v u)) (fst snd : βα), Limits.HasLimitsOfShape (Limits.WalkingMulticospan fst snd) D] [∀ (α β : Type (max v u)) (fst snd : βα), Limits.HasLimitsOfShape (Limits.WalkingMulticospan fst snd) E] [∀ (X : C) (W : J.Cover X) (P : Functor Cᵒᵖ D), Limits.PreservesLimit (W.index P).multicospan F] (P : Functor Cᵒᵖ D) (X : C) (W : (J.Cover X)ᵒᵖ) (i : (Opposite.unop W).Arrow) :
    @[simp]
    theorem CategoryTheory.GrothendieckTopology.diagramCompIso_hom_ι_assoc {C : Type u} [Category.{v, u} C] (J : GrothendieckTopology C) {D : Type w₁} [Category.{max v u, w₁} D] {E : Type w₂} [Category.{max v u, w₂} E] (F : Functor D E) [∀ (α β : Type (max v u)) (fst snd : βα), Limits.HasLimitsOfShape (Limits.WalkingMulticospan fst snd) D] [∀ (α β : Type (max v u)) (fst snd : βα), Limits.HasLimitsOfShape (Limits.WalkingMulticospan fst snd) E] [∀ (X : C) (W : J.Cover X) (P : Functor Cᵒᵖ D), Limits.PreservesLimit (W.index P).multicospan F] (P : Functor Cᵒᵖ D) (X : C) (W : (J.Cover X)ᵒᵖ) (i : (Opposite.unop W).Arrow) {Z : E} (h : ((Opposite.unop W).index (P.comp F)).left i Z) :
    def CategoryTheory.GrothendieckTopology.plusCompIso {C : Type u} [Category.{v, u} C] (J : GrothendieckTopology C) {D : Type w₁} [Category.{max v u, w₁} D] {E : Type w₂} [Category.{max v u, w₂} E] (F : Functor D E) [∀ (α β : Type (max v u)) (fst snd : βα), Limits.HasLimitsOfShape (Limits.WalkingMulticospan fst snd) D] [∀ (α β : Type (max v u)) (fst snd : βα), Limits.HasLimitsOfShape (Limits.WalkingMulticospan fst snd) E] [∀ (X : C) (W : J.Cover X) (P : Functor Cᵒᵖ D), Limits.PreservesLimit (W.index P).multicospan F] (P : Functor Cᵒᵖ D) [∀ (X : C), Limits.HasColimitsOfShape (J.Cover X)ᵒᵖ D] [∀ (X : C), Limits.HasColimitsOfShape (J.Cover X)ᵒᵖ E] [∀ (X : C), Limits.PreservesColimitsOfShape (J.Cover X)ᵒᵖ F] :
    (J.plusObj P).comp F J.plusObj (P.comp F)

    The isomorphism between P⁺ ⋙ F and (P ⋙ F)⁺.

    Equations
    • One or more equations did not get rendered due to their size.
    Instances For
      @[simp]
      theorem CategoryTheory.GrothendieckTopology.ι_plusCompIso_hom {C : Type u} [Category.{v, u} C] (J : GrothendieckTopology C) {D : Type w₁} [Category.{max v u, w₁} D] {E : Type w₂} [Category.{max v u, w₂} E] (F : Functor D E) [∀ (α β : Type (max v u)) (fst snd : βα), Limits.HasLimitsOfShape (Limits.WalkingMulticospan fst snd) D] [∀ (α β : Type (max v u)) (fst snd : βα), Limits.HasLimitsOfShape (Limits.WalkingMulticospan fst snd) E] [∀ (X : C) (W : J.Cover X) (P : Functor Cᵒᵖ D), Limits.PreservesLimit (W.index P).multicospan F] (P : Functor Cᵒᵖ D) [∀ (X : C), Limits.HasColimitsOfShape (J.Cover X)ᵒᵖ D] [∀ (X : C), Limits.HasColimitsOfShape (J.Cover X)ᵒᵖ E] [∀ (X : C), Limits.PreservesColimitsOfShape (J.Cover X)ᵒᵖ F] (X : Cᵒᵖ) (W : (J.Cover (Opposite.unop X))ᵒᵖ) :
      @[simp]
      theorem CategoryTheory.GrothendieckTopology.ι_plusCompIso_hom_assoc {C : Type u} [Category.{v, u} C] (J : GrothendieckTopology C) {D : Type w₁} [Category.{max v u, w₁} D] {E : Type w₂} [Category.{max v u, w₂} E] (F : Functor D E) [∀ (α β : Type (max v u)) (fst snd : βα), Limits.HasLimitsOfShape (Limits.WalkingMulticospan fst snd) D] [∀ (α β : Type (max v u)) (fst snd : βα), Limits.HasLimitsOfShape (Limits.WalkingMulticospan fst snd) E] [∀ (X : C) (W : J.Cover X) (P : Functor Cᵒᵖ D), Limits.PreservesLimit (W.index P).multicospan F] (P : Functor Cᵒᵖ D) [∀ (X : C), Limits.HasColimitsOfShape (J.Cover X)ᵒᵖ D] [∀ (X : C), Limits.HasColimitsOfShape (J.Cover X)ᵒᵖ E] [∀ (X : C), Limits.PreservesColimitsOfShape (J.Cover X)ᵒᵖ F] (X : Cᵒᵖ) (W : (J.Cover (Opposite.unop X))ᵒᵖ) {Z : E} (h : (J.plusObj (P.comp F)).obj X Z) :
      @[simp]
      theorem CategoryTheory.GrothendieckTopology.plusCompIso_whiskerLeft {C : Type u} [Category.{v, u} C] (J : GrothendieckTopology C) {D : Type w₁} [Category.{max v u, w₁} D] {E : Type w₂} [Category.{max v u, w₂} E] [∀ (α β : Type (max v u)) (fst snd : βα), Limits.HasLimitsOfShape (Limits.WalkingMulticospan fst snd) D] [∀ (α β : Type (max v u)) (fst snd : βα), Limits.HasLimitsOfShape (Limits.WalkingMulticospan fst snd) E] [∀ (X : C), Limits.HasColimitsOfShape (J.Cover X)ᵒᵖ D] [∀ (X : C), Limits.HasColimitsOfShape (J.Cover X)ᵒᵖ E] {F G : Functor D E} (η : F G) (P : Functor Cᵒᵖ D) [∀ (X : C), Limits.PreservesColimitsOfShape (J.Cover X)ᵒᵖ F] [∀ (X : C) (W : J.Cover X) (P : Functor Cᵒᵖ D), Limits.PreservesLimit (W.index P).multicospan F] [∀ (X : C), Limits.PreservesColimitsOfShape (J.Cover X)ᵒᵖ G] [∀ (X : C) (W : J.Cover X) (P : Functor Cᵒᵖ D), Limits.PreservesLimit (W.index P).multicospan G] :
      @[simp]
      theorem CategoryTheory.GrothendieckTopology.plusCompIso_whiskerLeft_assoc {C : Type u} [Category.{v, u} C] (J : GrothendieckTopology C) {D : Type w₁} [Category.{max v u, w₁} D] {E : Type w₂} [Category.{max v u, w₂} E] [∀ (α β : Type (max v u)) (fst snd : βα), Limits.HasLimitsOfShape (Limits.WalkingMulticospan fst snd) D] [∀ (α β : Type (max v u)) (fst snd : βα), Limits.HasLimitsOfShape (Limits.WalkingMulticospan fst snd) E] [∀ (X : C), Limits.HasColimitsOfShape (J.Cover X)ᵒᵖ D] [∀ (X : C), Limits.HasColimitsOfShape (J.Cover X)ᵒᵖ E] {F G : Functor D E} (η : F G) (P : Functor Cᵒᵖ D) [∀ (X : C), Limits.PreservesColimitsOfShape (J.Cover X)ᵒᵖ F] [∀ (X : C) (W : J.Cover X) (P : Functor Cᵒᵖ D), Limits.PreservesLimit (W.index P).multicospan F] [∀ (X : C), Limits.PreservesColimitsOfShape (J.Cover X)ᵒᵖ G] [∀ (X : C) (W : J.Cover X) (P : Functor Cᵒᵖ D), Limits.PreservesLimit (W.index P).multicospan G] {Z : Functor Cᵒᵖ E} (h : J.plusObj (P.comp G) Z) :
      def CategoryTheory.GrothendieckTopology.plusFunctorWhiskerLeftIso {C : Type u} [Category.{v, u} C] (J : GrothendieckTopology C) {D : Type w₁} [Category.{max v u, w₁} D] {E : Type w₂} [Category.{max v u, w₂} E] [∀ (α β : Type (max v u)) (fst snd : βα), Limits.HasLimitsOfShape (Limits.WalkingMulticospan fst snd) D] [∀ (α β : Type (max v u)) (fst snd : βα), Limits.HasLimitsOfShape (Limits.WalkingMulticospan fst snd) E] [∀ (X : C), Limits.HasColimitsOfShape (J.Cover X)ᵒᵖ D] [∀ (X : C), Limits.HasColimitsOfShape (J.Cover X)ᵒᵖ E] (P : Functor Cᵒᵖ D) [∀ (F : Functor D E) (X : C), Limits.PreservesColimitsOfShape (J.Cover X)ᵒᵖ F] [∀ (F : Functor D E) (X : C) (W : J.Cover X) (P : Functor Cᵒᵖ D), Limits.PreservesLimit (W.index P).multicospan F] :

      The isomorphism between P⁺ ⋙ F and (P ⋙ F)⁺, functorially in F.

      Equations
      Instances For
        @[simp]
        theorem CategoryTheory.GrothendieckTopology.plusFunctorWhiskerLeftIso_inv_app {C : Type u} [Category.{v, u} C] (J : GrothendieckTopology C) {D : Type w₁} [Category.{max v u, w₁} D] {E : Type w₂} [Category.{max v u, w₂} E] [∀ (α β : Type (max v u)) (fst snd : βα), Limits.HasLimitsOfShape (Limits.WalkingMulticospan fst snd) D] [∀ (α β : Type (max v u)) (fst snd : βα), Limits.HasLimitsOfShape (Limits.WalkingMulticospan fst snd) E] [∀ (X : C), Limits.HasColimitsOfShape (J.Cover X)ᵒᵖ D] [∀ (X : C), Limits.HasColimitsOfShape (J.Cover X)ᵒᵖ E] (P : Functor Cᵒᵖ D) [∀ (F : Functor D E) (X : C), Limits.PreservesColimitsOfShape (J.Cover X)ᵒᵖ F] [∀ (F : Functor D E) (X : C) (W : J.Cover X) (P : Functor Cᵒᵖ D), Limits.PreservesLimit (W.index P).multicospan F] (X : Functor D E) :
        @[simp]
        theorem CategoryTheory.GrothendieckTopology.plusFunctorWhiskerLeftIso_hom_app {C : Type u} [Category.{v, u} C] (J : GrothendieckTopology C) {D : Type w₁} [Category.{max v u, w₁} D] {E : Type w₂} [Category.{max v u, w₂} E] [∀ (α β : Type (max v u)) (fst snd : βα), Limits.HasLimitsOfShape (Limits.WalkingMulticospan fst snd) D] [∀ (α β : Type (max v u)) (fst snd : βα), Limits.HasLimitsOfShape (Limits.WalkingMulticospan fst snd) E] [∀ (X : C), Limits.HasColimitsOfShape (J.Cover X)ᵒᵖ D] [∀ (X : C), Limits.HasColimitsOfShape (J.Cover X)ᵒᵖ E] (P : Functor Cᵒᵖ D) [∀ (F : Functor D E) (X : C), Limits.PreservesColimitsOfShape (J.Cover X)ᵒᵖ F] [∀ (F : Functor D E) (X : C) (W : J.Cover X) (P : Functor Cᵒᵖ D), Limits.PreservesLimit (W.index P).multicospan F] (X : Functor D E) :
        @[simp]
        theorem CategoryTheory.GrothendieckTopology.plusCompIso_whiskerRight {C : Type u} [Category.{v, u} C] (J : GrothendieckTopology C) {D : Type w₁} [Category.{max v u, w₁} D] {E : Type w₂} [Category.{max v u, w₂} E] (F : Functor D E) [∀ (α β : Type (max v u)) (fst snd : βα), Limits.HasLimitsOfShape (Limits.WalkingMulticospan fst snd) D] [∀ (α β : Type (max v u)) (fst snd : βα), Limits.HasLimitsOfShape (Limits.WalkingMulticospan fst snd) E] [∀ (X : C) (W : J.Cover X) (P : Functor Cᵒᵖ D), Limits.PreservesLimit (W.index P).multicospan F] [∀ (X : C), Limits.HasColimitsOfShape (J.Cover X)ᵒᵖ D] [∀ (X : C), Limits.HasColimitsOfShape (J.Cover X)ᵒᵖ E] [∀ (X : C), Limits.PreservesColimitsOfShape (J.Cover X)ᵒᵖ F] {P Q : Functor Cᵒᵖ D} (η : P Q) :
        @[simp]
        theorem CategoryTheory.GrothendieckTopology.plusCompIso_whiskerRight_assoc {C : Type u} [Category.{v, u} C] (J : GrothendieckTopology C) {D : Type w₁} [Category.{max v u, w₁} D] {E : Type w₂} [Category.{max v u, w₂} E] (F : Functor D E) [∀ (α β : Type (max v u)) (fst snd : βα), Limits.HasLimitsOfShape (Limits.WalkingMulticospan fst snd) D] [∀ (α β : Type (max v u)) (fst snd : βα), Limits.HasLimitsOfShape (Limits.WalkingMulticospan fst snd) E] [∀ (X : C) (W : J.Cover X) (P : Functor Cᵒᵖ D), Limits.PreservesLimit (W.index P).multicospan F] [∀ (X : C), Limits.HasColimitsOfShape (J.Cover X)ᵒᵖ D] [∀ (X : C), Limits.HasColimitsOfShape (J.Cover X)ᵒᵖ E] [∀ (X : C), Limits.PreservesColimitsOfShape (J.Cover X)ᵒᵖ F] {P Q : Functor Cᵒᵖ D} (η : P Q) {Z : Functor Cᵒᵖ E} (h : J.plusObj (Q.comp F) Z) :
        def CategoryTheory.GrothendieckTopology.plusFunctorWhiskerRightIso {C : Type u} [Category.{v, u} C] (J : GrothendieckTopology C) {D : Type w₁} [Category.{max v u, w₁} D] {E : Type w₂} [Category.{max v u, w₂} E] (F : Functor D E) [∀ (α β : Type (max v u)) (fst snd : βα), Limits.HasLimitsOfShape (Limits.WalkingMulticospan fst snd) D] [∀ (α β : Type (max v u)) (fst snd : βα), Limits.HasLimitsOfShape (Limits.WalkingMulticospan fst snd) E] [∀ (X : C) (W : J.Cover X) (P : Functor Cᵒᵖ D), Limits.PreservesLimit (W.index P).multicospan F] [∀ (X : C), Limits.HasColimitsOfShape (J.Cover X)ᵒᵖ D] [∀ (X : C), Limits.HasColimitsOfShape (J.Cover X)ᵒᵖ E] [∀ (X : C), Limits.PreservesColimitsOfShape (J.Cover X)ᵒᵖ F] :

        The isomorphism between P⁺ ⋙ F and (P ⋙ F)⁺, functorially in P.

        Equations
        Instances For
          @[simp]
          theorem CategoryTheory.GrothendieckTopology.plusFunctorWhiskerRightIso_hom_app {C : Type u} [Category.{v, u} C] (J : GrothendieckTopology C) {D : Type w₁} [Category.{max v u, w₁} D] {E : Type w₂} [Category.{max v u, w₂} E] (F : Functor D E) [∀ (α β : Type (max v u)) (fst snd : βα), Limits.HasLimitsOfShape (Limits.WalkingMulticospan fst snd) D] [∀ (α β : Type (max v u)) (fst snd : βα), Limits.HasLimitsOfShape (Limits.WalkingMulticospan fst snd) E] [∀ (X : C) (W : J.Cover X) (P : Functor Cᵒᵖ D), Limits.PreservesLimit (W.index P).multicospan F] [∀ (X : C), Limits.HasColimitsOfShape (J.Cover X)ᵒᵖ D] [∀ (X : C), Limits.HasColimitsOfShape (J.Cover X)ᵒᵖ E] [∀ (X : C), Limits.PreservesColimitsOfShape (J.Cover X)ᵒᵖ F] (X : Functor Cᵒᵖ D) :
          @[simp]
          theorem CategoryTheory.GrothendieckTopology.plusFunctorWhiskerRightIso_inv_app {C : Type u} [Category.{v, u} C] (J : GrothendieckTopology C) {D : Type w₁} [Category.{max v u, w₁} D] {E : Type w₂} [Category.{max v u, w₂} E] (F : Functor D E) [∀ (α β : Type (max v u)) (fst snd : βα), Limits.HasLimitsOfShape (Limits.WalkingMulticospan fst snd) D] [∀ (α β : Type (max v u)) (fst snd : βα), Limits.HasLimitsOfShape (Limits.WalkingMulticospan fst snd) E] [∀ (X : C) (W : J.Cover X) (P : Functor Cᵒᵖ D), Limits.PreservesLimit (W.index P).multicospan F] [∀ (X : C), Limits.HasColimitsOfShape (J.Cover X)ᵒᵖ D] [∀ (X : C), Limits.HasColimitsOfShape (J.Cover X)ᵒᵖ E] [∀ (X : C), Limits.PreservesColimitsOfShape (J.Cover X)ᵒᵖ F] (X : Functor Cᵒᵖ D) :
          @[simp]
          theorem CategoryTheory.GrothendieckTopology.whiskerRight_toPlus_comp_plusCompIso_hom {C : Type u} [Category.{v, u} C] (J : GrothendieckTopology C) {D : Type w₁} [Category.{max v u, w₁} D] {E : Type w₂} [Category.{max v u, w₂} E] (F : Functor D E) [∀ (α β : Type (max v u)) (fst snd : βα), Limits.HasLimitsOfShape (Limits.WalkingMulticospan fst snd) D] [∀ (α β : Type (max v u)) (fst snd : βα), Limits.HasLimitsOfShape (Limits.WalkingMulticospan fst snd) E] [∀ (X : C) (W : J.Cover X) (P : Functor Cᵒᵖ D), Limits.PreservesLimit (W.index P).multicospan F] (P : Functor Cᵒᵖ D) [∀ (X : C), Limits.HasColimitsOfShape (J.Cover X)ᵒᵖ D] [∀ (X : C), Limits.HasColimitsOfShape (J.Cover X)ᵒᵖ E] [∀ (X : C), Limits.PreservesColimitsOfShape (J.Cover X)ᵒᵖ F] :
          @[simp]
          theorem CategoryTheory.GrothendieckTopology.whiskerRight_toPlus_comp_plusCompIso_hom_assoc {C : Type u} [Category.{v, u} C] (J : GrothendieckTopology C) {D : Type w₁} [Category.{max v u, w₁} D] {E : Type w₂} [Category.{max v u, w₂} E] (F : Functor D E) [∀ (α β : Type (max v u)) (fst snd : βα), Limits.HasLimitsOfShape (Limits.WalkingMulticospan fst snd) D] [∀ (α β : Type (max v u)) (fst snd : βα), Limits.HasLimitsOfShape (Limits.WalkingMulticospan fst snd) E] [∀ (X : C) (W : J.Cover X) (P : Functor Cᵒᵖ D), Limits.PreservesLimit (W.index P).multicospan F] (P : Functor Cᵒᵖ D) [∀ (X : C), Limits.HasColimitsOfShape (J.Cover X)ᵒᵖ D] [∀ (X : C), Limits.HasColimitsOfShape (J.Cover X)ᵒᵖ E] [∀ (X : C), Limits.PreservesColimitsOfShape (J.Cover X)ᵒᵖ F] {Z : Functor Cᵒᵖ E} (h : J.plusObj (P.comp F) Z) :
          @[simp]
          theorem CategoryTheory.GrothendieckTopology.toPlus_comp_plusCompIso_inv {C : Type u} [Category.{v, u} C] (J : GrothendieckTopology C) {D : Type w₁} [Category.{max v u, w₁} D] {E : Type w₂} [Category.{max v u, w₂} E] (F : Functor D E) [∀ (α β : Type (max v u)) (fst snd : βα), Limits.HasLimitsOfShape (Limits.WalkingMulticospan fst snd) D] [∀ (α β : Type (max v u)) (fst snd : βα), Limits.HasLimitsOfShape (Limits.WalkingMulticospan fst snd) E] [∀ (X : C) (W : J.Cover X) (P : Functor Cᵒᵖ D), Limits.PreservesLimit (W.index P).multicospan F] (P : Functor Cᵒᵖ D) [∀ (X : C), Limits.HasColimitsOfShape (J.Cover X)ᵒᵖ D] [∀ (X : C), Limits.HasColimitsOfShape (J.Cover X)ᵒᵖ E] [∀ (X : C), Limits.PreservesColimitsOfShape (J.Cover X)ᵒᵖ F] :
          theorem CategoryTheory.GrothendieckTopology.plusCompIso_inv_eq_plusLift {C : Type u} [Category.{v, u} C] (J : GrothendieckTopology C) {D : Type w₁} [Category.{max v u, w₁} D] {E : Type w₂} [Category.{max v u, w₂} E] (F : Functor D E) [∀ (α β : Type (max v u)) (fst snd : βα), Limits.HasLimitsOfShape (Limits.WalkingMulticospan fst snd) D] [∀ (α β : Type (max v u)) (fst snd : βα), Limits.HasLimitsOfShape (Limits.WalkingMulticospan fst snd) E] [∀ (X : C) (W : J.Cover X) (P : Functor Cᵒᵖ D), Limits.PreservesLimit (W.index P).multicospan F] (P : Functor Cᵒᵖ D) [∀ (X : C), Limits.HasColimitsOfShape (J.Cover X)ᵒᵖ D] [∀ (X : C), Limits.HasColimitsOfShape (J.Cover X)ᵒᵖ E] [∀ (X : C), Limits.PreservesColimitsOfShape (J.Cover X)ᵒᵖ F] (hP : Presheaf.IsSheaf J ((J.plusObj P).comp F)) :
          (J.plusCompIso F P).inv = J.plusLift (whiskerRight (J.toPlus P) F) hP