Bundled exact functors #

We say that a functor F is left exact if it preserves finite limits, it is right exact if it preserves finite colimits, and it is exact if it is both left exact and right exact.

In this file, we define the categories of bundled left exact, right exact and exact functors.

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def CategoryTheory.LeftExactFunctor (C : Type u₁) [Category.{v₁, u₁} C] (D : Type u₂) [Category.{v₂, u₂} D] :

Type (max (max (max u₁ u₂) v₁) v₂)

Bundled left-exact functors.

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instance CategoryTheory.instCategoryLeftExactFunctor (C : Type u₁) [Category.{v₁, u₁} C] (D : Type u₂) [Category.{v₂, u₂} D] :

Category.{max u₁ v₂, max (max (max u₂ u₁) v₂) v₁} (C ⥤ₗ D)

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def CategoryTheory.«term_⥤ₗ_» :

Lean.TrailingParserDescr

C ⥤ₗ D denotes left exact functors C ⥤ D

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def CategoryTheory.LeftExactFunctor.forget (C : Type u₁) [Category.{v₁, u₁} C] (D : Type u₂) [Category.{v₂, u₂} D] :

Functor (C ⥤ₗ D) (Functor C D)

A left exact functor is in particular a functor.

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instance CategoryTheory.instFullLeftExactFunctorFunctorForget (C : Type u₁) [Category.{v₁, u₁} C] (D : Type u₂) [Category.{v₂, u₂} D] :

(LeftExactFunctor.forget C D).Full

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instance CategoryTheory.instFaithfulLeftExactFunctorFunctorForget (C : Type u₁) [Category.{v₁, u₁} C] (D : Type u₂) [Category.{v₂, u₂} D] :

(LeftExactFunctor.forget C D).Faithful

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@[reducible, inline]

abbrev CategoryTheory.LeftExactFunctor.fullyFaithful (C : Type u₁) [Category.{v₁, u₁} C] (D : Type u₂) [Category.{v₂, u₂} D] :

(forget C D).FullyFaithful

The inclusion of left exact functors into functors is fully faithful.

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def CategoryTheory.RightExactFunctor (C : Type u₁) [Category.{v₁, u₁} C] (D : Type u₂) [Category.{v₂, u₂} D] :

Type (max (max (max u₁ u₂) v₁) v₂)

Bundled right-exact functors.

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instance CategoryTheory.instCategoryRightExactFunctor (C : Type u₁) [Category.{v₁, u₁} C] (D : Type u₂) [Category.{v₂, u₂} D] :

Category.{max u₁ v₂, max (max (max u₂ u₁) v₂) v₁} (C ⥤ᵣ D)

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def CategoryTheory.«term_⥤ᵣ_» :

Lean.TrailingParserDescr

C ⥤ᵣ D denotes right exact functors C ⥤ D

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def CategoryTheory.RightExactFunctor.forget (C : Type u₁) [Category.{v₁, u₁} C] (D : Type u₂) [Category.{v₂, u₂} D] :

Functor (C ⥤ᵣ D) (Functor C D)

A right exact functor is in particular a functor.

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instance CategoryTheory.instFullRightExactFunctorFunctorForget (C : Type u₁) [Category.{v₁, u₁} C] (D : Type u₂) [Category.{v₂, u₂} D] :

(RightExactFunctor.forget C D).Full

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instance CategoryTheory.instFaithfulRightExactFunctorFunctorForget (C : Type u₁) [Category.{v₁, u₁} C] (D : Type u₂) [Category.{v₂, u₂} D] :

(RightExactFunctor.forget C D).Faithful

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@[reducible, inline]

abbrev CategoryTheory.RightExactFunctor.fullyFaithful (C : Type u₁) [Category.{v₁, u₁} C] (D : Type u₂) [Category.{v₂, u₂} D] :

(forget C D).FullyFaithful

The inclusion of right exact functors into functors is fully faithful.

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def CategoryTheory.ExactFunctor (C : Type u₁) [Category.{v₁, u₁} C] (D : Type u₂) [Category.{v₂, u₂} D] :

Type (max (max (max u₁ u₂) v₁) v₂)

Bundled exact functors.

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instance CategoryTheory.instCategoryExactFunctor (C : Type u₁) [Category.{v₁, u₁} C] (D : Type u₂) [Category.{v₂, u₂} D] :

Category.{max u₁ v₂, max (max (max u₂ u₁) v₂) v₁} (C ⥤ₑ D)

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def CategoryTheory.«term_⥤ₑ_» :

Lean.TrailingParserDescr

C ⥤ₑ D denotes exact functors C ⥤ D

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def CategoryTheory.ExactFunctor.forget (C : Type u₁) [Category.{v₁, u₁} C] (D : Type u₂) [Category.{v₂, u₂} D] :

Functor (C ⥤ₑ D) (Functor C D)

An exact functor is in particular a functor.

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instance CategoryTheory.instFullExactFunctorFunctorForget (C : Type u₁) [Category.{v₁, u₁} C] (D : Type u₂) [Category.{v₂, u₂} D] :

(ExactFunctor.forget C D).Full

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instance CategoryTheory.instFaithfulExactFunctorFunctorForget (C : Type u₁) [Category.{v₁, u₁} C] (D : Type u₂) [Category.{v₂, u₂} D] :

(ExactFunctor.forget C D).Faithful

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def CategoryTheory.LeftExactFunctor.ofExact (C : Type u₁) [Category.{v₁, u₁} C] (D : Type u₂) [Category.{v₂, u₂} D] :

Functor (C ⥤ₑ D) (C ⥤ₗ D)

Turn an exact functor into a left exact functor.

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instance CategoryTheory.instFullExactFunctorLeftExactFunctorOfExact (C : Type u₁) [Category.{v₁, u₁} C] (D : Type u₂) [Category.{v₂, u₂} D] :

(LeftExactFunctor.ofExact C D).Full

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instance CategoryTheory.instFaithfulExactFunctorLeftExactFunctorOfExact (C : Type u₁) [Category.{v₁, u₁} C] (D : Type u₂) [Category.{v₂, u₂} D] :

(LeftExactFunctor.ofExact C D).Faithful

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def CategoryTheory.RightExactFunctor.ofExact (C : Type u₁) [Category.{v₁, u₁} C] (D : Type u₂) [Category.{v₂, u₂} D] :

Functor (C ⥤ₑ D) (C ⥤ᵣ D)

Turn an exact functor into a left exact functor.

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instance CategoryTheory.instFullExactFunctorRightExactFunctorOfExact (C : Type u₁) [Category.{v₁, u₁} C] (D : Type u₂) [Category.{v₂, u₂} D] :

(RightExactFunctor.ofExact C D).Full

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instance CategoryTheory.instFaithfulExactFunctorRightExactFunctorOfExact (C : Type u₁) [Category.{v₁, u₁} C] (D : Type u₂) [Category.{v₂, u₂} D] :

(RightExactFunctor.ofExact C D).Faithful

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@[simp]

theorem CategoryTheory.LeftExactFunctor.ofExact_obj {C : Type u₁} [Category.{v₁, u₁} C] {D : Type u₂} [Category.{v₂, u₂} D] (F : C ⥤ₑ D) :

(ofExact C D).obj F = { obj := F.obj, property := ⋯ }

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@[simp]

theorem CategoryTheory.RightExactFunctor.ofExact_obj {C : Type u₁} [Category.{v₁, u₁} C] {D : Type u₂} [Category.{v₂, u₂} D] (F : C ⥤ₑ D) :

(ofExact C D).obj F = { obj := F.obj, property := ⋯ }

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@[simp]

theorem CategoryTheory.LeftExactFunctor.ofExact_map {C : Type u₁} [Category.{v₁, u₁} C] {D : Type u₂} [Category.{v₂, u₂} D] {F G : C ⥤ₑ D} (α : F ⟶ G) :

(ofExact C D).map α = α

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@[simp]

theorem CategoryTheory.RightExactFunctor.ofExact_map {C : Type u₁} [Category.{v₁, u₁} C] {D : Type u₂} [Category.{v₂, u₂} D] {F G : C ⥤ₑ D} (α : F ⟶ G) :

(ofExact C D).map α = α

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@[simp]

theorem CategoryTheory.LeftExactFunctor.forget_obj {C : Type u₁} [Category.{v₁, u₁} C] {D : Type u₂} [Category.{v₂, u₂} D] (F : C ⥤ₗ D) :

(forget C D).obj F = F.obj

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@[simp]

theorem CategoryTheory.RightExactFunctor.forget_obj {C : Type u₁} [Category.{v₁, u₁} C] {D : Type u₂} [Category.{v₂, u₂} D] (F : C ⥤ᵣ D) :

(forget C D).obj F = F.obj

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@[simp]

theorem CategoryTheory.ExactFunctor.forget_obj {C : Type u₁} [Category.{v₁, u₁} C] {D : Type u₂} [Category.{v₂, u₂} D] (F : C ⥤ₑ D) :

(forget C D).obj F = F.obj

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@[simp]

theorem CategoryTheory.LeftExactFunctor.forget_map {C : Type u₁} [Category.{v₁, u₁} C] {D : Type u₂} [Category.{v₂, u₂} D] {F G : C ⥤ₗ D} (α : F ⟶ G) :

(forget C D).map α = α

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@[simp]

theorem CategoryTheory.RightExactFunctor.forget_map {C : Type u₁} [Category.{v₁, u₁} C] {D : Type u₂} [Category.{v₂, u₂} D] {F G : C ⥤ᵣ D} (α : F ⟶ G) :

(forget C D).map α = α

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@[simp]

theorem CategoryTheory.ExactFunctor.forget_map {C : Type u₁} [Category.{v₁, u₁} C] {D : Type u₂} [Category.{v₂, u₂} D] {F G : C ⥤ₑ D} (α : F ⟶ G) :

(forget C D).map α = α

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def CategoryTheory.LeftExactFunctor.of {C : Type u₁} [Category.{v₁, u₁} C] {D : Type u₂} [Category.{v₂, u₂} D] (F : Functor C D) [Limits.PreservesFiniteLimits F] :

C ⥤ₗ D

Turn a left exact functor into an object of the category LeftExactFunctor C D.

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def CategoryTheory.RightExactFunctor.of {C : Type u₁} [Category.{v₁, u₁} C] {D : Type u₂} [Category.{v₂, u₂} D] (F : Functor C D) [Limits.PreservesFiniteColimits F] :

C ⥤ᵣ D

Turn a right exact functor into an object of the category RightExactFunctor C D.

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def CategoryTheory.ExactFunctor.of {C : Type u₁} [Category.{v₁, u₁} C] {D : Type u₂} [Category.{v₂, u₂} D] (F : Functor C D) [Limits.PreservesFiniteLimits F] [Limits.PreservesFiniteColimits F] :

C ⥤ₑ D

Turn an exact functor into an object of the category ExactFunctor C D.

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theorem CategoryTheory.LeftExactFunctor.of_fst {C : Type u₁} [Category.{v₁, u₁} C] {D : Type u₂} [Category.{v₂, u₂} D] (F : Functor C D) [Limits.PreservesFiniteLimits F] :

(of F).obj = F

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@[simp]

theorem CategoryTheory.RightExactFunctor.of_fst {C : Type u₁} [Category.{v₁, u₁} C] {D : Type u₂} [Category.{v₂, u₂} D] (F : Functor C D) [Limits.PreservesFiniteColimits F] :

(of F).obj = F

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@[simp]

theorem CategoryTheory.ExactFunctor.of_fst {C : Type u₁} [Category.{v₁, u₁} C] {D : Type u₂} [Category.{v₂, u₂} D] (F : Functor C D) [Limits.PreservesFiniteLimits F] [Limits.PreservesFiniteColimits F] :

(of F).obj = F

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theorem CategoryTheory.LeftExactFunctor.forget_obj_of {C : Type u₁} [Category.{v₁, u₁} C] {D : Type u₂} [Category.{v₂, u₂} D] (F : Functor C D) [Limits.PreservesFiniteLimits F] :

(forget C D).obj (of F) = F

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theorem CategoryTheory.RightExactFunctor.forget_obj_of {C : Type u₁} [Category.{v₁, u₁} C] {D : Type u₂} [Category.{v₂, u₂} D] (F : Functor C D) [Limits.PreservesFiniteColimits F] :

(forget C D).obj (of F) = F

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theorem CategoryTheory.ExactFunctor.forget_obj_of {C : Type u₁} [Category.{v₁, u₁} C] {D : Type u₂} [Category.{v₂, u₂} D] (F : Functor C D) [Limits.PreservesFiniteLimits F] [Limits.PreservesFiniteColimits F] :

(forget C D).obj (of F) = F

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instance CategoryTheory.instPreservesFiniteLimitsObjFunctor {C : Type u₁} [Category.{v₁, u₁} C] {D : Type u₂} [Category.{v₂, u₂} D] (F : C ⥤ₗ D) :

Limits.PreservesFiniteLimits F.obj

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instance CategoryTheory.instPreservesFiniteColimitsObjFunctor {C : Type u₁} [Category.{v₁, u₁} C] {D : Type u₂} [Category.{v₂, u₂} D] (F : C ⥤ᵣ D) :

Limits.PreservesFiniteColimits F.obj

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instance CategoryTheory.instPreservesFiniteLimitsObjFunctorAndPreservesFiniteColimits {C : Type u₁} [Category.{v₁, u₁} C] {D : Type u₂} [Category.{v₂, u₂} D] (F : C ⥤ₑ D) :

Limits.PreservesFiniteLimits F.obj

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instance CategoryTheory.instPreservesFiniteColimitsObjFunctorAndPreservesFiniteLimits {C : Type u₁} [Category.{v₁, u₁} C] {D : Type u₂} [Category.{v₂, u₂} D] (F : C ⥤ₑ D) :

Limits.PreservesFiniteColimits F.obj

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def CategoryTheory.LeftExactFunctor.whiskeringLeft (C : Type u₁) [Category.{v₁, u₁} C] (D : Type u₂) [Category.{v₂, u₂} D] (E : Type u₃) [Category.{v₃, u₃} E] :

Functor (C ⥤ₗ D) (Functor (D ⥤ₗ E) (C ⥤ₗ E))

Whiskering a left exact functor by a left exact functor yields a left exact functor.

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theorem CategoryTheory.LeftExactFunctor.whiskeringLeft_obj_obj (C : Type u₁) [Category.{v₁, u₁} C] (D : Type u₂) [Category.{v₂, u₂} D] (E : Type u₃) [Category.{v₃, u₃} E] (F : C ⥤ₗ D) (X : D ⥤ₗ E) :

((whiskeringLeft C D E).obj F).obj X = { obj := F.obj.comp X.obj, property := ⋯ }

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@[simp]

theorem CategoryTheory.LeftExactFunctor.whiskeringLeft_map_app_app (C : Type u₁) [Category.{v₁, u₁} C] (D : Type u₂) [Category.{v₂, u₂} D] (E : Type u₃) [Category.{v₃, u₃} E] {F G : C ⥤ₗ D} (η : F ⟶ G) (H : D ⥤ₗ E) (c : C) :

(((whiskeringLeft C D E).map η).app H).app c = H.obj.map (η.app c)

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@[simp]

theorem CategoryTheory.LeftExactFunctor.whiskeringLeft_obj_map (C : Type u₁) [Category.{v₁, u₁} C] (D : Type u₂) [Category.{v₂, u₂} D] (E : Type u₃) [Category.{v₃, u₃} E] (F : C ⥤ₗ D) {X✝ Y✝ : D ⥤ₗ E} (f : X✝ ⟶ Y✝) :

((whiskeringLeft C D E).obj F).map f = F.obj.whiskerLeft f

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def CategoryTheory.LeftExactFunctor.whiskeringRight (C : Type u₁) [Category.{v₁, u₁} C] (D : Type u₂) [Category.{v₂, u₂} D] (E : Type u₃) [Category.{v₃, u₃} E] :

Functor (D ⥤ₗ E) (Functor (C ⥤ₗ D) (C ⥤ₗ E))

Whiskering a left exact functor by a left exact functor yields a left exact functor.

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@[simp]

theorem CategoryTheory.LeftExactFunctor.whiskeringRight_map_app_app (C : Type u₁) [Category.{v₁, u₁} C] (D : Type u₂) [Category.{v₂, u₂} D] (E : Type u₃) [Category.{v₃, u₃} E] {F G : D ⥤ₗ E} (η : F ⟶ G) (H : C ⥤ₗ D) (c : C) :

(((whiskeringRight C D E).map η).app H).app c = η.app (H.obj.obj c)

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@[simp]

theorem CategoryTheory.LeftExactFunctor.whiskeringRight_obj_map (C : Type u₁) [Category.{v₁, u₁} C] (D : Type u₂) [Category.{v₂, u₂} D] (E : Type u₃) [Category.{v₃, u₃} E] (F : D ⥤ₗ E) {X✝ Y✝ : C ⥤ₗ D} (f : X✝ ⟶ Y✝) :

((whiskeringRight C D E).obj F).map f = Functor.whiskerRight f F.obj

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@[simp]

theorem CategoryTheory.LeftExactFunctor.whiskeringRight_obj_obj (C : Type u₁) [Category.{v₁, u₁} C] (D : Type u₂) [Category.{v₂, u₂} D] (E : Type u₃) [Category.{v₃, u₃} E] (F : D ⥤ₗ E) (X : C ⥤ₗ D) :

((whiskeringRight C D E).obj F).obj X = { obj := X.obj.comp F.obj, property := ⋯ }

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def CategoryTheory.RightExactFunctor.whiskeringLeft (C : Type u₁) [Category.{v₁, u₁} C] (D : Type u₂) [Category.{v₂, u₂} D] (E : Type u₃) [Category.{v₃, u₃} E] :

Functor (C ⥤ᵣ D) (Functor (D ⥤ᵣ E) (C ⥤ᵣ E))

Whiskering a right exact functor by a right exact functor yields a right exact functor.

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@[simp]

theorem CategoryTheory.RightExactFunctor.whiskeringLeft_obj_map (C : Type u₁) [Category.{v₁, u₁} C] (D : Type u₂) [Category.{v₂, u₂} D] (E : Type u₃) [Category.{v₃, u₃} E] (F : C ⥤ᵣ D) {X✝ Y✝ : D ⥤ᵣ E} (f : X✝ ⟶ Y✝) :

((whiskeringLeft C D E).obj F).map f = F.obj.whiskerLeft f

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@[simp]

theorem CategoryTheory.RightExactFunctor.whiskeringLeft_obj_obj (C : Type u₁) [Category.{v₁, u₁} C] (D : Type u₂) [Category.{v₂, u₂} D] (E : Type u₃) [Category.{v₃, u₃} E] (F : C ⥤ᵣ D) (X : D ⥤ᵣ E) :

((whiskeringLeft C D E).obj F).obj X = { obj := F.obj.comp X.obj, property := ⋯ }

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@[simp]

theorem CategoryTheory.RightExactFunctor.whiskeringLeft_map_app_app (C : Type u₁) [Category.{v₁, u₁} C] (D : Type u₂) [Category.{v₂, u₂} D] (E : Type u₃) [Category.{v₃, u₃} E] {F G : C ⥤ᵣ D} (η : F ⟶ G) (H : D ⥤ᵣ E) (c : C) :

(((whiskeringLeft C D E).map η).app H).app c = H.obj.map (η.app c)

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def CategoryTheory.RightExactFunctor.whiskeringRight (C : Type u₁) [Category.{v₁, u₁} C] (D : Type u₂) [Category.{v₂, u₂} D] (E : Type u₃) [Category.{v₃, u₃} E] :

Functor (D ⥤ᵣ E) (Functor (C ⥤ᵣ D) (C ⥤ᵣ E))

Whiskering a right exact functor by a right exact functor yields a right exact functor.

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@[simp]

theorem CategoryTheory.RightExactFunctor.whiskeringRight_obj_map (C : Type u₁) [Category.{v₁, u₁} C] (D : Type u₂) [Category.{v₂, u₂} D] (E : Type u₃) [Category.{v₃, u₃} E] (F : D ⥤ᵣ E) {X✝ Y✝ : C ⥤ᵣ D} (f : X✝ ⟶ Y✝) :

((whiskeringRight C D E).obj F).map f = Functor.whiskerRight f F.obj

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@[simp]

theorem CategoryTheory.RightExactFunctor.whiskeringRight_obj_obj (C : Type u₁) [Category.{v₁, u₁} C] (D : Type u₂) [Category.{v₂, u₂} D] (E : Type u₃) [Category.{v₃, u₃} E] (F : D ⥤ᵣ E) (X : C ⥤ᵣ D) :

((whiskeringRight C D E).obj F).obj X = { obj := X.obj.comp F.obj, property := ⋯ }

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@[simp]

theorem CategoryTheory.RightExactFunctor.whiskeringRight_map_app_app (C : Type u₁) [Category.{v₁, u₁} C] (D : Type u₂) [Category.{v₂, u₂} D] (E : Type u₃) [Category.{v₃, u₃} E] {F G : D ⥤ᵣ E} (η : F ⟶ G) (H : C ⥤ᵣ D) (c : C) :

(((whiskeringRight C D E).map η).app H).app c = η.app (H.obj.obj c)

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def CategoryTheory.ExactFunctor.whiskeringLeft (C : Type u₁) [Category.{v₁, u₁} C] (D : Type u₂) [Category.{v₂, u₂} D] (E : Type u₃) [Category.{v₃, u₃} E] :

Functor (C ⥤ₑ D) (Functor (D ⥤ₑ E) (C ⥤ₑ E))

Whiskering an exact functor by an exact functor yields an exact functor.

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@[simp]

theorem CategoryTheory.ExactFunctor.whiskeringLeft_obj_map (C : Type u₁) [Category.{v₁, u₁} C] (D : Type u₂) [Category.{v₂, u₂} D] (E : Type u₃) [Category.{v₃, u₃} E] (F : C ⥤ₑ D) {X✝ Y✝ : D ⥤ₑ E} (f : X✝ ⟶ Y✝) :

((whiskeringLeft C D E).obj F).map f = F.obj.whiskerLeft f

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@[simp]

theorem CategoryTheory.ExactFunctor.whiskeringLeft_obj_obj (C : Type u₁) [Category.{v₁, u₁} C] (D : Type u₂) [Category.{v₂, u₂} D] (E : Type u₃) [Category.{v₃, u₃} E] (F : C ⥤ₑ D) (X : D ⥤ₑ E) :

((whiskeringLeft C D E).obj F).obj X = { obj := F.obj.comp X.obj, property := ⋯ }

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@[simp]

theorem CategoryTheory.ExactFunctor.whiskeringLeft_map_app_app (C : Type u₁) [Category.{v₁, u₁} C] (D : Type u₂) [Category.{v₂, u₂} D] (E : Type u₃) [Category.{v₃, u₃} E] {F G : C ⥤ₑ D} (η : F ⟶ G) (H : D ⥤ₑ E) (c : C) :

(((whiskeringLeft C D E).map η).app H).app c = H.obj.map (η.app c)

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def CategoryTheory.ExactFunctor.whiskeringRight (C : Type u₁) [Category.{v₁, u₁} C] (D : Type u₂) [Category.{v₂, u₂} D] (E : Type u₃) [Category.{v₃, u₃} E] :

Functor (D ⥤ₑ E) (Functor (C ⥤ₑ D) (C ⥤ₑ E))

Whiskering an exact functor by an exact functor yields an exact functor.

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Instances For

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@[simp]

theorem CategoryTheory.ExactFunctor.whiskeringRight_obj_map (C : Type u₁) [Category.{v₁, u₁} C] (D : Type u₂) [Category.{v₂, u₂} D] (E : Type u₃) [Category.{v₃, u₃} E] (F : D ⥤ₑ E) {X✝ Y✝ : C ⥤ₑ D} (f : X✝ ⟶ Y✝) :

((whiskeringRight C D E).obj F).map f = Functor.whiskerRight f F.obj

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@[simp]

theorem CategoryTheory.ExactFunctor.whiskeringRight_map_app_app (C : Type u₁) [Category.{v₁, u₁} C] (D : Type u₂) [Category.{v₂, u₂} D] (E : Type u₃) [Category.{v₃, u₃} E] {F G : D ⥤ₑ E} (η : F ⟶ G) (H : C ⥤ₑ D) (c : C) :

(((whiskeringRight C D E).map η).app H).app c = η.app (H.obj.obj c)

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@[simp]

theorem CategoryTheory.ExactFunctor.whiskeringRight_obj_obj (C : Type u₁) [Category.{v₁, u₁} C] (D : Type u₂) [Category.{v₂, u₂} D] (E : Type u₃) [Category.{v₃, u₃} E] (F : D ⥤ₑ E) (X : C ⥤ₑ D) :

((whiskeringRight C D E).obj F).obj X = { obj := X.obj.comp F.obj, property := ⋯ }

Documentation

Mathlib.CategoryTheory.Limits.ExactFunctor

Bundled exact functors #