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

Type (max (max (max u₁ u₂) v₁) v₂)

Bundled left-exact functors.

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(C ⥤ₗ D) = CategoryTheory.FullSubcategory fun (F : CategoryTheory.Functor C D) => Nonempty (CategoryTheory.Limits.PreservesFiniteLimits F)

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

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

Equations

CategoryTheory.instCategoryLeftExactFunctor C D = CategoryTheory.FullSubcategory.category fun (F : CategoryTheory.Functor C D) => Nonempty (CategoryTheory.Limits.PreservesFiniteLimits F)

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def CategoryTheory.«term_⥤ₗ_» :

Lean.TrailingParserDescr

C ⥤ₗ D denotes left exact functors C ⥤ D

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One or more equations did not get rendered due to their size.

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

CategoryTheory.Functor (C ⥤ₗ D) (CategoryTheory.Functor C D)

A left exact functor is in particular a functor.

Equations

CategoryTheory.LeftExactFunctor.forget C D = CategoryTheory.fullSubcategoryInclusion fun (F : CategoryTheory.Functor C D) => Nonempty (CategoryTheory.Limits.PreservesFiniteLimits F)

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

CategoryTheory.Functor.Full (CategoryTheory.LeftExactFunctor.forget C D)

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⋯ = ⋯

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

CategoryTheory.Functor.Faithful (CategoryTheory.LeftExactFunctor.forget C D)

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⋯ = ⋯

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

Type (max (max (max u₁ u₂) v₁) v₂)

Bundled right-exact functors.

Equations

(C ⥤ᵣ D) = CategoryTheory.FullSubcategory fun (F : CategoryTheory.Functor C D) => Nonempty (CategoryTheory.Limits.PreservesFiniteColimits F)

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

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

Equations

CategoryTheory.instCategoryRightExactFunctor C D = CategoryTheory.FullSubcategory.category fun (F : CategoryTheory.Functor C D) => Nonempty (CategoryTheory.Limits.PreservesFiniteColimits F)

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def CategoryTheory.«term_⥤ᵣ_» :

Lean.TrailingParserDescr

C ⥤ᵣ D denotes right exact functors C ⥤ D

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One or more equations did not get rendered due to their size.

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

CategoryTheory.Functor (C ⥤ᵣ D) (CategoryTheory.Functor C D)

A right exact functor is in particular a functor.

Equations

CategoryTheory.RightExactFunctor.forget C D = CategoryTheory.fullSubcategoryInclusion fun (F : CategoryTheory.Functor C D) => Nonempty (CategoryTheory.Limits.PreservesFiniteColimits F)

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

CategoryTheory.Functor.Full (CategoryTheory.RightExactFunctor.forget C D)

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⋯ = ⋯

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

CategoryTheory.Functor.Faithful (CategoryTheory.RightExactFunctor.forget C D)

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⋯ = ⋯

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

Type (max (max (max u₁ u₂) v₁) v₂)

Bundled exact functors.

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One or more equations did not get rendered due to their size.

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

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

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One or more equations did not get rendered due to their size.

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def CategoryTheory.«term_⥤ₑ_» :

Lean.TrailingParserDescr

C ⥤ₑ D denotes exact functors C ⥤ D

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One or more equations did not get rendered due to their size.

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

CategoryTheory.Functor (C ⥤ₑ D) (CategoryTheory.Functor C D)

An exact functor is in particular a functor.

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One or more equations did not get rendered due to their size.

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

CategoryTheory.Functor.Full (CategoryTheory.ExactFunctor.forget C D)

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⋯ = ⋯

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

CategoryTheory.Functor.Faithful (CategoryTheory.ExactFunctor.forget C D)

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⋯ = ⋯

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

CategoryTheory.Functor (C ⥤ₑ D) (C ⥤ₗ D)

Turn an exact functor into a left exact functor.

Equations

CategoryTheory.LeftExactFunctor.ofExact C D = CategoryTheory.FullSubcategory.map ⋯

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

CategoryTheory.Functor.Full (CategoryTheory.LeftExactFunctor.ofExact C D)

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⋯ = ⋯

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

CategoryTheory.Functor.Faithful (CategoryTheory.LeftExactFunctor.ofExact C D)

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⋯ = ⋯

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

CategoryTheory.Functor (C ⥤ₑ D) (C ⥤ᵣ D)

Turn an exact functor into a left exact functor.

Equations

CategoryTheory.RightExactFunctor.ofExact C D = CategoryTheory.FullSubcategory.map ⋯

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

CategoryTheory.Functor.Full (CategoryTheory.RightExactFunctor.ofExact C D)

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⋯ = ⋯

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

CategoryTheory.Functor.Faithful (CategoryTheory.RightExactFunctor.ofExact C D)

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⋯ = ⋯

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@[simp]

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

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

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@[simp]

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

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

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@[simp]

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

(CategoryTheory.LeftExactFunctor.ofExact C D).map α = α

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@[simp]

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

(CategoryTheory.RightExactFunctor.ofExact C D).map α = α

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@[simp]

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

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

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@[simp]

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

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

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@[simp]

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

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

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@[simp]

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

(CategoryTheory.LeftExactFunctor.forget C D).map α = α

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@[simp]

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

(CategoryTheory.RightExactFunctor.forget C D).map α = α

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@[simp]

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

(CategoryTheory.ExactFunctor.forget C D).map α = α

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

C ⥤ₗ D

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

Equations

CategoryTheory.LeftExactFunctor.of F = { obj := F, property := ⋯ }

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

C ⥤ᵣ D

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

Equations

CategoryTheory.RightExactFunctor.of F = { obj := F, property := ⋯ }

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

C ⥤ₑ D

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

Equations

CategoryTheory.ExactFunctor.of F = { obj := F, property := ⋯ }

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@[simp]

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

(CategoryTheory.LeftExactFunctor.of F).obj = F

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@[simp]

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

(CategoryTheory.RightExactFunctor.of F).obj = F

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@[simp]

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

(CategoryTheory.ExactFunctor.of F).obj = F

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

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

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

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

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

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

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

CategoryTheory.Limits.PreservesFiniteLimits F.obj

Equations

CategoryTheory.instPreservesFiniteLimitsObjFunctorNonempty F = Nonempty.some ⋯

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

CategoryTheory.Limits.PreservesFiniteColimits F.obj

Equations

CategoryTheory.instPreservesFiniteColimitsObjFunctorNonempty F = Nonempty.some ⋯

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

CategoryTheory.Limits.PreservesFiniteLimits F.obj

Equations

CategoryTheory.instPreservesFiniteLimitsObjFunctorAndNonemptyPreservesFiniteColimits F = Nonempty.some ⋯

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

CategoryTheory.Limits.PreservesFiniteColimits F.obj

Equations

CategoryTheory.instPreservesFiniteColimitsObjFunctorAndNonemptyPreservesFiniteLimits F = Nonempty.some ⋯

Documentation

Mathlib.CategoryTheory.Limits.ExactFunctor

Bundled exact functors #