Expressions for monoidal categories

This file converts lean expressions representing morphisms in monoidal categories into Mor₂Iso or Mor terms. The converted expressions are used in the coherence tactics and the string diagram widgets.

def Mathlib.Tactic.Monoidal.srcExpr (η : Lean.Expr) : Lean.MetaM Lean.Expr

The domain of a morphism.

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def Mathlib.Tactic.Monoidal.tgtExpr (η : Lean.Expr) : Lean.MetaM Lean.Expr

The codomain of a morphism.

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def Mathlib.Tactic.Monoidal.srcExprOfIso (η : Lean.Expr) : Lean.MetaM Lean.Expr

The domain of an isomorphism.

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def Mathlib.Tactic.Monoidal.tgtExprOfIso (η : Lean.Expr) : Lean.MetaM Lean.Expr

The codomain of an isomorphism.

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structure Mathlib.Tactic.Monoidal.Context : Type

The context for evaluating expressions.

• level₂ : Lean.Level

  The level for morphisms.

• level₁ : Lean.Level

  The level for objects.

• C : Q(Type udummy._uniq.15)

  The expression for the underlying category.

• instCat : Q(CategoryTheory.Category.{udummy._uniq.16, udummy._uniq.17} unknown_1)

  The category instance.

• instMonoidal? : Option Q(CategoryTheory.MonoidalCategory unknown_1)

  The monoidal category instance.

def Mathlib.Tactic.Monoidal.mkContext? (e : Lean.Expr) : Lean.MetaM (Option Context)

Populate a context object for evaluating e.

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instance Mathlib.Tactic.Monoidal.instContextContext : BicategoryLike.Context Context

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• Mathlib.Tactic.Monoidal.instContextContext = { mkContext? := Mathlib.Tactic.Monoidal.mkContext? }

@[reducible, inline]

abbrev Mathlib.Tactic.Monoidal.MonoidalM (α : Type) : Type

The monad for the normalization of 2-morphisms.

Equations

• Mathlib.Tactic.Monoidal.MonoidalM = Mathlib.Tactic.BicategoryLike.CoherenceM Mathlib.Tactic.Monoidal.Context

def Mathlib.Tactic.Monoidal.synthMonoidalError {α : Type} : Lean.MetaM α

Throw an error if the monoidal category instance is not found.

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• Mathlib.Tactic.Monoidal.synthMonoidalError = Lean.throwError (Lean.toMessageData "failed to find monoidal category instance")

instance Mathlib.Tactic.Monoidal.instMonadMor₁MonoidalM : BicategoryLike.MonadMor₁ MonoidalM

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theorem Mathlib.Tactic.Monoidal.structuralIsoOfExpr_comp {C : Type u} [CategoryTheory.Category.{v, u} C] {f g h : C} (η : f ⟶ g) (η' : f ≅ g) (ih_η : η'.hom = η) (θ : g ⟶ h) (θ' : g ≅ h) (ih_θ : θ'.hom = θ) : (η' ≪≫ θ').hom = CategoryTheory.CategoryStruct.comp η θ

theorem Mathlib.Tactic.Monoidal.StructuralOfExpr_monoidalComp {C : Type u} [CategoryTheory.Category.{v, u} C] {f g h i : C} [CategoryTheory.MonoidalCoherence g h] (η : f ⟶ g) (η' : f ≅ g) (ih_η : η'.hom = η) (θ : h ⟶ i) (θ' : h ≅ i) (ih_θ : θ'.hom = θ) : (CategoryTheory.monoidalIsoComp η' θ').hom = CategoryTheory.monoidalComp η θ

theorem Mathlib.Tactic.Monoidal.structuralIsoOfExpr_whiskerLeft {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.MonoidalCategory C] (f : C) {g h : C} (η : g ⟶ h) (η' : g ≅ h) (ih_η : η'.hom = η) : (CategoryTheory.MonoidalCategory.whiskerLeftIso f η').hom = CategoryTheory.MonoidalCategoryStruct.whiskerLeft f η

theorem Mathlib.Tactic.Monoidal.structuralIsoOfExpr_whiskerRight {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.MonoidalCategory C] {f g : C} (h : C) (η : f ⟶ g) (η' : f ≅ g) (ih_η : η'.hom = η) : (CategoryTheory.MonoidalCategory.whiskerRightIso η' h).hom = CategoryTheory.MonoidalCategoryStruct.whiskerRight η h

theorem Mathlib.Tactic.Monoidal.structuralIsoOfExpr_horizontalComp {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.MonoidalCategory C] {f₁ g₁ f₂ g₂ : C} (η : f₁ ⟶ g₁) (η' : f₁ ≅ g₁) (ih_η : η'.hom = η) (θ : f₂ ⟶ g₂) (θ' : f₂ ≅ g₂) (ih_θ : θ'.hom = θ) : (CategoryTheory.MonoidalCategory.tensorIso η' θ').hom = CategoryTheory.MonoidalCategoryStruct.tensorHom η θ

instance Mathlib.Tactic.Monoidal.instMonadMor₂IsoMonoidalM : BicategoryLike.MonadMor₂Iso MonoidalM

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instance Mathlib.Tactic.Monoidal.instMonadMor₂MonoidalM : BicategoryLike.MonadMor₂ MonoidalM

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def Mathlib.Tactic.Monoidal.id₁? (e : Lean.Expr) : MonoidalM (Option BicategoryLike.Obj)

Check that e is definitionally equal to 𝟙_ C.

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def Mathlib.Tactic.Monoidal.comp? (e : Lean.Expr) : MonoidalM (Option (BicategoryLike.Mor₁ × BicategoryLike.Mor₁))

Return (f, g) if e is definitionally equal to f ⊗ g.

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partial def Mathlib.Tactic.Monoidal.mor₁OfExpr (e : Lean.Expr) : MonoidalM BicategoryLike.Mor₁

Construct a Mor₁ expression from a Lean expression.

instance Mathlib.Tactic.Monoidal.instMkMor₁MonoidalM : BicategoryLike.MkMor₁ MonoidalM

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• Mathlib.Tactic.Monoidal.instMkMor₁MonoidalM = { ofExpr := Mathlib.Tactic.Monoidal.mor₁OfExpr }

partial def Mathlib.Tactic.Monoidal.Mor₂IsoOfExpr (e : Lean.Expr) : MonoidalM BicategoryLike.Mor₂Iso

Construct a Mor₂Iso term from a Lean expression.

partial def Mathlib.Tactic.Monoidal.Mor₂OfExpr (e : Lean.Expr) : MonoidalM BicategoryLike.Mor₂

Construct a Mor₂ term from a Lean expression.

instance Mathlib.Tactic.Monoidal.instMkMor₂MonoidalM : BicategoryLike.MkMor₂ MonoidalM

Equations

• Mathlib.Tactic.Monoidal.instMkMor₂MonoidalM = { ofExpr := Mathlib.Tactic.Monoidal.Mor₂OfExpr }

instance Mathlib.Tactic.Monoidal.instMonadCoherehnceHomMonoidalM : BicategoryLike.MonadCoherehnceHom MonoidalM

Equations

• Mathlib.Tactic.Monoidal.instMonadCoherehnceHomMonoidalM = { unfoldM := fun (α : Mathlib.Tactic.BicategoryLike.CoherenceHom) => Mathlib.Tactic.Monoidal.Mor₂IsoOfExpr α.unfold }