Expressions for bicategories

This file converts lean expressions representing 2-morphisms in bicategories into Mor₂Iso or Mor terms. The converted expressions are used in the coherence tactics and the string diagram widgets.

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

The domain of a morphism.

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

The codomain of a morphism.

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

The domain of an isomorphism.

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

The codomain of an isomorphism.

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structure Mathlib.Tactic.Bicategory.Context : Type

The context for evaluating expressions.

• level₂ : Lean.Level

  The level for 2-morphisms.

• level₁ : Lean.Level

  The level for 1-morphisms.

• level₀ : Lean.Level

  The level for objects.

• B : Q(Type udummy._uniq.14)

  The expression for the underlying category.

• instBicategory : Q(CategoryTheory.Bicategory unknown_1)

  The bicategory instance.

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

Populate a context object for evaluating e.

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

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

@[reducible, inline]

abbrev Mathlib.Tactic.Bicategory.BicategoryM (α : Type) : Type

The monad for the normalization of 2-morphisms.

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• Mathlib.Tactic.Bicategory.BicategoryM = Mathlib.Tactic.BicategoryLike.CoherenceM Mathlib.Tactic.Bicategory.Context

instance Mathlib.Tactic.Bicategory.instMonadMor₁BicategoryM : BicategoryLike.MonadMor₁ BicategoryM

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theorem Mathlib.Tactic.Bicategory.structuralIso_inv {B : Type u} [CategoryTheory.Bicategory B] {a b : B} {f g : a ⟶ b} (η : f ≅ g) : η.symm.hom = η.inv

theorem Mathlib.Tactic.Bicategory.structuralIsoOfExpr_comp {B : Type u} [CategoryTheory.Bicategory B] {a b : B} {f g h : a ⟶ b} (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.Bicategory.structuralIsoOfExpr_whiskerLeft {B : Type u} [CategoryTheory.Bicategory B] {a b c : B} (f : a ⟶ b) {g h : b ⟶ c} (η : g ⟶ h) (η' : g ≅ h) (ih_η : η'.hom = η) : (CategoryTheory.Bicategory.whiskerLeftIso f η').hom = CategoryTheory.Bicategory.whiskerLeft f η

theorem Mathlib.Tactic.Bicategory.structuralIsoOfExpr_whiskerRight {B : Type u} [CategoryTheory.Bicategory B] {a b c : B} {f g : a ⟶ b} (h : b ⟶ c) (η : f ⟶ g) (η' : f ≅ g) (ih_η : η'.hom = η) : (CategoryTheory.Bicategory.whiskerRightIso η' h).hom = CategoryTheory.Bicategory.whiskerRight η h

theorem Mathlib.Tactic.Bicategory.StructuralOfExpr_bicategoricalComp {B : Type u} [CategoryTheory.Bicategory B] {a b : B} {f g h i : a ⟶ b} [CategoryTheory.BicategoricalCoherence 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.bicategoricalIsoComp η' θ').hom = CategoryTheory.bicategoricalComp η θ

instance Mathlib.Tactic.Bicategory.instMonadMor₂IsoBicategoryM : BicategoryLike.MonadMor₂Iso BicategoryM

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instance Mathlib.Tactic.Bicategory.instMonadMor₂BicategoryM : BicategoryLike.MonadMor₂ BicategoryM

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def Mathlib.Tactic.Bicategory.id₁? (e : Lean.Expr) : BicategoryM (Option BicategoryLike.Obj)

Check that e is definitionally equal to 𝟙 a.

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

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

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partial def Mathlib.Tactic.Bicategory.mor₁OfExpr (e : Lean.Expr) : BicategoryM BicategoryLike.Mor₁

Construct a Mor₁ expression from a Lean expression.

instance Mathlib.Tactic.Bicategory.instMkMor₁BicategoryM : BicategoryLike.MkMor₁ BicategoryM

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• Mathlib.Tactic.Bicategory.instMkMor₁BicategoryM = { ofExpr := Mathlib.Tactic.Bicategory.mor₁OfExpr }

partial def Mathlib.Tactic.Bicategory.Mor₂IsoOfExpr (e : Lean.Expr) : BicategoryM BicategoryLike.Mor₂Iso

Construct a Mor₂Iso term from a Lean expression.

partial def Mathlib.Tactic.Bicategory.Mor₂OfExpr (e : Lean.Expr) : BicategoryM BicategoryLike.Mor₂

Construct a Mor₂ term from a Lean expression.

instance Mathlib.Tactic.Bicategory.instMkMor₂BicategoryM : BicategoryLike.MkMor₂ BicategoryM

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• Mathlib.Tactic.Bicategory.instMkMor₂BicategoryM = { ofExpr := Mathlib.Tactic.Bicategory.Mor₂OfExpr }

instance Mathlib.Tactic.Bicategory.instMonadCoherehnceHomBicategoryM : BicategoryLike.MonadCoherehnceHom BicategoryM

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