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BicategoricalComp.lean
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BicategoricalComp.lean
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/-
Copyright (c) 2022 Yuma Mizuno. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yuma Mizuno
-/
import Mathlib.CategoryTheory.Bicategory.Basic
/-!
# Bicategorical composition `⊗≫` (composition up to associators)
We provide `f ⊗≫ g`, the `bicategoricalComp` operation,
which automatically inserts associators and unitors as needed
to make the target of `f` match the source of `g`.
-/
universe w v u
open CategoryTheory Bicategory
namespace CategoryTheory
variable {B : Type u} [Bicategory.{w, v} B] {a b c d : B}
/-- A typeclass carrying a choice of bicategorical structural isomorphism between two objects.
Used by the `⊗≫` bicategorical composition operator, and the `coherence` tactic.
-/
class BicategoricalCoherence (f g : a ⟶ b) where
/-- The chosen structural isomorphism between to 1-morphisms. -/
hom : f ⟶ g
[isIso : IsIso hom]
/-- Notation for identities up to unitors and associators. -/
scoped[CategoryTheory.Bicategory] notation " ⊗𝟙 " =>
BicategoricalCoherence.hom -- type as \ot 𝟙
attribute [instance] BicategoricalCoherence.isIso
noncomputable section
/-- Construct an isomorphism between two objects in a bicategorical category
out of unitors and associators. -/
def bicategoricalIso (f g : a ⟶ b) [BicategoricalCoherence f g] : f ≅ g :=
asIso ⊗𝟙
/-- Compose two morphisms in a bicategorical category,
inserting unitors and associators between as necessary. -/
def bicategoricalComp {f g h i : a ⟶ b} [BicategoricalCoherence g h]
(η : f ⟶ g) (θ : h ⟶ i) : f ⟶ i :=
η ≫ ⊗𝟙 ≫ θ
-- type as \ot \gg
@[inherit_doc bicategoricalComp]
scoped[CategoryTheory.Bicategory] infixr:80 " ⊗≫ " => bicategoricalComp
/-- Compose two isomorphisms in a bicategorical category,
inserting unitors and associators between as necessary. -/
def bicategoricalIsoComp {f g h i : a ⟶ b} [BicategoricalCoherence g h]
(η : f ≅ g) (θ : h ≅ i) : f ≅ i :=
η ≪≫ asIso ⊗𝟙 ≪≫ θ
@[inherit_doc bicategoricalIsoComp]
scoped[CategoryTheory.Bicategory] infixr:80 " ≪⊗≫ " =>
bicategoricalIsoComp -- type as \ll \ot \gg
namespace BicategoricalCoherence
@[simps]
instance refl (f : a ⟶ b) : BicategoricalCoherence f f :=
⟨𝟙 _⟩
@[simps]
instance whiskerLeft (f : a ⟶ b) (g h : b ⟶ c)
[BicategoricalCoherence g h] : BicategoricalCoherence (f ≫ g) (f ≫ h) :=
⟨f ◁ ⊗𝟙⟩
@[simps]
instance whiskerRight (f g : a ⟶ b) (h : b ⟶ c)
[BicategoricalCoherence f g] : BicategoricalCoherence (f ≫ h) (g ≫ h) :=
⟨⊗𝟙 ▷ h⟩
@[simps]
instance tensorRight (f : a ⟶ b) (g : b ⟶ b)
[BicategoricalCoherence (𝟙 b) g] : BicategoricalCoherence f (f ≫ g) :=
⟨(ρ_ f).inv ≫ f ◁ ⊗𝟙⟩
@[simps]
instance tensorRight' (f : a ⟶ b) (g : b ⟶ b)
[BicategoricalCoherence g (𝟙 b)] : BicategoricalCoherence (f ≫ g) f :=
⟨f ◁ ⊗𝟙 ≫ (ρ_ f).hom⟩
@[simps]
instance left (f g : a ⟶ b) [BicategoricalCoherence f g] :
BicategoricalCoherence (𝟙 a ≫ f) g :=
⟨(λ_ f).hom ≫ ⊗𝟙⟩
@[simps]
instance left' (f g : a ⟶ b) [BicategoricalCoherence f g] :
BicategoricalCoherence f (𝟙 a ≫ g) :=
⟨⊗𝟙 ≫ (λ_ g).inv⟩
@[simps]
instance right (f g : a ⟶ b) [BicategoricalCoherence f g] :
BicategoricalCoherence (f ≫ 𝟙 b) g :=
⟨(ρ_ f).hom ≫ ⊗𝟙⟩
@[simps]
instance right' (f g : a ⟶ b) [BicategoricalCoherence f g] :
BicategoricalCoherence f (g ≫ 𝟙 b) :=
⟨⊗𝟙 ≫ (ρ_ g).inv⟩
@[simps]
instance assoc (f : a ⟶ b) (g : b ⟶ c) (h : c ⟶ d) (i : a ⟶ d)
[BicategoricalCoherence (f ≫ g ≫ h) i] :
BicategoricalCoherence ((f ≫ g) ≫ h) i :=
⟨(α_ f g h).hom ≫ ⊗𝟙⟩
@[simps]
instance assoc' (f : a ⟶ b) (g : b ⟶ c) (h : c ⟶ d) (i : a ⟶ d)
[BicategoricalCoherence i (f ≫ g ≫ h)] :
BicategoricalCoherence i ((f ≫ g) ≫ h) :=
⟨⊗𝟙 ≫ (α_ f g h).inv⟩
end BicategoricalCoherence
@[simp]
theorem bicategoricalComp_refl {f g h : a ⟶ b} (η : f ⟶ g) (θ : g ⟶ h) : η ⊗≫ θ = η ≫ θ := by
dsimp [bicategoricalComp]; simp
example {f' : a ⟶ d} {f : a ⟶ b} {g : b ⟶ c} {h : c ⟶ d} {h' : a ⟶ d} (η : f' ⟶ f ≫ g ≫ h)
(θ : (f ≫ g) ≫ h ⟶ h') : f' ⟶ h' :=
η ⊗≫ θ
-- To automatically insert unitors/associators at the beginning or end,
-- you can use `η ⊗≫ 𝟙 _`
example {f' : a ⟶ d} {f : a ⟶ b} {g : b ⟶ c} {h : c ⟶ d} (η : f' ⟶ (f ≫ g) ≫ h) :
f' ⟶ f ≫ g ≫ h :=
η ⊗≫ 𝟙 _
end
end CategoryTheory