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Unitor.lean
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Unitor.lean
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/-
Copyright (c) 2024 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.CategoryTheory.GradedObject.Associator
import Mathlib.CategoryTheory.GradedObject.Single
/-!
# The left and right unitors
Given a bifunctor `F : C ⥤ D ⥤ D`, an object `X : C` such that `F.obj X ≅ 𝟭 D` and a
map `p : I × J → J` such that `hp : ∀ (j : J), p ⟨0, j⟩ = j`,
we define an isomorphism of `J`-graded objects for any `Y : GradedObject J D`.
`mapBifunctorLeftUnitor F X e p hp Y : mapBifunctorMapObj F p ((single₀ I).obj X) Y ≅ Y`.
Under similar assumptions, we also obtain a right unitor isomorphism
`mapBifunctorMapObj F p X ((single₀ I).obj Y) ≅ X`. Finally,
the lemma `mapBifunctor_triangle` promotes a triangle identity involving functors
to a triangle identity for the induced functors on graded objects.
-/
namespace CategoryTheory
open Category Limits
namespace GradedObject
section LeftUnitor
variable {C D I J : Type*} [Category C] [Category D]
[Zero I] [DecidableEq I] [HasInitial C]
(F : C ⥤ D ⥤ D) (X : C) (e : F.obj X ≅ 𝟭 D)
[∀ (Y : D), PreservesColimit (Functor.empty.{0} C) (F.flip.obj Y)]
(p : I × J → J) (hp : ∀ (j : J), p ⟨0, j⟩ = j)
(Y Y' : GradedObject J D) (φ : Y ⟶ Y')
/-- Given `F : C ⥤ D ⥤ D`, `X : C`, `e : F.obj X ≅ 𝟭 D` and `Y : GradedObject J D`,
this is the isomorphism `((mapBifunctor F I J).obj ((single₀ I).obj X)).obj Y a ≅ Y a.2`
when `a : I × J` is such that `a.1 = 0`. -/
@[simps!]
noncomputable def mapBifunctorObjSingle₀ObjIso (a : I × J) (ha : a.1 = 0) :
((mapBifunctor F I J).obj ((single₀ I).obj X)).obj Y a ≅ Y a.2 :=
(F.mapIso (singleObjApplyIsoOfEq _ X _ ha)).app _ ≪≫ e.app (Y a.2)
/-- Given `F : C ⥤ D ⥤ D`, `X : C` and `Y : GradedObject J D`,
`((mapBifunctor F I J).obj ((single₀ I).obj X)).obj Y a` is an initial object
when `a : I × J` is such that `a.1 ≠ 0`. -/
noncomputable def mapBifunctorObjSingle₀ObjIsInitial (a : I × J) (ha : a.1 ≠ 0) :
IsInitial (((mapBifunctor F I J).obj ((single₀ I).obj X)).obj Y a) :=
IsInitial.isInitialObj (F.flip.obj (Y a.2)) _ (isInitialSingleObjApply _ _ _ ha)
/-- Given `F : C ⥤ D ⥤ D`, `X : C`, `e : F.obj X ≅ 𝟭 D`, `Y : GradedObject J D` and
`p : I × J → J` such that `p ⟨0, j⟩ = j` for all `j`,
this is the (colimit) cofan which shall be used to construct the isomorphism
`mapBifunctorMapObj F p ((single₀ I).obj X) Y ≅ Y`, see `mapBifunctorLeftUnitor`. -/
noncomputable def mapBifunctorLeftUnitorCofan (j : J) :
(((mapBifunctor F I J).obj ((single₀ I).obj X)).obj Y).CofanMapObjFun p j :=
CofanMapObjFun.mk _ _ _ (Y j) (fun a ha =>
if ha : a.1 = 0 then
(mapBifunctorObjSingle₀ObjIso F X e Y a ha).hom ≫ eqToHom (by aesop)
else
(mapBifunctorObjSingle₀ObjIsInitial F X Y a ha).to _)
@[simp, reassoc]
lemma mapBifunctorLeftUnitorCofan_inj (j : J) :
(mapBifunctorLeftUnitorCofan F X e p hp Y j).inj ⟨⟨0, j⟩, hp j⟩ =
(F.map (singleObjApplyIso (0 : I) X).hom).app (Y j) ≫ e.hom.app (Y j) := by
simp [mapBifunctorLeftUnitorCofan]
/-- The cofan `mapBifunctorLeftUnitorCofan F X e p hp Y j` is a colimit. -/
noncomputable def mapBifunctorLeftUnitorCofanIsColimit (j : J) :
IsColimit (mapBifunctorLeftUnitorCofan F X e p hp Y j) :=
mkCofanColimit _
(fun s => e.inv.app (Y j) ≫
(F.map (singleObjApplyIso (0 : I) X).inv).app (Y j) ≫ s.inj ⟨⟨0, j⟩, hp j⟩)
(fun s => by
rintro ⟨⟨i, j'⟩, h⟩
by_cases hi : i = 0
· subst hi
simp only [Set.mem_preimage, hp, Set.mem_singleton_iff] at h
subst h
simp
· apply IsInitial.hom_ext
exact mapBifunctorObjSingle₀ObjIsInitial _ _ _ _ hi)
(fun s m hm => by simp [← hm ⟨⟨0, j⟩, hp j⟩])
lemma mapBifunctorLeftUnitor_hasMap :
HasMap (((mapBifunctor F I J).obj ((single₀ I).obj X)).obj Y) p :=
CofanMapObjFun.hasMap _ _ _ (mapBifunctorLeftUnitorCofanIsColimit F X e p hp Y)
variable [HasMap (((mapBifunctor F I J).obj ((single₀ I).obj X)).obj Y) p]
[HasMap (((mapBifunctor F I J).obj ((single₀ I).obj X)).obj Y') p]
/-- Given `F : C ⥤ D ⥤ D`, `X : C`, `e : F.obj X ≅ 𝟭 D`, `Y : GradedObject J D` and
`p : I × J → J` such that `p ⟨0, j⟩ = j` for all `j`,
this is the left unitor isomorphism `mapBifunctorMapObj F p ((single₀ I).obj X) Y ≅ Y`. -/
noncomputable def mapBifunctorLeftUnitor : mapBifunctorMapObj F p ((single₀ I).obj X) Y ≅ Y :=
isoMk _ _ (fun j => (CofanMapObjFun.iso
(mapBifunctorLeftUnitorCofanIsColimit F X e p hp Y j)).symm)
@[reassoc (attr := simp)]
lemma ι_mapBifunctorLeftUnitor_hom_apply (j : J) :
ιMapBifunctorMapObj F p ((single₀ I).obj X) Y 0 j j (hp j) ≫
(mapBifunctorLeftUnitor F X e p hp Y).hom j =
(F.map (singleObjApplyIso (0 : I) X).hom).app _ ≫ e.hom.app (Y j) := by
dsimp [mapBifunctorLeftUnitor]
erw [CofanMapObjFun.ιMapObj_iso_inv]
rw [mapBifunctorLeftUnitorCofan_inj]
lemma mapBifunctorLeftUnitor_inv_apply (j : J) :
(mapBifunctorLeftUnitor F X e p hp Y).inv j =
e.inv.app (Y j) ≫ (F.map (singleObjApplyIso (0 : I) X).inv).app (Y j) ≫
ιMapBifunctorMapObj F p ((single₀ I).obj X) Y 0 j j (hp j) := rfl
variable {Y Y'}
@[reassoc]
lemma mapBifunctorLeftUnitor_inv_naturality :
φ ≫ (mapBifunctorLeftUnitor F X e p hp Y').inv =
(mapBifunctorLeftUnitor F X e p hp Y).inv ≫ mapBifunctorMapMap F p (𝟙 _) φ := by
ext j
dsimp
rw [mapBifunctorLeftUnitor_inv_apply, mapBifunctorLeftUnitor_inv_apply, assoc, assoc,
ι_mapBifunctorMapMap]
dsimp
rw [Functor.map_id, NatTrans.id_app, id_comp]
erw [← NatTrans.naturality_assoc, ← NatTrans.naturality_assoc]
rfl
@[reassoc]
lemma mapBifunctorLeftUnitor_naturality :
mapBifunctorMapMap F p (𝟙 _) φ ≫ (mapBifunctorLeftUnitor F X e p hp Y').hom =
(mapBifunctorLeftUnitor F X e p hp Y).hom ≫ φ := by
rw [← cancel_mono (mapBifunctorLeftUnitor F X e p hp Y').inv, assoc, assoc, Iso.hom_inv_id,
comp_id, mapBifunctorLeftUnitor_inv_naturality, Iso.hom_inv_id_assoc]
end LeftUnitor
section RightUnitor
variable {C D I J : Type*} [Category C] [Category D]
[Zero I] [DecidableEq I] [HasInitial C]
(F : D ⥤ C ⥤ D) (Y : C) (e : F.flip.obj Y ≅ 𝟭 D)
[∀ (X : D), PreservesColimit (Functor.empty.{0} C) (F.obj X)]
(p : J × I → J)
(hp : ∀ (j : J), p ⟨j, 0⟩ = j) (X X' : GradedObject J D) (φ : X ⟶ X')
/-- Given `F : D ⥤ C ⥤ D`, `Y : C`, `e : F.flip.obj X ≅ 𝟭 D` and `X : GradedObject J D`,
this is the isomorphism `((mapBifunctor F J I).obj X).obj ((single₀ I).obj Y) a ≅ Y a.2`
when `a : J × I` is such that `a.2 = 0`. -/
@[simps!]
noncomputable def mapBifunctorObjObjSingle₀Iso (a : J × I) (ha : a.2 = 0) :
((mapBifunctor F J I).obj X).obj ((single₀ I).obj Y) a ≅ X a.1 :=
Functor.mapIso _ (singleObjApplyIsoOfEq _ Y _ ha) ≪≫ e.app (X a.1)
/-- Given `F : D ⥤ C ⥤ D`, `Y : C` and `X : GradedObject J D`,
`((mapBifunctor F J I).obj X).obj ((single₀ I).obj X) a` is an initial when `a : J × I`
is such that `a.2 ≠ 0`. -/
noncomputable def mapBifunctorObjObjSingle₀IsInitial (a : J × I) (ha : a.2 ≠ 0) :
IsInitial (((mapBifunctor F J I).obj X).obj ((single₀ I).obj Y) a) :=
IsInitial.isInitialObj (F.obj (X a.1)) _ (isInitialSingleObjApply _ _ _ ha)
/-- Given `F : D ⥤ C ⥤ D`, `Y : C`, `e : F.flip.obj Y ≅ 𝟭 D`, `X : GradedObject J D` and
`p : J × I → J` such that `p ⟨j, 0⟩ = j` for all `j`,
this is the (colimit) cofan which shall be used to construct the isomorphism
`mapBifunctorMapObj F p X ((single₀ I).obj Y) ≅ X`, see `mapBifunctorRightUnitor`. -/
noncomputable def mapBifunctorRightUnitorCofan (j : J) :
(((mapBifunctor F J I).obj X).obj ((single₀ I).obj Y)).CofanMapObjFun p j :=
CofanMapObjFun.mk _ _ _ (X j) (fun a ha =>
if ha : a.2 = 0 then
(mapBifunctorObjObjSingle₀Iso F Y e X a ha).hom ≫ eqToHom (by aesop)
else
(mapBifunctorObjObjSingle₀IsInitial F Y X a ha).to _)
@[simp, reassoc]
lemma mapBifunctorRightUnitorCofan_inj (j : J) :
(mapBifunctorRightUnitorCofan F Y e p hp X j).inj ⟨⟨j, 0⟩, hp j⟩ =
(F.obj (X j)).map (singleObjApplyIso (0 : I) Y).hom ≫ e.hom.app (X j) := by
simp [mapBifunctorRightUnitorCofan]
/-- The cofan `mapBifunctorRightUnitorCofan F Y e p hp X j` is a colimit. -/
noncomputable def mapBifunctorRightUnitorCofanIsColimit (j : J) :
IsColimit (mapBifunctorRightUnitorCofan F Y e p hp X j) :=
mkCofanColimit _
(fun s => e.inv.app (X j) ≫
(F.obj (X j)).map (singleObjApplyIso (0 : I) Y).inv ≫ s.inj ⟨⟨j, 0⟩, hp j⟩)
(fun s => by
rintro ⟨⟨j', i⟩, h⟩
by_cases hi : i = 0
· subst hi
simp only [Set.mem_preimage, hp, Set.mem_singleton_iff] at h
subst h
dsimp
rw [mapBifunctorRightUnitorCofan_inj, assoc, Iso.hom_inv_id_app_assoc,
← Functor.map_comp_assoc, Iso.hom_inv_id, Functor.map_id, id_comp]
· apply IsInitial.hom_ext
exact mapBifunctorObjObjSingle₀IsInitial _ _ _ _ hi)
(fun s m hm => by
dsimp
rw [← hm ⟨⟨j, 0⟩, hp j⟩, mapBifunctorRightUnitorCofan_inj, assoc, ← Functor.map_comp_assoc,
Iso.inv_hom_id, Functor.map_id, id_comp, Iso.inv_hom_id_app_assoc])
lemma mapBifunctorRightUnitor_hasMap :
HasMap (((mapBifunctor F J I).obj X).obj ((single₀ I).obj Y)) p :=
CofanMapObjFun.hasMap _ _ _ (mapBifunctorRightUnitorCofanIsColimit F Y e p hp X)
variable [HasMap (((mapBifunctor F J I).obj X).obj ((single₀ I).obj Y)) p]
[HasMap (((mapBifunctor F J I).obj X').obj ((single₀ I).obj Y)) p]
/-- Given `F : D ⥤ C ⥤ D`, `Y : C`, `e : F.flip.obj Y ≅ 𝟭 D`, `X : GradedObject J D` and
`p : J × I → J` such that `p ⟨j, 0⟩ = j` for all `j`,
this is the right unitor isomorphism `mapBifunctorMapObj F p X ((single₀ I).obj Y) ≅ X`. -/
noncomputable def mapBifunctorRightUnitor : mapBifunctorMapObj F p X ((single₀ I).obj Y) ≅ X :=
isoMk _ _ (fun j => (CofanMapObjFun.iso
(mapBifunctorRightUnitorCofanIsColimit F Y e p hp X j)).symm)
@[reassoc (attr := simp)]
lemma ι_mapBifunctorRightUnitor_hom_apply (j : J) :
ιMapBifunctorMapObj F p X ((single₀ I).obj Y) j 0 j (hp j) ≫
(mapBifunctorRightUnitor F Y e p hp X).hom j =
(F.obj (X j)).map (singleObjApplyIso (0 : I) Y).hom ≫ e.hom.app (X j) := by
dsimp [mapBifunctorRightUnitor]
erw [CofanMapObjFun.ιMapObj_iso_inv]
rw [mapBifunctorRightUnitorCofan_inj]
lemma mapBifunctorRightUnitor_inv_apply (j : J) :
(mapBifunctorRightUnitor F Y e p hp X).inv j =
e.inv.app (X j) ≫ (F.obj (X j)).map (singleObjApplyIso (0 : I) Y).inv ≫
ιMapBifunctorMapObj F p X ((single₀ I).obj Y) j 0 j (hp j) := rfl
variable {Y Y'}
@[reassoc]
lemma mapBifunctorRightUnitor_inv_naturality :
φ ≫ (mapBifunctorRightUnitor F Y e p hp X').inv =
(mapBifunctorRightUnitor F Y e p hp X).inv ≫ mapBifunctorMapMap F p φ (𝟙 _) := by
ext j
dsimp
rw [mapBifunctorRightUnitor_inv_apply, mapBifunctorRightUnitor_inv_apply, assoc, assoc,
ι_mapBifunctorMapMap]
dsimp
rw [Functor.map_id, id_comp, NatTrans.naturality_assoc]
erw [← NatTrans.naturality_assoc]
rfl
@[reassoc]
lemma mapBifunctorRightUnitor_naturality :
mapBifunctorMapMap F p φ (𝟙 _) ≫ (mapBifunctorRightUnitor F Y e p hp X').hom =
(mapBifunctorRightUnitor F Y e p hp X).hom ≫ φ := by
rw [← cancel_mono (mapBifunctorRightUnitor F Y e p hp X').inv, assoc, assoc, Iso.hom_inv_id,
comp_id, mapBifunctorRightUnitor_inv_naturality, Iso.hom_inv_id_assoc]
end RightUnitor
section
variable {I₁ I₂ I₃ J : Type*} [Zero I₂]
/-- Given two maps `r : I₁ × I₂ × I₃ → J` and `π : I₁ × I₃ → J`, this structure is the
input in the formulation of the triangle equality `mapBifunctor_triangle` which
relates the left and right unitor and the associator for `GradedObject.mapBifunctor`. -/
structure TriangleIndexData (r : I₁ × I₂ × I₃ → J) (π : I₁ × I₃ → J) where
/-- a map `I₁ × I₂ → I₁` -/
p₁₂ : I₁ × I₂ → I₁
hp₁₂ (i : I₁ × I₂ × I₃) : π ⟨p₁₂ ⟨i.1, i.2.1⟩, i.2.2⟩ = r i
/-- a map `I₂ × I₃ → I₃` -/
p₂₃ : I₂ × I₃ → I₃
hp₂₃ (i : I₁ × I₂ × I₃) : π ⟨i.1, p₂₃ i.2⟩ = r i
h₁ (i₁ : I₁) : p₁₂ (i₁, 0) = i₁
h₃ (i₃ : I₃) : p₂₃ (0, i₃) = i₃
variable {r : I₁ × I₂ × I₃ → J} {π : I₁ × I₃ → J}
(τ : TriangleIndexData r π)
namespace TriangleIndexData
attribute [simp] h₁ h₃
lemma r_zero (i₁ : I₁) (i₃ : I₃) : r ⟨i₁, 0, i₃⟩ = π ⟨i₁, i₃⟩ := by
rw [← τ.hp₂₃, τ.h₃ i₃]
/-- The `BifunctorComp₁₂IndexData r` attached to a `TriangleIndexData r π`. -/
@[reducible]
def ρ₁₂ : BifunctorComp₁₂IndexData r where
I₁₂ := I₁
p := τ.p₁₂
q := π
hpq := τ.hp₁₂
/-- The `BifunctorComp₂₃IndexData r` attached to a `TriangleIndexData r π`. -/
@[reducible]
def ρ₂₃ : BifunctorComp₂₃IndexData r where
I₂₃ := I₃
p := τ.p₂₃
q := π
hpq := τ.hp₂₃
end TriangleIndexData
end
section Triangle
variable {C₁ C₂ C₃ D I₁ I₂ I₃ J : Type*} [Category C₁] [Category C₂] [Category C₃] [Category D]
[Zero I₂] [DecidableEq I₂] [HasInitial C₂]
{F₁ : C₁ ⥤ C₂ ⥤ C₁} {F₂ : C₂ ⥤ C₃ ⥤ C₃} {G : C₁ ⥤ C₃ ⥤ D}
(associator : bifunctorComp₁₂ F₁ G ≅ bifunctorComp₂₃ G F₂)
(X₂ : C₂) (e₁ : F₁.flip.obj X₂ ≅ 𝟭 C₁) (e₂ : F₂.obj X₂ ≅ 𝟭 C₃)
[∀ (X₁ : C₁), PreservesColimit (Functor.empty.{0} C₂) (F₁.obj X₁)]
[∀ (X₃ : C₃), PreservesColimit (Functor.empty.{0} C₂) (F₂.flip.obj X₃)]
{r : I₁ × I₂ × I₃ → J} {π : I₁ × I₃ → J}
(τ : TriangleIndexData r π)
(X₁ : GradedObject I₁ C₁) (X₃ : GradedObject I₃ C₃)
[HasMap (((mapBifunctor F₁ I₁ I₂).obj X₁).obj ((single₀ I₂).obj X₂)) τ.p₁₂]
[HasMap (((mapBifunctor G I₁ I₃).obj
(mapBifunctorMapObj F₁ τ.p₁₂ X₁ ((single₀ I₂).obj X₂))).obj X₃) π]
[HasMap (((mapBifunctor F₂ I₂ I₃).obj ((single₀ I₂).obj X₂)).obj X₃) τ.p₂₃]
[HasMap (((mapBifunctor G I₁ I₃).obj X₁).obj
(mapBifunctorMapObj F₂ τ.p₂₃ ((single₀ I₂).obj X₂) X₃)) π]
[HasGoodTrifunctor₁₂Obj F₁ G τ.ρ₁₂ X₁ ((single₀ I₂).obj X₂) X₃]
[HasGoodTrifunctor₂₃Obj G F₂ τ.ρ₂₃ X₁ ((single₀ I₂).obj X₂) X₃]
[HasMap (((mapBifunctor G I₁ I₃).obj X₁).obj X₃) π]
(triangle : ∀ (X₁ : C₁) (X₃ : C₃), ((associator.hom.app X₁).app X₂).app X₃ ≫
(G.obj X₁).map (e₂.hom.app X₃) = (G.map (e₁.hom.app X₁)).app X₃)
lemma mapBifunctor_triangle :
(mapBifunctorAssociator associator τ.ρ₁₂ τ.ρ₂₃ X₁ ((single₀ I₂).obj X₂) X₃).hom ≫
mapBifunctorMapMap G π (𝟙 X₁) (mapBifunctorLeftUnitor F₂ X₂ e₂ τ.p₂₃ τ.h₃ X₃).hom =
mapBifunctorMapMap G π (mapBifunctorRightUnitor F₁ X₂ e₁ τ.p₁₂ τ.h₁ X₁).hom (𝟙 X₃) := by
rw [← cancel_epi ((mapBifunctorMapMap G π
(mapBifunctorRightUnitor F₁ X₂ e₁ τ.p₁₂ τ.h₁ X₁).inv (𝟙 X₃)))]
ext j i₁ i₃ hj
simp only [categoryOfGradedObjects_comp, ι_mapBifunctorMapMap_assoc,
mapBifunctorRightUnitor_inv_apply, Functor.id_obj, Functor.flip_obj_obj, Functor.map_comp,
NatTrans.comp_app, categoryOfGradedObjects_id, Functor.map_id, id_comp, assoc,
ι_mapBifunctorMapMap]
congr 2
rw [← ιMapBifunctor₁₂BifunctorMapObj_eq_assoc F₁ G τ.ρ₁₂ _ _ _ i₁ 0 i₃ j
(by rw [τ.r_zero, hj]) i₁ (by simp), ι_mapBifunctorAssociator_hom_assoc,
ιMapBifunctorBifunctor₂₃MapObj_eq_assoc G F₂ τ.ρ₂₃ _ _ _ i₁ 0 i₃ j
(by rw [τ.r_zero, hj]) i₃ (by simp), ι_mapBifunctorMapMap]
dsimp
rw [Functor.map_id, NatTrans.id_app, id_comp,
← Functor.map_comp_assoc, ← NatTrans.comp_app_assoc, ← Functor.map_comp,
ι_mapBifunctorLeftUnitor_hom_apply F₂ X₂ e₂ τ.p₂₃ τ.h₃ X₃ i₃,
ι_mapBifunctorRightUnitor_hom_apply F₁ X₂ e₁ τ.p₁₂ τ.h₁ X₁ i₁]
dsimp
simp only [Functor.map_comp, NatTrans.comp_app, ← triangle (X₁ i₁) (X₃ i₃), ← assoc]
congr 2
symm
apply NatTrans.naturality_app (associator.hom.app (X₁ i₁))
end Triangle
end GradedObject
end CategoryTheory