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import OAI.GroupTheory.SimpleAmenable.Simplicial.MonoidalCongruence
import OAI.GroupTheory.SimpleAmenable.Simplicial.DiagramResolutionNatural
namespace OAI
section
open _root_.CategoryTheory _root_.OAI.CategoryTheory MonoidalCategory SimplicialObject Simplicial Opposite
namespace RestrictedNerve
open IntervalBar IntervalBar.Diagram
variable {C:Type} [Groupoid.{0} C] (W:MorphismProperty C)
[Fact W.StableUnderInverse] [MonoidalCategory C] [SymmetricCategory C]
[W.IsStableUnderBraiding]
variable {I J K:Type} [Preorder I] [Preorder J] [Preorder K]
omit [Fact W.StableUnderInverse] in
@[simp] lemma inclusion_η : Functor.OplaxMonoidal.η (inclusion W) = 𝟙 _ := rfl
omit [Fact W.StableUnderInverse] in
@[simp] lemma inclusion_μ (A B:WideSubcategory W) :
Functor.LaxMonoidal.μ (inclusion W) A B = 𝟙 _ := rfl
@[simp] lemma widePosReindex_η (u:I→oJ) :
Functor.OplaxMonoidal.η (widePosReindex W u) = 𝟙 _ := rfl
@[simp] lemma widePosReindex_μ (u:I→oJ) (A B:WideSubcategory (diagramProperty W J)) :
Functor.LaxMonoidal.μ (widePosReindex W u) A B = 𝟙 _ := rfl
@[simp] lemma posReindex_η (u:I→oJ) :
Functor.OplaxMonoidal.η (posReindex W u) = 𝟙 _ := rfl
@[simp] lemma posReindex_μ (u:I→oJ) (A B:PosDiagrams W J) :
Functor.LaxMonoidal.μ (posReindex W u) A B = 𝟙 _ := rfl
@[simp] lemma eqToHom_pos_app {A B:PosDiagrams W I} (e:A=B) (i j:I) (h:i≤j) :
(eqToHom e).hom.app i j h =
eqToHom (congrArg (fun A:PosDiagrams W I=>((diagramInclusion W I).obj A).obj i j h) e) := by
cases e; rfl
noncomputable def nestedPosReindex (u:I→oJ) :
PosDiagrams (diagramProperty W J) K ⥤ PosDiagrams (diagramProperty W I) K where
obj A := (Diagram.map (I:=K) (widePosReindex W u)).obj A
map {A B} f := InducedCategory.homMk {
app i j h := (posReindex W u).map (f.hom.app i j h)
unit i := by
apply InducedCategory.hom_ext; apply Hom.ext; intro x y h
have he := congrArg (fun z => z.hom.app (u x) (u y) (u.monotone h)) (f.hom.unit i)
change (f.hom.app i i (le_refl i)).hom.app (u x) (u y) (u.monotone h) ≫
((B.unit i).hom.hom.hom.app (u x) (u y) (u.monotone h) ≫ 𝟙 _) =
(A.unit i).hom.hom.hom.app (u x) (u y) (u.monotone h) ≫ 𝟙 _ at he
change (f.hom.app i i (le_refl i)).hom.app (u x) (u y) (u.monotone h) ≫
(((B.unit i).hom.hom.hom.app (u x) (u y) (u.monotone h) ≫ 𝟙 _) ≫ 𝟙 _) =
((A.unit i).hom.hom.hom.app (u x) (u y) (u.monotone h) ≫ 𝟙 _) ≫ 𝟙 _
simpa only [Category.comp_id] using he
cut i j k hij hjk := by
apply InducedCategory.hom_ext; apply Hom.ext; intro x y h
have he := congrArg (fun z => z.hom.app (u x) (u y) (u.monotone h)) (f.hom.cut i j k hij hjk)
change ((f.hom.app i j hij ⊗ₘ f.hom.app j k hjk).hom.app (u x) (u y) (u.monotone h)) ≫
(𝟙 _ ≫ (B.cut i j k hij hjk).hom.hom.hom.app (u x) (u y) (u.monotone h)) =
(𝟙 _ ≫ (A.cut i j k hij hjk).hom.hom.hom.app (u x) (u y) (u.monotone h)) ≫
(f.hom.app i k (le_trans hij hjk)).hom.app (u x) (u y) (u.monotone h) at he
erw [posTensor_app] at he
change (((posReindex W u).map (f.hom.app i j hij) ⊗ₘ
(posReindex W u).map (f.hom.app j k hjk)).hom.app x y h) ≫
(𝟙 _ ≫ 𝟙 _ ≫ (B.cut i j k hij hjk).hom.hom.hom.app (u x) (u y) (u.monotone h)) =
(𝟙 _ ≫ 𝟙 _ ≫ (A.cut i j k hij hjk).hom.hom.hom.app (u x) (u y) (u.monotone h)) ≫
(f.hom.app i k (le_trans hij hjk)).hom.app (u x) (u y) (u.monotone h)
erw [posTensor_app]
simpa only [posReindex, Category.id_comp] using! he }
map_id A := by
apply InducedCategory.hom_ext; apply Hom.ext; intro i j h
exact (posReindex W u).map_id _
map_comp f g := by
apply InducedCategory.hom_ext; apply Hom.ext; intro i j h
exact (posReindex W u).map_comp _ _
lemma nestedPosReindex_property (u:I→oJ) {A B:PosDiagrams (diagramProperty W J) K} (f:A⟶B)
(hf:diagramProperty (diagramProperty W J) K f) :
diagramProperty (diagramProperty W I) K ((nestedPosReindex W u).map f) :=
fun i j h => posReindex_property W u (f.hom.app i j h) (hf i j h)
private lemma nestedPosReindex_inclusion_obj (u:I→oJ) (A:PosDiagrams (diagramProperty W J) K) :
(nestedPosReindex (K:=K) W u ⋙ posDiagramInclusion (diagramProperty W I) K).obj A =
(posDiagramInclusion (diagramProperty W J) K ⋙ Diagram.map (posReindex W u)).obj A := by
refine ext_heq rfl ?_ ?_
· apply heq_of_eq; funext i; apply Iso.ext
apply InducedCategory.hom_ext; apply Hom.ext; intro x y h
simp [nestedPosReindex,diagramInclusion,mapObj]
simp [posReindex,widePosReindex]
rfl
· apply heq_of_eq; funext i j k hij hjk; apply Iso.ext
apply InducedCategory.hom_ext; apply Hom.ext; intro x y h
simp [nestedPosReindex,diagramInclusion,mapObj]
simp [posReindex,widePosReindex]
rfl
lemma nestedPosReindex_inclusion (u:I→oJ) :
nestedPosReindex (K:=K) W u ⋙ posDiagramInclusion (diagramProperty W I) K =
posDiagramInclusion (diagramProperty W J) K ⋙ Diagram.map (posReindex W u) := by
refine CategoryTheory.Functor.ext (nestedPosReindex_inclusion_obj W u) ?_
intro A B f
apply Hom.ext; intro i j h
apply InducedCategory.hom_ext; apply Hom.ext; intro x y hxy
simp [nestedPosReindex,posReindex,Diagram.map]
erw [Diagram.eqToHom_app, Diagram.eqToHom_app]
erw [comp_app, comp_app, eqToHom_pos_app, eqToHom_pos_app]
change (f.hom.app i j h).hom.app (u x) (u y) (u.monotone hxy) =
𝟙 _ ≫ (f.hom.app i j h).hom.app (u x) (u y) (u.monotone hxy) ≫ 𝟙 _
simp only [Category.id_comp, Category.comp_id]
@[simp] lemma transpose_η_app (n:ℕ) (t:Fin (n+1)) (i j:I) (h:i≤j) :
((Functor.OplaxMonoidal.η (transposeDiagram (I:=I) W n)).hom.app t).hom.app i j h = 𝟙 _ := by
have he := congrArg (fun z=> (z.hom.app t).hom.app i j h)
(Functor.Monoidal.ε_η (transposeDiagram (I:=I) W n))
change 𝟙 (𝟙_ C) ≫ ((Functor.OplaxMonoidal.η (transposeDiagram (I:=I) W n)).hom.app t).hom.app i j h = 𝟙 (𝟙_ C) at he
change ((Functor.OplaxMonoidal.η (transposeDiagram (I:=I) W n)).hom.app t).hom.app i j h = 𝟙 (𝟙_ C)
exact (Category.id_comp _).symm.trans he
@[simp] lemma transpose_obj (n:ℕ) (A:Diagram (Strings W n) I) :
(transposeDiagram W n).obj A=transposeDiagramObj W A := rfl
@[simp] lemma transpose_map (n:ℕ) {A B:Diagram (Strings W n) I} (f:A⟶B) :
(transposeDiagram W n).map f=transposeDiagramHom W f := rfl
@[simp] lemma diagramReindex_η (u:I→oJ) :
Functor.OplaxMonoidal.η (Diagram.reindex (C:=C) u)=𝟙 _ := rfl
@[simp] lemma diagramReindex_μ (u:I→oJ) (A B:Diagram C J) :
Functor.LaxMonoidal.μ (Diagram.reindex u) A B=𝟙 _ := rfl
private noncomputable abbrev innerReindexLeft (u:I→oJ) (n:ℕ) :=
Diagram.map (I:=K) (Diagram.reindex (C:=Strings W n) u) ⋙
Diagram.map (I:=K) (transposeDiagram (I:=I) W n) ⋙
transposeDiagram (I:=K) (diagramProperty W I) n
private noncomputable abbrev innerReindexRight (u:I→oJ) (n:ℕ) :=
Diagram.map (I:=K) (transposeDiagram (I:=J) W n) ⋙
transposeDiagram (I:=K) (diagramProperty W J) n ⋙
stringsMap (diagramProperty (diagramProperty W J) K) (diagramProperty (diagramProperty W I) K)
(nestedPosReindex W u) (nestedPosReindex_property W u) n
private lemma doubleTranspose_innerReindex_vertex (u:I→oJ) (n:ℕ)
(A:Diagram (Diagram (Strings W n) J) K) (t:Fin (n+1)) :
(((innerReindexLeft (K:=K) W u n).obj A).obj.obj t) =
(((innerReindexRight (K:=K) W u n).obj A).obj.obj t) := by
refine ext_heq rfl ?_ ?_
· apply heq_of_eq; funext i; apply Iso.ext; apply WideSubcategory.hom_ext
apply InducedCategory.hom_ext; apply Hom.ext; intro x y h
simp [stringsMap,nestedPosReindex,transposeDiagramObj,stringVertex,mapObj]
simp [widePosReindex,posReindex,Diagram.reindex]
erw [comp_app, transpose_η_app]
change (((A.unit i).hom.app (u x) (u y) (u.monotone h)).hom.app t ≫ 𝟙 _) ≫ 𝟙 _ =
(((A.unit i).hom.app (u x) (u y) (u.monotone h)).hom.app t ≫
((Functor.OplaxMonoidal.η (transposeDiagram (I:=J) W n)).hom.app t).hom.app
(u x) (u y) (u.monotone h)) ≫ 𝟙 _
rw [transpose_η_app]
rfl
· apply heq_of_eq; funext i j k hij hjk; apply Iso.ext; apply WideSubcategory.hom_ext
apply InducedCategory.hom_ext; apply Hom.ext; intro x y h
simp [stringsMap,nestedPosReindex,transposeDiagramObj,stringVertex,mapObj]
simp [widePosReindex,posReindex,Diagram.reindex]
change 𝟙 _ ≫ 𝟙 _ ≫ ((A.cut i j k hij hjk).hom.app (u x) (u y) (u.monotone h)).hom.app t =
𝟙 _ ≫ 𝟙 _ ≫ ((A.cut i j k hij hjk).hom.app (u x) (u y) (u.monotone h)).hom.app t
simp
private lemma nested_comp_app {A B D:PosDiagrams (diagramProperty W I) K}
(f:A⟶B) (g:B⟶D) (i j:K) (h:i≤j) (x y:I) (hxy:x≤y) :
((f ≫ g).hom.app i j h).hom.app x y hxy =
(f.hom.app i j h).hom.app x y hxy ≫ (g.hom.app i j h).hom.app x y hxy := rfl
private lemma nested_eqToHom_app {A B:PosDiagrams (diagramProperty W I) K}
(e:A=B) (i j:K) (h:i≤j) (x y:I) (hxy:x≤y) :
((eqToHom e).hom.app i j h).hom.app x y hxy =
eqToHom (congrArg (fun A:PosDiagrams (diagramProperty W I) K =>
((diagramInclusion W I).obj (A.obj i j h).obj).obj x y hxy) e) := by
cases e
rfl
private lemma twiceTransposed_comp_app {n:ℕ}
{A B D:Strings (diagramProperty (diagramProperty W I) K) n}
(f:A⟶B) (g:B⟶D) (t:Fin (n+1)) (i j:K) (h:i≤j) (x y:I) (hxy:x≤y) :
(((f ≫ g).hom.app t).hom.app i j h).hom.app x y hxy =
((f.hom.app t).hom.app i j h).hom.app x y hxy ≫
((g.hom.app t).hom.app i j h).hom.app x y hxy := rfl
private lemma twiceTransposed_eqToHom_app {n:ℕ}
{A B:Strings (diagramProperty (diagramProperty W I) K) n}
(e:A=B) (t:Fin (n+1)) (i j:K) (h:i≤j) (x y:I) (hxy:x≤y) :
(((eqToHom e).hom.app t).hom.app i j h).hom.app x y hxy =
eqToHom (congrArg (fun A:Strings (diagramProperty (diagramProperty W I) K) n =>
(((A.obj.obj t).obj i j h).obj.obj x y hxy).obj) e) := by
cases e
rfl
private lemma doubleTranspose_innerReindex_obj (u:I→oJ) (n:ℕ)
(A:Diagram (Diagram (Strings W n) J) K) :
(innerReindexLeft (K:=K) W u n).obj A = (innerReindexRight (K:=K) W u n).obj A := by
apply WideSubcategory.ext
refine CategoryTheory.Functor.ext (doubleTranspose_innerReindex_vertex W u n A) ?_
intro t s f
apply InducedCategory.hom_ext; apply Hom.ext; intro i j h
apply InducedCategory.hom_ext; apply Hom.ext; intro x y hxy
erw [nested_comp_app, nested_comp_app, nested_eqToHom_app, nested_eqToHom_app]
change ((A.obj i j h).obj (u x) (u y) (u.monotone hxy)).obj.map f =
𝟙 _ ≫ ((A.obj i j h).obj (u x) (u y) (u.monotone hxy)).obj.map f ≫ 𝟙 _
simp only [Category.id_comp, Category.comp_id]
private lemma twiceTransposed_transport_ext {n:ℕ}
{A A' B B':Strings (diagramProperty (diagramProperty W I) K) n}
(eA:A=A') (eB:B'=B) (f:A⟶B) (g:A'⟶B')
(hf:∀ (t:Fin (n+1)) (i j:K) (h:i≤j) (x y:I) (hxy:x≤y),
((f.hom.app t).hom.app i j h).hom.app x y hxy =
eqToHom (congrArg (fun A:Strings (diagramProperty (diagramProperty W I) K) n =>
(((A.obj.obj t).obj i j h).obj.obj x y hxy).obj) eA) ≫
((g.hom.app t).hom.app i j h).hom.app x y hxy ≫
eqToHom (congrArg (fun A:Strings (diagramProperty (diagramProperty W I) K) n =>
(((A.obj.obj t).obj i j h).obj.obj x y hxy).obj) eB)) :
f = eqToHom eA ≫ g ≫ eqToHom eB := by
apply WideSubcategory.hom_ext; apply NatTrans.ext; funext t
apply InducedCategory.hom_ext; apply Hom.ext; intro i j h
apply InducedCategory.hom_ext; apply Hom.ext; intro x y hxy
erw [twiceTransposed_comp_app, twiceTransposed_comp_app,
twiceTransposed_eqToHom_app, twiceTransposed_eqToHom_app]
exact hf t i j h x y hxy
private lemma doubleTranspose_innerReindex_eq (u:I→oJ) (n:ℕ) :
innerReindexLeft (K:=K) W u n = innerReindexRight (K:=K) W u n := by
refine CategoryTheory.Functor.ext (doubleTranspose_innerReindex_obj W u n) ?_
intro A B f
apply twiceTransposed_transport_ext
intro t i j h x y hxy
change ((f.app i j h).app (u x) (u y) (u.monotone hxy)).hom.app t =
𝟙 _ ≫ ((f.app i j h).app (u x) (u y) (u.monotone hxy)).hom.app t ≫ 𝟙 _
simp only [Category.id_comp, Category.comp_id]
lemma doubleTranspose_innerReindex (u:I→oJ) (n:ℕ) :
Diagram.map (I:=K) (Diagram.reindex (C:=Strings W n) u) ⋙
Diagram.map (I:=K) (transposeDiagram (I:=I) W n) ⋙
transposeDiagram (I:=K) (diagramProperty W I) n =
Diagram.map (I:=K) (transposeDiagram (I:=J) W n) ⋙
transposeDiagram (I:=K) (diagramProperty W J) n ⋙
stringsMap (diagramProperty (diagramProperty W J) K) (diagramProperty (diagramProperty W I) K)
(nestedPosReindex W u) (nestedPosReindex_property W u) n := by
exact doubleTranspose_innerReindex_eq W u n
end RestrictedNerve
end
open _root_.CategoryTheory _root_.OAI.CategoryTheory MonoidalCategory
namespace IntervalBar.Diagram
variable {C D:Type} [Groupoid.{0} C] [Groupoid.{0} D]
[MonoidalCategory C] [MonoidalCategory D]
[SymmetricCategory C] [SymmetricCategory D]
variable {I J K:Type} [Preorder I] [Preorder J] [Preorder K]
@[simp] lemma map_ε (F:C⥤D) [F.Braided] :
Functor.LaxMonoidal.ε (map (I:=I) F)=mapεHom F := rfl
@[simp] lemma map_μ (F:C⥤D) [F.Braided] (A B:Diagram C I) :
Functor.LaxMonoidal.μ (map F) A B=mapμHom F A B := rfl
@[simp] lemma reindex_ε (u:I→oJ) :
Functor.LaxMonoidal.ε (reindex (C:=C) u)=𝟙 _ := rfl
@[simp] lemma reindex_μ (u:I→oJ) (A B:Diagram C J) :
Functor.LaxMonoidal.μ (reindex u) A B=𝟙 _ := rfl
noncomputable instance reindex_map_monoidal (F:C⥤D) [F.Braided] (u:I→oJ) :
NatTrans.IsMonoidal (eqToHom (reindex_map F u)) where
unit := by
apply Hom.ext; intro i j h
simp only [CategoryTheory.eqToHom_app]
simp only [Functor.LaxMonoidal.comp_ε,map_ε,reindex_ε]
simp [mapεHom,map,reindex]
erw [Diagram.eqToHom_app]
change Functor.LaxMonoidal.ε F ≫ F.map (𝟙 (𝟙_ C)) ≫ 𝟙 _ = 𝟙 _ ≫ Functor.LaxMonoidal.ε F
simp only [CategoryTheory.Functor.map_id, Category.id_comp, Category.comp_id]
tensor A B := by
apply Hom.ext; intro i j h
simp only [CategoryTheory.eqToHom_app]
simp only [Functor.LaxMonoidal.comp_μ,map_μ,reindex_μ]
simp [mapμHom,map,reindex]
erw [Diagram.eqToHom_app, Diagram.eqToHom_app, Diagram.eqToHom_app]
change Functor.LaxMonoidal.μ F (A.obj (u i) (u j) (u.monotone h)) (B.obj (u i) (u j) (u.monotone h)) ≫
F.map (𝟙 _) ≫ 𝟙 _ = (𝟙 _ ⊗ₘ 𝟙 _) ≫ 𝟙 _ ≫
Functor.LaxMonoidal.μ F (A.obj (u i) (u j) (u.monotone h)) (B.obj (u i) (u j) (u.monotone h))
simp only [CategoryTheory.Functor.map_id, id_tensorHom_id, Category.id_comp, Category.comp_id]
lemma map_reindex_map (F:C⥤D) [F.Braided] (u:I→oJ) :
map (I:=K) (reindex (C:=C) u) ⋙ map (I:=K) (map F) =
map (I:=K) (map F) ⋙ map (I:=K) (reindex u) := by
have he := map_congr (I:=K) (reindex_map F u) (reindex_map_monoidal F u)
simpa only [map_comp] using he
end IntervalBar.Diagram
end OAI
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