| import Mathlib |
| 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 |
|
|
