| import Mathlib |
| import OAI.GroupTheory.SimpleAmenable.Homology.DoubleCoordinates |
|
|
| namespace OAI |
|
|
| section |
| open _root_.CategoryTheory _root_.OAI.CategoryTheory Limits MonoidalCategory Simplicial Opposite |
| namespace IntervalBar.Diagram |
| |
| variable {C:Type} [Groupoid.{0} C] [MonoidalCategory C] |
| noncomputable def interval {n:ℕ} (i j:Fin (n+1)) (h:i≤j) : Diagram C (Fin (n+1)) ⥤ C where |
| obj D := D.obj i j h |
| map f := f.app i j h |
| noncomputable def pairAt {n:ℕ} (i j:Fin n) : (Fin n→C) ⥤ C where |
| obj X := X i ⊗ X j |
| map f := f i ⊗ₘ f j |
| map_id X := by simp |
| map_comp f g := by simp [tensorHom_comp_tensorHom] |
| lemma interval_congr {n:ℕ} {i j i' j':Fin (n+1)} {h:i≤j} {h':i'≤j'} (hi:i=i') (hj:j=j') : |
| interval (C:=C) i j h=interval i' j' h' := by subst i' |
| noncomputable def faceEdge {n:ℕ} (k:Fin (n+2)) (j:Fin n) : (Fin (n+1)→C) ⥤ C := |
| if j.val+1<k.val then Pi.eval _ j.castSucc else |
| if j.val+1=k.val then pairAt j.castSucc j.succ else Pi.eval _ j.succ |
| lemma faceEdge_left {n:ℕ} (k:Fin (n+2)) (j:Fin n) (h:j.val+1<k.val) : |
| reindex (C:=C) (SimplexCategory.δ k).toOrderHom ⋙ eval n ⋙ Pi.eval _ j = |
| eval (n+1) ⋙ Pi.eval _ j.castSucc := by |
| apply interval_congr |
| · exact Fin.succAbove_of_castSucc_lt _ _ (by change j.val<k.val; omega) |
| · exact Fin.succAbove_of_castSucc_lt _ _ (by change j.val+1<k.val; exact h) |
| lemma faceEdge_right {n:ℕ} (k:Fin (n+2)) (j:Fin n) (h:k.val≤j.val) : |
| reindex (C:=C) (SimplexCategory.δ k).toOrderHom ⋙ eval n ⋙ Pi.eval _ j = |
| eval (n+1) ⋙ Pi.eval _ j.succ := by |
| apply interval_congr |
| · exact Fin.succAbove_of_le_castSucc _ _ (by change k.val≤j.val; exact h) |
| · exact Fin.succAbove_of_le_castSucc _ _ (by change k.val≤j.val+1; omega) |
| noncomputable def faceEdge_middle {n:ℕ} (k:Fin (n+2)) (j:Fin n) (h:j.val+1=k.val) : |
| eval (C:=C) (n+1) ⋙ pairAt j.castSucc j.succ ≅ |
| reindex (SimplexCategory.δ k).toOrderHom ⋙ eval n ⋙ Pi.eval _ j := by |
| have he:reindex (C:=C) (SimplexCategory.δ k).toOrderHom ⋙ eval n ⋙ Pi.eval _ j = |
| interval j.castSucc.castSucc j.succ.succ (by change j.val≤j.val+2; omega) := by |
| apply interval_congr |
| · exact Fin.succAbove_of_castSucc_lt _ _ (by change j.val<k.val; omega) |
| · exact Fin.succAbove_of_le_castSucc _ _ (by change k.val≤j.val+1; omega) |
| let e : eval (C:=C) (n+1) ⋙ pairAt j.castSucc j.succ ≅ |
| interval j.castSucc.castSucc j.succ.succ (by change j.val≤j.val+2; omega) := |
| NatIso.ofComponents (fun D=>D.cut j.castSucc.castSucc j.succ.castSucc j.succ.succ |
| (Fin.castSucc_le_castSucc_iff.mpr (Fin.castSucc_le_succ _)) (Fin.castSucc_le_succ _)) |
| (by intro D E f |
| exact e ≪≫ (eqToIso he).symm |
| noncomputable def faceEdgeIso {n:ℕ} (k:Fin (n+2)) (j:Fin n) : |
| eval (C:=C) (n+1) ⋙ faceEdge k j ≅ |
| reindex (SimplexCategory.δ k).toOrderHom ⋙ eval n ⋙ Pi.eval _ j := by |
| dsimp only [faceEdge] |
| split_ifs with h h' |
| · exact (eqToIso (faceEdge_left k j h)).symm |
| · exact faceEdge_middle k j h' |
| · exact (eqToIso (faceEdge_right k j (by omega))).symm |
| end IntervalBar.Diagram |
| |
| end |
|
|
| open _root_.CategoryTheory _root_.OAI.CategoryTheory Limits MonoidalCategory Simplicial Opposite |
| namespace NerveHomology |
| open FreeChains |
|
|
| variable {C D E:Type} [Category.{0} C] [Category.{0} D] [Category.{0} E] |
| noncomputable abbrev map (F:C⥤D) (q:ℕ) := SSet.homologyMap (nerveMap F) Z q |
| lemma comp (F:C⥤D) (G:D⥤E) (q:ℕ) : map (F⋙G) q=map F q ≫ map G q := by |
| change SSet.homologyMap (nerveMap F ≫ nerveMap G) Z q=_ |
| rw [SSet.homologyMap_comp] |
| lemma iso {F G:C⥤D} (e:F≅G) (q:ℕ) : map F q=map G q := |
| (NerveHomotopy.ofNatTrans e.hom).congr_homologyMap Z q |
| lemma const (U:D) (q:ℕ) (hq:q≠0) : map ((Functor.const C).obj U) q=0 := by |
| have h : nerveMap ((Functor.const C).obj U)=SSet.const (nerveEquiv.symm U) := by ext n x |
| rw [map,h] |
| exact ConnectedProduct.const_homology_zero (X:=nerve C) (Y:=nerve D) (nerveEquiv.symm U) q hq |
| end NerveHomology |
| namespace MarkedH1 |
| open FreeChains ComponentTranslation IntervalBar.Diagram NerveHomology |
|
|
| variable {C:Type} [Groupoid.{0} C] [MonoidalCategory C] [SymmetricCategory C] |
| omit [SymmetricCategory C] in |
| @[reassoc] lemma primitive_eval {n:ℕ} (p:Fin n→Skeleton C) (i:Fin n) : |
| primitive p i ≫ NerveHomology.map (eval n) 1=NerveHomology.map (wordInclude p i) 1 := by |
| rw [primitive_eq,←NerveHomology.comp] |
| exact NerveHomology.iso (Functor.isoWhiskerLeft (wordInclude p i) (eval n).asEquivalence.counitIso) 1 |
| omit [SymmetricCategory C] in |
| lemma recognition {D:Type} [Category.{0} D] {n:ℕ} (p:Fin n→Skeleton C) (i:Fin n) |
| (F:D⥤IntervalBar.Diagram C (Fin (n+1))) (T:D⥤Fiber (p i)) |
| (e:F⋙eval n≅T⋙wordInclude p i) : |
| NerveHomology.map F 1=NerveHomology.map T 1 ≫ primitive p i := by |
| have := NerveHomotopy.homologyMap_isIso (eval (C:=C) n) Z 1 |
| apply (cancel_mono (NerveHomology.map (eval n) 1)).mp |
| rw [Category.assoc,primitive_eval,←NerveHomology.comp,←NerveHomology.comp] |
| exact NerveHomology.iso e 1 |
| omit [SymmetricCategory C] in |
| lemma recognition_const {D:Type} [Category.{0} D] {n:ℕ} (F:D⥤IntervalBar.Diagram C (Fin (n+1))) |
| (U:Fin n→C) (e:F⋙eval n≅(Functor.const D).obj U) : NerveHomology.map F 1=0 := by |
| have := NerveHomotopy.homologyMap_isIso (eval (C:=C) n) Z 1 |
| apply (cancel_mono (NerveHomology.map (eval n) 1)).mp |
| rw [zero_comp,←NerveHomology.comp,NerveHomology.iso e 1] |
| exact NerveHomology.const U 1 (by decide) |
| noncomputable def wordDiagram_eval {n:ℕ} (p:Fin n→Skeleton C) (i:Fin n) : |
| wordDiagram p i ⋙ eval n ≅ wordInclude p i := |
| Functor.isoWhiskerLeft (wordInclude p i) (eval n).asEquivalence.counitIso |
| noncomputable def primitiveFace_eval {n:ℕ} (p:Fin (n+1)→Skeleton C) (i:Fin (n+1)) |
| (k:Fin (n+2)) (j:Fin n) : |
| (wordDiagram p i ⋙ reindex (SimplexCategory.δ k).toOrderHom) ⋙ eval n ⋙ Pi.eval _ j ≅ |
| wordInclude p i ⋙ faceEdge k j := |
| Functor.isoWhiskerLeft (wordDiagram p i) (faceEdgeIso k j).symm ≪≫ |
| Functor.isoWhiskerRight (wordDiagram_eval p i) (faceEdge k j) |
| noncomputable def includeProjIso {n:ℕ} (p:Fin n→Skeleton C) (i j:Fin n) : |
| wordInclude p i ⋙ Pi.eval _ j ≅ |
| (if i=j then (property (p i)).ι else (Functor.const _).obj (repr (p j))) := |
| eqToIso (wordInclude_proj p i j) |
| noncomputable def wordPair_left {n:ℕ} (p:Fin n→Skeleton C) (i j:Fin n) (h:i≠j) : |
| wordInclude p i ⋙ pairAt i j ≅ (property (p i)).ι ⋙ tensorRight (repr (p j)) := by |
| classical |
| change ((wordInclude p i ⋙ Pi.eval _ i).prod' (wordInclude p i ⋙ Pi.eval _ j)) ⋙ tensor C ≅ _ |
| rw [wordInclude_proj,wordInclude_proj,ite_eq_left rfl,ite_eq_right h] |
| exact NatIso.ofComponents (fun _=>Iso.refl _) (by intro X Y f; simp) |
| noncomputable def wordPair_right {n:ℕ} (p:Fin n→Skeleton C) (i j:Fin n) (h:i≠j) : |
| wordInclude p j ⋙ pairAt i j ≅ (property (p j)).ι ⋙ tensorLeft (repr (p i)) := by |
| classical |
| change ((wordInclude p j ⋙ Pi.eval _ i).prod' (wordInclude p j ⋙ Pi.eval _ j)) ⋙ tensor C ≅ _ |
| rw [wordInclude_proj,wordInclude_proj,ite_eq_right h.symm,ite_eq_left rfl] |
| exact NatIso.ofComponents (fun _=>Iso.refl _) (by intro X Y f |
| noncomputable def wordPair_other {n:ℕ} (p:Fin n→Skeleton C) (i j k:Fin n) (hi:k≠i) (hj:k≠j) : |
| wordInclude p k ⋙ pairAt i j ≅ (Functor.const _).obj (repr (p i) ⊗ repr (p j)) := by |
| classical |
| change ((wordInclude p k ⋙ Pi.eval _ i).prod' (wordInclude p k ⋙ Pi.eval _ j)) ⋙ tensor C ≅ _ |
| rw [wordInclude_proj,wordInclude_proj,ite_eq_right hi,ite_eq_right hj] |
| exact NatIso.ofComponents (fun _=>Iso.refl _) (by intro X Y f; simp) |
| omit [SymmetricCategory C] in |
| lemma face_recognition {n:ℕ} (p:Fin (n+1)→Skeleton C) (i:Fin (n+1)) (k:Fin (n+2)) |
| (p':Fin n→Skeleton C) (i':Fin n) (T:Fiber (p i)⥤Fiber (p' i')) |
| (e:∀j:Fin n,wordInclude p i ⋙ faceEdge k j ≅ (T⋙wordInclude p' i') ⋙ Pi.eval _ j) : |
| primitive p i ≫ NerveHomology.map (reindex (SimplexCategory.δ k).toOrderHom) 1 = |
| NerveHomology.map T 1 ≫ primitive p' i' := by |
| rw [primitive_eq,←NerveHomology.comp] |
| apply recognition p' i' _ T |
| exact NatIso.pi' (fun j=>primitiveFace_eval p i k j ≪≫ e j) |
| omit [SymmetricCategory C] in |
| lemma face_recognition_const {n:ℕ} (p:Fin (n+1)→Skeleton C) (i:Fin (n+1)) (k:Fin (n+2)) |
| (U:Fin n→C) |
| (e:∀j:Fin n,wordInclude p i ⋙ faceEdge k j ≅ (Functor.const (Fiber (p i))).obj (U j)) : |
| primitive p i ≫ NerveHomology.map (reindex (SimplexCategory.δ k).toOrderHom) 1 =0 := by |
| rw [primitive_eq,←NerveHomology.comp] |
| apply recognition_const _ U |
| exact NatIso.pi' (fun j=>primitiveFace_eval p i k j ≪≫ e j) |
| end MarkedH1 |
| |
| end OAI |
| |
