| import Mathlib |
| import OAI.GroupTheory.SimpleAmenable.Simplicial.MonoidNerveCoordinates |
|
|
| namespace OAI |
|
|
| section |
| open CategoryTheory Simplicial SimplicialObject Opposite |
| namespace MonoidNerveCoordinates |
|
|
| variable {P:Type} [CommMonoid P] {n:ℕ} |
| def faceLabels (k:Fin (n+2)) (x:Fin (n+1)→P) : Fin n→P := fun j=> |
| if j.val+1 < k.val then x j.castSucc else |
| if j.val+1 = k.val then x j.castSucc * x j.succ else x j.succ |
| lemma labels_face (s:Simplex P (n+1)) (k:Fin (n+2)) : |
| labels ((nerve _).δ k s)=faceLabels k (labels s) := by |
| funext j |
| dsimp only [faceLabels] |
| change (s.map (homOfLE (show k.succAbove j.castSucc ≤ k.succAbove j.succ from |
| (Fin.succAboveOrderEmb k).monotone (Fin.castSucc_le_succ j)))).unop = _ |
| split_ifs with h h' |
| · apply map_congr s |
| · 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) |
| · have hh : k.succAbove j.castSucc=j.castSucc.castSucc := |
| Fin.succAbove_of_castSucc_lt _ _ (by change j.val < k.val; omega) |
| have hh' : k.succAbove j.succ=j.succ.succ := |
| Fin.succAbove_of_le_castSucc _ _ (by change k.val ≤ j.val+1; omega) |
| rw [map_congr s hh hh' _ (homOfLE (show j.castSucc.castSucc ≤ j.succ.succ from by |
| change j.val ≤ j.val+2; omega))] |
| have hc:=congrArg Quiver.Hom.unop (s.map_comp |
| (homOfLE (Fin.castSucc_le_succ j.castSucc)) |
| (homOfLE (Fin.castSucc_le_succ j.succ))) |
| exact hc |
| · apply map_congr s |
| · exact Fin.succAbove_of_le_castSucc _ _ (by change k.val ≤ j.val; omega) |
| · exact Fin.succAbove_of_le_castSucc _ _ (by change k.val ≤ j.val+1; omega) |
| lemma tail_face_val (s:Simplex P (n+1)) (k:Fin (n+2)) (hk:k≠Fin.last (n+1)) : |
| (s.map (homOfLE (Fin.le_last (k.succAbove (Fin.last n))))).unop=(1:P) := by |
| have h:k.succAbove (Fin.last n)=Fin.last (n+1) := by |
| apply Fin.ext |
| rw [Fin.succAbove_of_le_castSucc _ _ (by change k.val ≤ n; have:=k.isLt; have hn:=Fin.val_ne_of_ne hk; dsimp at hn; omega)] |
| rfl |
| have he:=map_congr s h rfl (homOfLE (Fin.le_last (k.succAbove (Fin.last n)))) (𝟙 _) |
| rw [he,s.map_id] |
| rfl |
| lemma tail_face_last_val (s:Simplex P (n+1)) : |
| (s.map (homOfLE (Fin.le_last ((Fin.last (n+1)).succAbove (Fin.last n))))).unop=labels s (Fin.last n) := by |
| apply map_congr s |
| · simp |
| · rfl |
| end MonoidNerveCoordinates |
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| end |
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| end OAI |
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