begin LESs

homotopy/long_exact_sequence.hlean

@@ -0,0 +1,72 @@
+import types.pointed types.int types.fiber
+
+open algebra nat int pointed unit sigma fiber sigma.ops eq
+
+namespace LES
+
+  structure LES : Type :=
+    (car : ℤ → Type*)
+    (fn : Π{n : ℤ}, car (n + 1) →* car n)
+    (chain_complex : Π{n : ℤ} (x : car ((n + 1) + 1)), fn (fn x) = pt)
+    (is_exact : Π{n : ℤ} (x : car (n + 1)), fn x = pt → Σ(y : car ((n + 1) + 1)), fn y = x)
+
+  structure LLES : Type := -- "left" long exact sequence
+    (car : ℕ → Type*)
+    (fn : Π{n : ℕ}, car (n + 1) →* car n)
+    (chain_complex : Π{n : ℕ} (x : car ((n + 1) + 1)), fn (fn x) = pt)
+    (is_exact : Π{n : ℕ} (x : car (n + 1)), fn x = pt → Σ(y : car ((n + 1) + 1)), fn y = x)
+
+  structure RLES : Type := -- "right" long exact sequence
+    (car : ℕ → Type*)
+    (fn : Π{n : ℕ}, car n →* car (n + 1))
+    (chain_complex : Π{n : ℕ} (x : car n), fn (fn x) = pt)
+    (is_exact : Π{n : ℕ} (x : car (n + 1)), fn x = pt → Σ(y : car n), fn y = x)
+
+  open LES LLES RLES
+
+  /-
+    this construction is currently wrong. We need to add one element between the sequence
+  -/
+  definition LES_of_LLES (X : LLES) : LES :=
+  LES.mk (int.rec (car X) (λn, Unit))
+    begin
+      intro n, fconstructor,
+      { induction n with n n,
+        { exact @fn X n},
+        { esimp, intro x, exact star}},
+      { induction n with n n,
+        { apply respect_pt},
+        { reflexivity}}
+    end
+    begin
+      intro n, induction n with n n,
+      { exact chain_complex X},
+      { esimp, intro x, reflexivity}
+    end
+    begin
+      intro n, induction n with n n,
+      { exact is_exact X},
+      { esimp, intro x p, exact sorry}
+    end
+
+  definition fiber_sequence_helper (v : Σ(X Y : Type*), X →* Y) : Σ(Z X : Type*), Z →* X :=
+  ⟨pointed_fiber v.2.2 pt, v.1, pmap.mk point rfl⟩
+  -- match v with
+  -- | ⟨X, Y, f⟩ := ⟨pointed_fiber f pt, X, pmap.mk point rfl⟩
+  -- end
+
+exit
+  definition fiber_sequence.{u} {X Y : Pointed.{u}} (f : X →* Y) : LLES.{u} :=
+  begin
+    fconstructor,
+    { intro n, cases n with n,
+      { exact Y},
+      { exact (iterate fiber_sequence_helper n ⟨X, Y, f⟩).1}},
+    { intro n, cases n with n,
+      { exact f},
+      { exact pmap.mk point rfl}},
+    { intro n x, exact sorry},
+    { exact sorry}
+  end
+
+end LES