feat(Algebra/Homology): construction of the homotopy cofiber (#8969)

Mathlib.lean

@@ -245,6 +245,7 @@ import Mathlib.Algebra.Homology.Homotopy
 import Mathlib.Algebra.Homology.HomotopyCategory
 import Mathlib.Algebra.Homology.HomotopyCategory.HomComplex
 import Mathlib.Algebra.Homology.HomotopyCategory.Shift
+import Mathlib.Algebra.Homology.HomotopyCofiber
 import Mathlib.Algebra.Homology.ImageToKernel
 import Mathlib.Algebra.Homology.LocalCohomology
 import Mathlib.Algebra.Homology.Localization

Mathlib/Algebra/Homology/Homotopy.lean

@@ -56,6 +56,11 @@ theorem dNext_eq (f : ∀ i j, C.X i ⟶ D.X j) {i i' : ι} (w : c.Rel i i') :
   rfl
 #align d_next_eq dNext_eq
 
+lemma dNext_eq_zero (f : ∀ i j, C.X i ⟶ D.X j) (i : ι) (hi : ¬ c.Rel i (c.next i)) :
+    dNext i f = 0 := by
+  dsimp [dNext]
+  rw [shape _ _ _ hi, zero_comp]
+
 @[simp 1100]
 theorem dNext_comp_left (f : C ⟶ D) (g : ∀ i j, D.X i ⟶ E.X j) (i : ι) :
     (dNext i fun i j => f.f i ≫ g i j) = f.f i ≫ dNext i g :=
@@ -74,6 +79,11 @@ def prevD (j : ι) : (∀ i j, C.X i ⟶ D.X j) →+ (C.X j ⟶ D.X j) :=
     Preadditive.add_comp _ _ _ _ _ _
 #align prev_d prevD
 
+lemma prevD_eq_zero (f : ∀ i j, C.X i ⟶ D.X j) (i : ι) (hi : ¬ c.Rel (c.prev i) i) :
+    prevD i f = 0 := by
+  dsimp [prevD]
+  rw [shape _ _ _ hi, comp_zero]
+
 /-- `f j (c.prev j)`. -/
 def toPrev (j : ι) : (∀ i j, C.X i ⟶ D.X j) →+ (C.X j ⟶ D.xPrev j) :=
   AddMonoidHom.mk' (fun f => f j (c.prev j)) fun _ _ => rfl

Mathlib/Algebra/Homology/HomotopyCofiber.lean

@@ -0,0 +1,232 @@
+/-
+Copyright (c) 2023 Joël Riou. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Joël Riou
+-/
+import Mathlib.Algebra.Homology.Homotopy
+
+/-! The homotopy cofiber of a morphism of homological complexes
+
+In this file, we construct the homotopy cofiber of a morphism `φ : F ⟶ G`
+between homological complexes in `HomologicalComplex C c`. In degree `i`,
+it is isomorphic to `(F.X j) ⊞ (G.X i)` if there is a `j` such that `c.Rel i j`,
+and `G.X i` otherwise. (This is also known as the mapping cone of `φ`. Under
+the name `CochainComplex.mappingCone`, a specific API shall be developed
+for the case of cochain complexes indexed by `ℤ` (TODO).)
+
+When we assume `hc : ∀ j, ∃ i, c.Rel i j` (which holds in the case of chain complexes,
+or cochain complexes indexed by `ℤ`), then for any homological complex `K`,
+there is a bijection `HomologicalComplex.homotopyCofiber.descEquiv φ K hc`
+between `homotopyCofiber φ ⟶ K` and the tuples `(α, hα)` with
+`α : G ⟶ K` and `hα : Homotopy (φ ≫ α) 0` (TODO).
+
+We shall also study the cylinder of a homological complex `K`: this is the
+homotopy cofiber of the morphism  `biprod.lift (𝟙 K) (-𝟙 K) : K ⟶ K ⊞ K`.
+Then, a morphism `K.cylinder ⟶ M` is determined by the data of two
+morphisms `φ₀ φ₁ : K ⟶ M` and a homotopy `h : Homotopy φ₀ φ₁`,
+see `cylinder.desc`. There is also a homotopy equivalence
+`cylinder.homotopyEquiv K : HomotopyEquiv K.cylinder K`. From the construction of
+the cylinder, we deduce the lemma `Homotopy.map_eq_of_inverts_homotopyEquivalences`
+which assert that if a functor inverts homotopy equivalences, then the image of
+two homotopic maps are equal (TODO).
+
+-/
+
+
+open CategoryTheory Category Limits Preadditive
+
+variable {C : Type*} [Category C] [Preadditive C]
+
+namespace HomologicalComplex
+
+variable {ι : Type*} {c : ComplexShape ι} {F G K : HomologicalComplex C c} (φ : F ⟶ G)
+
+/-- A morphism of homological complexes `φ : F ⟶ G` has a homotopy cofiber if for all
+indices `i` and `j` such that `c.Rel i j`, the binary biproduct `F.X j ⊞ G.X i` exists. -/
+class HasHomotopyCofiber (φ : F ⟶ G) : Prop where
+  hasBinaryBiproduct (i j : ι) (hij : c.Rel i j) : HasBinaryBiproduct (F.X j) (G.X i)
+
+instance [HasBinaryBiproducts C] : HasHomotopyCofiber φ where
+  hasBinaryBiproduct _ _ _ := inferInstance
+
+variable [HasHomotopyCofiber φ] [DecidableRel c.Rel]
+
+namespace homotopyCofiber
+
+/-- The `X` field of the homological complex `homotopyCofiber φ`. -/
+noncomputable def X (i : ι) : C :=
+  if hi : c.Rel i (c.next i)
+  then
+    haveI := HasHomotopyCofiber.hasBinaryBiproduct φ _ _ hi
+    (F.X (c.next i)) ⊞ (G.X i)
+  else G.X i
+
+/-- The canonical isomorphism `(homotopyCofiber φ).X i ≅ F.X j ⊞ G.X i` when `c.Rel i j`. -/
+noncomputable def XIsoBiprod (i j : ι) (hij : c.Rel i j) [HasBinaryBiproduct (F.X j) (G.X i)] :
+    X φ i ≅ F.X j ⊞ G.X i :=
+  eqToIso (by
+    obtain rfl := c.next_eq' hij
+    apply dif_pos hij)
+
+/-- The canonical isomorphism `(homotopyCofiber φ).X i ≅ G.X i` when `¬ c.Rel i (c.next i)`. -/
+noncomputable def XIso (i : ι) (hi : ¬ c.Rel i (c.next i)) :
+    X φ i ≅ G.X i :=
+  eqToIso (dif_neg hi)
+
+/-- The second projection `(homotopyCofiber φ).X i ⟶ G.X i`. -/
+noncomputable def sndX (i : ι) : X φ i ⟶ G.X i :=
+  if hi : c.Rel i (c.next i)
+  then
+    haveI := HasHomotopyCofiber.hasBinaryBiproduct φ _ _ hi
+    (XIsoBiprod φ _ _ hi).hom ≫ biprod.snd
+  else
+    (XIso φ i hi).hom
+
+/-- The right inclusion `G.X i ⟶ (homotopyCofiber φ).X i`. -/
+noncomputable def inrX (i : ι) : G.X i ⟶ X φ i :=
+  if hi : c.Rel i (c.next i)
+  then
+    haveI := HasHomotopyCofiber.hasBinaryBiproduct φ _ _ hi
+    biprod.inr ≫ (XIsoBiprod φ _ _ hi).inv
+  else
+    (XIso φ i hi).inv
+
+@[reassoc (attr := simp)]
+lemma inrX_sndX (i : ι) : inrX φ i ≫ sndX φ i = 𝟙 _ := by
+  dsimp [sndX, inrX]
+  split_ifs with hi <;> simp
+
+@[reassoc]
+lemma sndX_inrX (i : ι) (hi : ¬ c.Rel i (c.next i)) :
+    sndX φ i ≫ inrX φ i = 𝟙 _ := by
+  dsimp [sndX, inrX]
+  simp only [dif_neg hi, Iso.hom_inv_id]
+
+/-- The first projection `(homotopyCofiber φ).X i ⟶ F.X j` when `c.Rel i j`. -/
+noncomputable def fstX (i j : ι) (hij : c.Rel i j) : X φ i ⟶ F.X j :=
+  haveI := HasHomotopyCofiber.hasBinaryBiproduct φ _ _ hij
+  (XIsoBiprod φ i j hij).hom ≫ biprod.fst
+
+/-- The left inclusion `F.X i ⟶ (homotopyCofiber φ).X j` when `c.Rel j i`. -/
+noncomputable def inlX (i j : ι) (hij : c.Rel j i) : F.X i ⟶ X φ j :=
+  haveI := HasHomotopyCofiber.hasBinaryBiproduct φ _ _ hij
+  biprod.inl ≫ (XIsoBiprod φ j i hij).inv
+
+@[reassoc (attr := simp)]
+lemma inlX_fstX (i j : ι ) (hij : c.Rel j i) :
+    inlX φ i j hij ≫ fstX φ j i hij = 𝟙 _ := by
+  simp [inlX, fstX]
+
+@[reassoc (attr := simp)]
+lemma inlX_sndX (i j : ι) (hij : c.Rel j i) :
+    inlX φ i j hij ≫ sndX φ j = 0 := by
+  obtain rfl := c.next_eq' hij
+  simp [inlX, sndX, dif_pos hij]
+
+@[reassoc (attr := simp)]
+lemma inrX_fstX (i j : ι) (hij : c.Rel i j) :
+    inrX φ i ≫ fstX φ i j hij = 0 := by
+  obtain rfl := c.next_eq' hij
+  simp [inrX, fstX, dif_pos hij]
+
+/-- The `d` field of the homological complex `homotopyCofiber φ`. -/
+noncomputable def d (i j : ι) : X φ i ⟶ X φ j :=
+  if hij : c.Rel i j
+  then
+    (if hj : c.Rel j (c.next j) then -fstX φ i j hij ≫ F.d _ _ ≫ inlX φ _ _ hj else 0) +
+      fstX φ i j hij ≫ φ.f j ≫ inrX φ j + sndX φ i ≫ G.d i j ≫ inrX φ j
+  else
+    0
+
+lemma ext_to_X (i j : ι) (hij : c.Rel i j) {A : C} {f g : A ⟶ X φ i}
+    (h₁ : f ≫ fstX φ i j hij = g ≫ fstX φ i j hij) (h₂ : f ≫ sndX φ i = g ≫ sndX φ i) :
+    f = g := by
+  haveI := HasHomotopyCofiber.hasBinaryBiproduct φ _ _ hij
+  rw [← cancel_mono (XIsoBiprod φ i j hij).hom]
+  apply biprod.hom_ext
+  · simpa using h₁
+  · obtain rfl := c.next_eq' hij
+    simpa [sndX, dif_pos hij] using h₂
+
+lemma ext_to_X' (i : ι) (hi : ¬ c.Rel i (c.next i)) {A : C} {f g : A ⟶ X φ i}
+    (h : f ≫ sndX φ i = g ≫ sndX φ i) : f = g := by
+  rw [← cancel_mono (XIso φ i hi).hom]
+  simpa only [sndX, dif_neg hi] using h
+
+lemma ext_from_X (i j : ι) (hij : c.Rel j i) {A : C} {f g : X φ j ⟶ A}
+    (h₁ : inlX φ i j hij ≫ f = inlX φ i j hij ≫ g) (h₂ : inrX φ j ≫ f = inrX φ j ≫ g) :
+    f = g := by
+  haveI := HasHomotopyCofiber.hasBinaryBiproduct φ _ _ hij
+  rw [← cancel_epi (XIsoBiprod φ j i hij).inv]
+  apply biprod.hom_ext'
+  · simpa [inlX] using h₁
+  · obtain rfl := c.next_eq' hij
+    simpa [inrX, dif_pos hij] using h₂
+
+lemma ext_from_X' (i : ι) (hi : ¬ c.Rel i (c.next i)) {A : C} {f g : X φ i ⟶ A}
+    (h : inrX φ i ≫ f = inrX φ i ≫ g) : f = g := by
+  rw [← cancel_epi (XIso φ i hi).inv]
+  simpa only [inrX, dif_neg hi] using h
+
+@[reassoc]
+lemma d_fstX (i j k : ι) (hij : c.Rel i j) (hjk : c.Rel j k) :
+    d φ i j ≫ fstX φ j k hjk = -fstX φ i j hij ≫ F.d j k := by
+  obtain rfl := c.next_eq' hjk
+  simp [d, dif_pos hij, dif_pos hjk]
+
+@[reassoc]
+lemma d_sndX (i j : ι) (hij : c.Rel i j) :
+    d φ i j ≫ sndX φ j = fstX φ i j hij ≫ φ.f j + sndX φ i ≫ G.d i j := by
+  dsimp [d]
+  split_ifs with hij <;> simp
+
+@[reassoc]
+lemma inlX_d (i j k : ι) (hij : c.Rel i j) (hjk : c.Rel j k) :
+    inlX φ j i hij ≫ d φ i j = -F.d j k ≫ inlX φ k j hjk + φ.f j ≫ inrX φ j := by
+  apply ext_to_X φ j k hjk
+  · dsimp
+    simp [d_fstX φ  _ _ _ hij hjk]
+  · simp [d_sndX φ _ _ hij]
+
+@[reassoc]
+lemma inlX_d' (i j : ι) (hij : c.Rel i j) (hj : ¬ c.Rel j (c.next j)):
+    inlX φ j i hij ≫ d φ i j = φ.f j ≫ inrX φ j := by
+  apply ext_to_X' _ _ hj
+  simp [d_sndX φ i j hij]
+
+lemma shape (i j : ι) (hij : ¬ c.Rel i j) :
+    d φ i j = 0 :=
+  dif_neg hij
+
+@[reassoc (attr := simp)]
+lemma inrX_d (i j : ι) :
+    inrX φ i ≫ d φ i j = G.d i j ≫ inrX φ j := by
+  by_cases hij : c.Rel i j
+  · by_cases hj : c.Rel j (c.next j)
+    · apply ext_to_X _ _ _ hj
+      · simp [d_fstX φ _ _ _ hij]
+      · simp [d_sndX φ _ _ hij]
+    · apply ext_to_X' _ _ hj
+      simp [d_sndX φ _ _ hij]
+  · rw [shape φ _ _ hij, G.shape _ _ hij, zero_comp, comp_zero]
+
+end homotopyCofiber
+
+/-- The homotopy cofiber of a morphism of homological complexes,
+also known as the mapping cone. -/
+@[simps]
+noncomputable def homotopyCofiber : HomologicalComplex C c where
+  X i := homotopyCofiber.X φ i
+  d i j := homotopyCofiber.d φ i j
+  shape i j hij := homotopyCofiber.shape φ i j hij
+  d_comp_d' i j k hij hjk := by
+    apply homotopyCofiber.ext_from_X φ j i hij
+    · dsimp
+      simp only [comp_zero, homotopyCofiber.inlX_d_assoc φ i j k hij hjk,
+        add_comp, assoc, homotopyCofiber.inrX_d, Hom.comm_assoc, neg_comp]
+      by_cases hk : c.Rel k (c.next k)
+      · simp [homotopyCofiber.inlX_d φ j k _ hjk hk]
+      · simp [homotopyCofiber.inlX_d' φ j k hjk hk]
+    · simp
+
+end HomologicalComplex