File: Mathlib/AlgebraicTopology/DoldKan/HomotopyEquivalence.lean

full_name: AlgebraicTopology.DoldKan.homotopyPToId
code: /-- Inductive construction of homotopies from `P q` to `𝟙 _` -/
noncomputable def homotopyPToId : ∀ q : ℕ, Homotopy (P q : K[X] ⟶ _) (𝟙 _)
  
goal: Homotopy (AlgebraicTopology.DoldKan.P q) (𝟙 K[X])

full_name: AlgebraicTopology.DoldKan.homotopyQToZero
code: /-- The complement projection `Q q` to `P q` is homotopic to zero. -/
def homotopyQToZero (q : ℕ) : Homotopy (Q q : K[X] ⟶ _) 0 
goal: Homotopy (AlgebraicTopology.DoldKan.Q q) 0

full_name: AlgebraicTopology.DoldKan.homotopyPToId_eventually_constant
code: theorem homotopyPToId_eventually_constant {q n : ℕ} (hqn : n < q) :
    ((homotopyPToId X (q + 1)).hom n (n + 1) : X _[n] ⟶ X _[n + 1]) =
      (homotopyPToId X q).hom n (n + 1) 
goal: (AlgebraicTopology.DoldKan.homotopyPToId X (q + 1)).hom n (n + 1) =
  (AlgebraicTopology.DoldKan.homotopyPToId X q).hom n (n + 1)

full_name: AlgebraicTopology.DoldKan.homotopyPInftyToId
code: /-- Construction of the homotopy from `PInfty` to the identity using eventually
(termwise) constant homotopies from `P q` to the identity for all `q` -/
@[simps]
def homotopyPInftyToId : Homotopy (PInfty : K[X] ⟶ _) (𝟙 _) 
goal: Homotopy AlgebraicTopology.DoldKan.PInfty (𝟙 K[X])

full_name: AlgebraicTopology.DoldKan.homotopyPInftyToId_hom
code: /-- Construction of the homotopy from `PInfty` to the identity using eventually
(termwise) constant homotopies from `P q` to the identity for all `q` -/
@[simps]
def homotopyPInftyToId : Homotopy (PInfty : K[X] ⟶ _) (𝟙 _) 
goal: (AlgebraicTopology.DoldKan.homotopyPInftyToId X).hom i j = (AlgebraicTopology.DoldKan.homotopyPToId X (j + 1)).hom i j

full_name: AlgebraicTopology.DoldKan.homotopyEquivNormalizedMooreComplexAlternatingFaceMapComplex_homotopyHomInvId
code: /-- The inclusion of the Moore complex in the alternating face map complex
is a homotopy equivalence -/
@[simps]
def homotopyEquivNormalizedMooreComplexAlternatingFaceMapComplex {A : Type*} [Category A]
    [Abelian A] {Y : SimplicialObject A} :
    HomotopyEquiv ((normalizedMooreComplex A).obj Y) ((alternatingFaceMapComplex A).obj Y) 
goal: AlgebraicTopology.DoldKan.homotopyEquivNormalizedMooreComplexAlternatingFaceMapComplex.homotopyHomInvId =
  Homotopy.ofEq ⋯

full_name: AlgebraicTopology.DoldKan.homotopyEquivNormalizedMooreComplexAlternatingFaceMapComplex_inv
code: /-- The inclusion of the Moore complex in the alternating face map complex
is a homotopy equivalence -/
@[simps]
def homotopyEquivNormalizedMooreComplexAlternatingFaceMapComplex {A : Type*} [Category A]
    [Abelian A] {Y : SimplicialObject A} :
    HomotopyEquiv ((normalizedMooreComplex A).obj Y) ((alternatingFaceMapComplex A).obj Y) 
goal: AlgebraicTopology.DoldKan.homotopyEquivNormalizedMooreComplexAlternatingFaceMapComplex.inv =
  AlgebraicTopology.DoldKan.PInftyToNormalizedMooreComplex Y

full_name: AlgebraicTopology.DoldKan.homotopyEquivNormalizedMooreComplexAlternatingFaceMapComplex
code: /-- The inclusion of the Moore complex in the alternating face map complex
is a homotopy equivalence -/
@[simps]
def homotopyEquivNormalizedMooreComplexAlternatingFaceMapComplex {A : Type*} [Category A]
    [Abelian A] {Y : SimplicialObject A} :
    HomotopyEquiv ((normalizedMooreComplex A).obj Y) ((alternatingFaceMapComplex A).obj Y) 
goal: HomotopyEquiv ((AlgebraicTopology.normalizedMooreComplex A).obj Y)
  ((AlgebraicTopology.alternatingFaceMapComplex A).obj Y)

full_name: AlgebraicTopology.DoldKan.homotopyEquivNormalizedMooreComplexAlternatingFaceMapComplex_hom
code: /-- The inclusion of the Moore complex in the alternating face map complex
is a homotopy equivalence -/
@[simps]
def homotopyEquivNormalizedMooreComplexAlternatingFaceMapComplex {A : Type*} [Category A]
    [Abelian A] {Y : SimplicialObject A} :
    HomotopyEquiv ((normalizedMooreComplex A).obj Y) ((alternatingFaceMapComplex A).obj Y) 
goal: AlgebraicTopology.DoldKan.homotopyEquivNormalizedMooreComplexAlternatingFaceMapComplex.hom =
  AlgebraicTopology.inclusionOfMooreComplexMap Y

full_name: AlgebraicTopology.DoldKan.homotopyEquivNormalizedMooreComplexAlternatingFaceMapComplex_homotopyInvHomId
code: /-- The inclusion of the Moore complex in the alternating face map complex
is a homotopy equivalence -/
@[simps]
def homotopyEquivNormalizedMooreComplexAlternatingFaceMapComplex {A : Type*} [Category A]
    [Abelian A] {Y : SimplicialObject A} :
    HomotopyEquiv ((normalizedMooreComplex A).obj Y) ((alternatingFaceMapComplex A).obj Y) 
goal: AlgebraicTopology.DoldKan.homotopyEquivNormalizedMooreComplexAlternatingFaceMapComplex.homotopyInvHomId =
  (Homotopy.ofEq ⋯).trans (AlgebraicTopology.DoldKan.homotopyPInftyToId Y)