File: Mathlib/AlgebraicTopology/AlternatingFaceMapComplex.lean

full_name: AlgebraicTopology.AlternatingFaceMapComplex.objD
code: /-- The differential on the alternating face map complex is the alternate
sum of the face maps -/
@[simp]
def objD (n : ℕ) : X _[n + 1] ⟶ X _[n] 
goal: X _[n + 1] ⟶ X _[n]

full_name: AlgebraicTopology.AlternatingFaceMapComplex.d_squared
code: /-- ## The chain complex relation `d ≫ d`
-/
theorem d_squared (n : ℕ) : objD X (n + 1) ≫ objD X n = 0 
goal: AlgebraicTopology.AlternatingFaceMapComplex.objD X (n + 1) ≫ AlgebraicTopology.AlternatingFaceMapComplex.objD X n = 0

full_name: AlgebraicTopology.AlternatingFaceMapComplex.obj
code: /-- The alternating face map complex, on objects -/
def obj : ChainComplex C ℕ 
goal: ChainComplex C ℕ

full_name: AlgebraicTopology.AlternatingFaceMapComplex.obj_X
code: @[simp]
theorem obj_X (X : SimplicialObject C) (n : ℕ) : (AlternatingFaceMapComplex.obj X).X n = X _[n] 
goal: (AlgebraicTopology.AlternatingFaceMapComplex.obj X).X n = X _[n]

full_name: AlgebraicTopology.AlternatingFaceMapComplex.obj_d_eq
code: @[simp]
theorem obj_d_eq (X : SimplicialObject C) (n : ℕ) :
    (AlternatingFaceMapComplex.obj X).d (n + 1) n
      = ∑ i : Fin (n + 2), (-1 : ℤ) ^ (i : ℕ) • X.δ i 
goal: (AlgebraicTopology.AlternatingFaceMapComplex.obj X).d (n + 1) n = ∑ i : Fin (n + 2), (-1) ^ ↑i • X.δ i

full_name: AlgebraicTopology.AlternatingFaceMapComplex.map
code: /-- The alternating face map complex, on morphisms -/
def map (f : X ⟶ Y) : obj X ⟶ obj Y 
goal: AlgebraicTopology.AlternatingFaceMapComplex.obj X ⟶ AlgebraicTopology.AlternatingFaceMapComplex.obj Y

full_name: AlgebraicTopology.AlternatingFaceMapComplex.map_f
code: @[simp]
theorem map_f (f : X ⟶ Y) (n : ℕ) : (map f).f n = f.app (op [n]) 
goal: (AlgebraicTopology.AlternatingFaceMapComplex.map f).f n = f.app (Opposite.op [n])

full_name: AlgebraicTopology.alternatingFaceMapComplex
code: /-- The alternating face map complex, as a functor -/
def alternatingFaceMapComplex : SimplicialObject C ⥤ ChainComplex C ℕ 
goal: CategoryTheory.SimplicialObject C ⥤ ChainComplex C ℕ

full_name: AlgebraicTopology.alternatingFaceMapComplex_obj_X
code: @[simp]
theorem alternatingFaceMapComplex_obj_X (X : SimplicialObject C) (n : ℕ) :
    ((alternatingFaceMapComplex C).obj X).X n = X _[n] 
goal: ((AlgebraicTopology.alternatingFaceMapComplex C).obj X).X n = X _[n]

full_name: AlgebraicTopology.alternatingFaceMapComplex_obj_d
code: @[simp]
theorem alternatingFaceMapComplex_obj_d (X : SimplicialObject C) (n : ℕ) :
    ((alternatingFaceMapComplex C).obj X).d (n + 1) n = AlternatingFaceMapComplex.objD X n 
goal: ((AlgebraicTopology.alternatingFaceMapComplex C).obj X).d (n + 1) n =
  AlgebraicTopology.AlternatingFaceMapComplex.objD X n

full_name: AlgebraicTopology.alternatingFaceMapComplex_map_f
code: @[simp]
theorem alternatingFaceMapComplex_map_f {X Y : SimplicialObject C} (f : X ⟶ Y) (n : ℕ) :
    ((alternatingFaceMapComplex C).map f).f n = f.app (op [n]) 
goal: ((AlgebraicTopology.alternatingFaceMapComplex C).map f).f n = f.app (Opposite.op [n])

full_name: AlgebraicTopology.map_alternatingFaceMapComplex
code: theorem map_alternatingFaceMapComplex {D : Type*} [Category D] [Preadditive D] (F : C ⥤ D)
    [F.Additive] :
    alternatingFaceMapComplex C ⋙ F.mapHomologicalComplex _ =
      (SimplicialObject.whiskering C D).obj F ⋙ alternatingFaceMapComplex D 
goal: AlgebraicTopology.alternatingFaceMapComplex C ⋙ F.mapHomologicalComplex (ComplexShape.down ℕ) =
  (CategoryTheory.SimplicialObject.whiskering C D).obj F ⋙ AlgebraicTopology.alternatingFaceMapComplex D

full_name: AlgebraicTopology.karoubi_alternatingFaceMapComplex_d
code: theorem karoubi_alternatingFaceMapComplex_d (P : Karoubi (SimplicialObject C)) (n : ℕ) :
    ((AlternatingFaceMapComplex.obj (KaroubiFunctorCategoryEmbedding.obj P)).d (n + 1) n).f =
      P.p.app (op [n + 1]) ≫ (AlternatingFaceMapComplex.obj P.X).d (n + 1) n 
goal: ((AlgebraicTopology.AlternatingFaceMapComplex.obj (CategoryTheory.Idempotents.KaroubiFunctorCategoryEmbedding.obj P)).d
      (n + 1) n).f =
  P.p.app (Opposite.op [n + 1]) ≫ (AlgebraicTopology.AlternatingFaceMapComplex.obj P.X).d (n + 1) n

full_name: AlgebraicTopology.AlternatingFaceMapComplex.ε
code: /-- The natural transformation which gives the augmentation of the alternating face map
complex attached to an augmented simplicial object. -/
def ε [Limits.HasZeroObject C] :
    SimplicialObject.Augmented.drop ⋙ AlgebraicTopology.alternatingFaceMapComplex C ⟶
      SimplicialObject.Augmented.point ⋙ ChainComplex.single₀ C 
goal: CategoryTheory.SimplicialObject.Augmented.drop ⋙ AlgebraicTopology.alternatingFaceMapComplex C ⟶
  CategoryTheory.SimplicialObject.Augmented.point ⋙ ChainComplex.single₀ C

full_name: AlgebraicTopology.AlternatingFaceMapComplex.ε_app_f_zero
code: @[simp]
lemma ε_app_f_zero [Limits.HasZeroObject C] (X : SimplicialObject.Augmented C) :
    (ε.app X).f 0 = X.hom.app (op [0]) 
goal: (AlgebraicTopology.AlternatingFaceMapComplex.ε.app X).f 0 = X.hom.app (Opposite.op [0])

full_name: AlgebraicTopology.AlternatingFaceMapComplex.ε_app_f_succ
code: @[simp]
lemma ε_app_f_succ [Limits.HasZeroObject C] (X : SimplicialObject.Augmented C) (n : ℕ) :
    (ε.app X).f (n + 1) = 0 
goal: (AlgebraicTopology.AlternatingFaceMapComplex.ε.app X).f (n + 1) = 0

full_name: AlgebraicTopology.inclusionOfMooreComplexMap
code: /-- The inclusion map of the Moore complex in the alternating face map complex -/
def inclusionOfMooreComplexMap (X : SimplicialObject A) :
    (normalizedMooreComplex A).obj X ⟶ (alternatingFaceMapComplex A).obj X 
goal: (AlgebraicTopology.normalizedMooreComplex A).obj X ⟶ (AlgebraicTopology.alternatingFaceMapComplex A).obj X

full_name: AlgebraicTopology.inclusionOfMooreComplexMap_f
code: @[simp]
theorem inclusionOfMooreComplexMap_f (X : SimplicialObject A) (n : ℕ) :
    (inclusionOfMooreComplexMap X).f n = (NormalizedMooreComplex.objX X n).arrow 
goal: (AlgebraicTopology.inclusionOfMooreComplexMap X).f n = (AlgebraicTopology.NormalizedMooreComplex.objX X n).arrow

full_name: AlgebraicTopology.inclusionOfMooreComplex_app
code: /-- The inclusion map of the Moore complex in the alternating face map complex,
as a natural transformation -/
@[simps]
def inclusionOfMooreComplex : normalizedMooreComplex A ⟶ alternatingFaceMapComplex A 
goal: (AlgebraicTopology.inclusionOfMooreComplex A).app X = AlgebraicTopology.inclusionOfMooreComplexMap X

full_name: AlgebraicTopology.inclusionOfMooreComplex
code: /-- The inclusion map of the Moore complex in the alternating face map complex,
as a natural transformation -/
@[simps]
def inclusionOfMooreComplex : normalizedMooreComplex A ⟶ alternatingFaceMapComplex A 
goal: AlgebraicTopology.normalizedMooreComplex A ⟶ AlgebraicTopology.alternatingFaceMapComplex A

full_name: AlgebraicTopology.AlternatingCofaceMapComplex.objD
code: /-- The differential on the alternating coface map complex is the alternate
sum of the coface maps -/
@[simp]
def objD (n : ℕ) : X.obj [n] ⟶ X.obj [n + 1] 
goal: X.obj [n] ⟶ X.obj [n + 1]

full_name: AlgebraicTopology.AlternatingCofaceMapComplex.d_eq_unop_d
code: theorem d_eq_unop_d (n : ℕ) :
    objD X n =
      (AlternatingFaceMapComplex.objD ((cosimplicialSimplicialEquiv C).functor.obj (op X))
          n).unop 
goal: AlgebraicTopology.AlternatingCofaceMapComplex.objD X n =
  (AlgebraicTopology.AlternatingFaceMapComplex.objD
      ((CategoryTheory.cosimplicialSimplicialEquiv C).functor.obj (Opposite.op X)) n).unop

full_name: AlgebraicTopology.AlternatingCofaceMapComplex.d_squared
code: theorem d_squared (n : ℕ) : objD X n ≫ objD X (n + 1) = 0 
goal: AlgebraicTopology.AlternatingCofaceMapComplex.objD X n ≫ AlgebraicTopology.AlternatingCofaceMapComplex.objD X (n + 1) =
  0

full_name: AlgebraicTopology.AlternatingCofaceMapComplex.obj
code: /-- The alternating coface map complex, on objects -/
def obj : CochainComplex C ℕ 
goal: CochainComplex C ℕ

full_name: AlgebraicTopology.AlternatingCofaceMapComplex.map
code: /-- The alternating face map complex, on morphisms -/
@[simp]
def map (f : X ⟶ Y) : obj X ⟶ obj Y 
goal: AlgebraicTopology.AlternatingCofaceMapComplex.obj X ⟶ AlgebraicTopology.AlternatingCofaceMapComplex.obj Y

full_name: AlgebraicTopology.alternatingCofaceMapComplex_obj
code: /-- The alternating coface map complex, as a functor -/
@[simps]
def alternatingCofaceMapComplex : CosimplicialObject C ⥤ CochainComplex C ℕ 
goal: (AlgebraicTopology.alternatingCofaceMapComplex C).obj X = AlgebraicTopology.AlternatingCofaceMapComplex.obj X

full_name: AlgebraicTopology.alternatingCofaceMapComplex_map
code: /-- The alternating coface map complex, as a functor -/
@[simps]
def alternatingCofaceMapComplex : CosimplicialObject C ⥤ CochainComplex C ℕ 
goal: ∀ {X Y : CategoryTheory.CosimplicialObject C} (f : X ⟶ Y),
  (AlgebraicTopology.alternatingCofaceMapComplex C).map f = AlgebraicTopology.AlternatingCofaceMapComplex.map f

full_name: AlgebraicTopology.alternatingCofaceMapComplex
code: /-- The alternating coface map complex, as a functor -/
@[simps]
def alternatingCofaceMapComplex : CosimplicialObject C ⥤ CochainComplex C ℕ 
goal: CategoryTheory.CosimplicialObject C ⥤ CochainComplex C ℕ