File: Mathlib/AlgebraicTopology/DoldKan/Normalized.lean

full_name: AlgebraicTopology.DoldKan.HigherFacesVanish.inclusionOfMooreComplexMap
code: theorem HigherFacesVanish.inclusionOfMooreComplexMap (n : ℕ) :
    HigherFacesVanish (n + 1) ((inclusionOfMooreComplexMap X).f (n + 1)) 
goal: AlgebraicTopology.DoldKan.HigherFacesVanish (n + 1) ((AlgebraicTopology.inclusionOfMooreComplexMap X).f (n + 1))

full_name: AlgebraicTopology.DoldKan.factors_normalizedMooreComplex_PInfty
code: theorem factors_normalizedMooreComplex_PInfty (n : ℕ) :
    Subobject.Factors (NormalizedMooreComplex.objX X n) (PInfty.f n) 
goal: (AlgebraicTopology.NormalizedMooreComplex.objX X n).Factors (AlgebraicTopology.DoldKan.PInfty.f n)

full_name: AlgebraicTopology.DoldKan.PInftyToNormalizedMooreComplex
code: /-- `PInfty` factors through the normalized Moore complex -/
@[simps!]
def PInftyToNormalizedMooreComplex (X : SimplicialObject A) : K[X] ⟶ N[X] 
goal: K[X] ⟶ N[X]

full_name: AlgebraicTopology.DoldKan.PInftyToNormalizedMooreComplex_f
code: /-- `PInfty` factors through the normalized Moore complex -/
@[simps!]
def PInftyToNormalizedMooreComplex (X : SimplicialObject A) : K[X] ⟶ N[X] 
goal: (AlgebraicTopology.DoldKan.PInftyToNormalizedMooreComplex X).f n =
  (AlgebraicTopology.NormalizedMooreComplex.objX X n).factorThru (AlgebraicTopology.DoldKan.PInfty.f n) ⋯

full_name: AlgebraicTopology.DoldKan.PInftyToNormalizedMooreComplex_comp_inclusionOfMooreComplexMap
code: @[reassoc (attr := simp)]
theorem PInftyToNormalizedMooreComplex_comp_inclusionOfMooreComplexMap (X : SimplicialObject A) :
    PInftyToNormalizedMooreComplex X ≫ inclusionOfMooreComplexMap X = PInfty 
goal: AlgebraicTopology.DoldKan.PInftyToNormalizedMooreComplex X ≫ AlgebraicTopology.inclusionOfMooreComplexMap X =
  AlgebraicTopology.DoldKan.PInfty

full_name: AlgebraicTopology.DoldKan.PInftyToNormalizedMooreComplex_naturality
code: @[reassoc (attr := simp)]
theorem PInftyToNormalizedMooreComplex_naturality {X Y : SimplicialObject A} (f : X ⟶ Y) :
    AlternatingFaceMapComplex.map f ≫ PInftyToNormalizedMooreComplex Y =
      PInftyToNormalizedMooreComplex X ≫ NormalizedMooreComplex.map f 
goal: AlgebraicTopology.AlternatingFaceMapComplex.map f ≫ AlgebraicTopology.DoldKan.PInftyToNormalizedMooreComplex Y =
  AlgebraicTopology.DoldKan.PInftyToNormalizedMooreComplex X ≫ AlgebraicTopology.NormalizedMooreComplex.map f

full_name: AlgebraicTopology.DoldKan.PInfty_comp_PInftyToNormalizedMooreComplex
code: @[reassoc (attr := simp)]
theorem PInfty_comp_PInftyToNormalizedMooreComplex (X : SimplicialObject A) :
    PInfty ≫ PInftyToNormalizedMooreComplex X = PInftyToNormalizedMooreComplex X 
goal: AlgebraicTopology.DoldKan.PInfty ≫ AlgebraicTopology.DoldKan.PInftyToNormalizedMooreComplex X =
  AlgebraicTopology.DoldKan.PInftyToNormalizedMooreComplex X

full_name: AlgebraicTopology.DoldKan.inclusionOfMooreComplexMap_comp_PInfty
code: @[reassoc (attr := simp)]
theorem inclusionOfMooreComplexMap_comp_PInfty (X : SimplicialObject A) :
    inclusionOfMooreComplexMap X ≫ PInfty = inclusionOfMooreComplexMap X 
goal: AlgebraicTopology.inclusionOfMooreComplexMap X ≫ AlgebraicTopology.DoldKan.PInfty =
  AlgebraicTopology.inclusionOfMooreComplexMap X

full_name: AlgebraicTopology.DoldKan.instMonoChainComplexNatInclusionOfMooreComplexMap
code: instance : Mono (inclusionOfMooreComplexMap X) 
goal: CategoryTheory.Mono (AlgebraicTopology.inclusionOfMooreComplexMap X)

full_name: AlgebraicTopology.DoldKan.splitMonoInclusionOfMooreComplexMap
code: /-- `inclusionOfMooreComplexMap X` is a split mono. -/
def splitMonoInclusionOfMooreComplexMap (X : SimplicialObject A) :
    SplitMono (inclusionOfMooreComplexMap X) 
goal: CategoryTheory.SplitMono (AlgebraicTopology.inclusionOfMooreComplexMap X)

full_name: AlgebraicTopology.DoldKan.N₁_iso_normalizedMooreComplex_comp_toKaroubi
code: /-- When the category `A` is abelian,
the functor `N₁ : SimplicialObject A ⥤ Karoubi (ChainComplex A ℕ)` defined
using `PInfty` identifies to the composition of the normalized Moore complex functor
and the inclusion in the Karoubi envelope. -/
def N₁_iso_normalizedMooreComplex_comp_toKaroubi : N₁ ≅ normalizedMooreComplex A ⋙ toKaroubi _ 
goal: AlgebraicTopology.DoldKan.N₁ ≅
  AlgebraicTopology.normalizedMooreComplex A ⋙ CategoryTheory.Idempotents.toKaroubi (ChainComplex A ℕ)