Behaviour of P_infty with respect to degeneracies #
For any X : SimplicialObject C where C is an abelian category,
the projector PInfty : K[X] ⟶ K[X] is supposed to be the projection
on the normalized subcomplex, parallel to the degenerate subcomplex, i.e.
the subcomplex generated by the images of all X.σ i.
In this file, we obtain degeneracy_comp_P_infty which states that
if X : SimplicialObject C with C a preadditive category,
θ : [n] ⟶ Δ' is a non injective map in SimplexCategory, then
X.map θ.op ≫ P_infty.f n = 0. It follows from the more precise
statement vanishing statement σ_comp_P_eq_zero for the P q.
(See Equivalence.lean for the general strategy of proof of the Dold-Kan equivalence.)
theorem
AlgebraicTopology.DoldKan.HigherFacesVanish.comp_σ
{C : Type u_1}
[CategoryTheory.Category.{u_2, u_1} C]
[CategoryTheory.Preadditive C]
{Y : C}
{X : CategoryTheory.SimplicialObject C}
{n : ℕ}
{b : ℕ}
{q : ℕ}
{φ : Y ⟶ X.obj (Opposite.op (SimplexCategory.mk (n + 1)))}
(v : AlgebraicTopology.DoldKan.HigherFacesVanish q φ)
(hnbq : n + 1 = b + q)
:
AlgebraicTopology.DoldKan.HigherFacesVanish q
(CategoryTheory.CategoryStruct.comp φ (CategoryTheory.SimplicialObject.σ X { val := b, isLt := ⋯ }))
theorem
AlgebraicTopology.DoldKan.σ_comp_P_eq_zero
{C : Type u_1}
[CategoryTheory.Category.{u_2, u_1} C]
[CategoryTheory.Preadditive C]
(X : CategoryTheory.SimplicialObject C)
{n : ℕ}
{q : ℕ}
(i : Fin (n + 1))
(hi : n + 1 ≤ ↑i + q)
:
CategoryTheory.CategoryStruct.comp (CategoryTheory.SimplicialObject.σ X i) ((AlgebraicTopology.DoldKan.P q).f (n + 1)) = 0
@[simp]
theorem
AlgebraicTopology.DoldKan.σ_comp_PInfty_assoc
{C : Type u_1}
[CategoryTheory.Category.{u_2, u_1} C]
[CategoryTheory.Preadditive C]
(X : CategoryTheory.SimplicialObject C)
{n : ℕ}
(i : Fin (n + 1))
{Z : C}
(h : (AlgebraicTopology.AlternatingFaceMapComplex.obj X).X (n + 1) ⟶ Z)
:
CategoryTheory.CategoryStruct.comp (CategoryTheory.SimplicialObject.σ X i)
(CategoryTheory.CategoryStruct.comp (AlgebraicTopology.DoldKan.PInfty.f (n + 1)) h) = CategoryTheory.CategoryStruct.comp 0 h
@[simp]
theorem
AlgebraicTopology.DoldKan.σ_comp_PInfty
{C : Type u_1}
[CategoryTheory.Category.{u_2, u_1} C]
[CategoryTheory.Preadditive C]
(X : CategoryTheory.SimplicialObject C)
{n : ℕ}
(i : Fin (n + 1))
:
CategoryTheory.CategoryStruct.comp (CategoryTheory.SimplicialObject.σ X i)
(AlgebraicTopology.DoldKan.PInfty.f (n + 1)) = 0
theorem
AlgebraicTopology.DoldKan.degeneracy_comp_PInfty_assoc
{C : Type u_1}
[CategoryTheory.Category.{u_2, u_1} C]
[CategoryTheory.Preadditive C]
(X : CategoryTheory.SimplicialObject C)
(n : ℕ)
{Δ' : SimplexCategory}
(θ : SimplexCategory.mk n ⟶ Δ')
(hθ : ¬CategoryTheory.Mono θ)
{Z : C}
(h : (AlgebraicTopology.AlternatingFaceMapComplex.obj X).X n ⟶ Z)
:
CategoryTheory.CategoryStruct.comp (X.map θ.op)
(CategoryTheory.CategoryStruct.comp (AlgebraicTopology.DoldKan.PInfty.f n) h) = CategoryTheory.CategoryStruct.comp 0 h
theorem
AlgebraicTopology.DoldKan.degeneracy_comp_PInfty
{C : Type u_1}
[CategoryTheory.Category.{u_2, u_1} C]
[CategoryTheory.Preadditive C]
(X : CategoryTheory.SimplicialObject C)
(n : ℕ)
{Δ' : SimplexCategory}
(θ : SimplexCategory.mk n ⟶ Δ')
(hθ : ¬CategoryTheory.Mono θ)
:
CategoryTheory.CategoryStruct.comp (X.map θ.op) (AlgebraicTopology.DoldKan.PInfty.f n) = 0