Behaviour of P_infty with respect to degeneracies #

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For any X : simplicial_object C where C is an abelian category, the projector P_infty : 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 : simplicial_object C with C a preadditive category, θ : [n] ⟶ Δ' is a non injective map in simplex_category, 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.)

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theorem algebraic_topology.dold_kan.higher_faces_vanish.comp_σ {C : Type u_1} [category_theory.category C] [category_theory.preadditive C] {Y : C} {X : category_theory.simplicial_object C} {n b q : ℕ} {φ : Y ⟶ X.obj (opposite.op (simplex_category.mk (n + 1)))} (v : algebraic_topology.dold_kan.higher_faces_vanish q φ) (hnbq : n + 1 = b + q) :

algebraic_topology.dold_kan.higher_faces_vanish q (φ ≫ X.σ ⟨b, _⟩)

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theorem algebraic_topology.dold_kan.σ_comp_P_eq_zero {C : Type u_1} [category_theory.category C] [category_theory.preadditive C] (X : category_theory.simplicial_object C) {n q : ℕ} (i : fin (n + 1)) (hi : n + 1 ≤ ↑i + q) :

X.σ i ≫ (algebraic_topology.dold_kan.P q).f (n + 1) = 0

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@[simp]

theorem algebraic_topology.dold_kan.σ_comp_P_infty_assoc {C : Type u_1} [category_theory.category C] [category_theory.preadditive C] (X : category_theory.simplicial_object C) {n : ℕ} (i : fin (n + 1)) {X' : C} (f' : (algebraic_topology.alternating_face_map_complex.obj X).X (n + 1) ⟶ X') :

X.σ i ≫ algebraic_topology.dold_kan.P_infty.f (n + 1) ≫ f' = 0 ≫ f'

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@[simp]

theorem algebraic_topology.dold_kan.σ_comp_P_infty {C : Type u_1} [category_theory.category C] [category_theory.preadditive C] (X : category_theory.simplicial_object C) {n : ℕ} (i : fin (n + 1)) :

X.σ i ≫ algebraic_topology.dold_kan.P_infty.f (n + 1) = 0

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theorem algebraic_topology.dold_kan.degeneracy_comp_P_infty_assoc {C : Type u_1} [category_theory.category C] [category_theory.preadditive C] (X : category_theory.simplicial_object C) (n : ℕ) {Δ' : simplex_category} (θ : simplex_category.mk n ⟶ Δ') (hθ : ¬category_theory.mono θ) {X' : C} (f' : (algebraic_topology.alternating_face_map_complex.obj X).X n ⟶ X') :

X.map θ.op ≫ algebraic_topology.dold_kan.P_infty.f n ≫ f' = 0 ≫ f'

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theorem algebraic_topology.dold_kan.degeneracy_comp_P_infty {C : Type u_1} [category_theory.category C] [category_theory.preadditive C] (X : category_theory.simplicial_object C) (n : ℕ) {Δ' : simplex_category} (θ : simplex_category.mk n ⟶ Δ') (hθ : ¬category_theory.mono θ) :

X.map θ.op ≫ algebraic_topology.dold_kan.P_infty.f n = 0