[Merged by Bors] - feat(algebraic_topology/dold_kan): construction of idempotent endomorphisms #15616
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| def Q (q : ℕ) : K[X] ⟶ K[X] := 𝟙 _ - P q | ||
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| lemma P_add_Q (q : ℕ) : P q + Q q = 𝟙 K[X] := by { rw Q, abel, } | ||
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| lemma P_add_Q_f (q n : ℕ) : (P q).f n + (Q q).f n = 𝟙 (X _[n]) := | ||
| homological_complex.congr_hom (P_add_Q q) n | ||
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| @[simp] | ||
| lemma Q_eq_zero : (Q 0 : K[X] ⟶ _) = 0 := sub_self _ | ||
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| lemma Q_eq (q : ℕ) : (Q (q+1) : K[X] ⟶ _) = Q q - P q ≫ Hσ q := | ||
| by { unfold Q P, simp only [comp_add, comp_id], abel, } | ||
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| /-- All the `Q q` coincide with `0` in degree 0. -/ | ||
| @[simp] | ||
| lemma Q_f_0_eq (q : ℕ) : ((Q q).f 0 : X _[0] ⟶ X _[0]) = 0 := | ||
| by simp only [homological_complex.sub_f_apply, homological_complex.id_f, Q, P_f_0_eq, sub_self] |
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What is the purpose of introducing Q? Will you use it in later PRs?
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Thanks for the review. Indeed, for a few statements and proofs, it will be more convenient to have Q also.
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…phisms (#15616) This PR introduces a sequence of idempotent endomorphisms of the alternating face map complex of a simplicial object in a preadditive category. In a future PR, by taking the "limit" of this sequence, we shall get the projection on the normalized subcomplex, parallel to the degenerate sucomplex.
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…phisms (leanprover-community#15616) This PR introduces a sequence of idempotent endomorphisms of the alternating face map complex of a simplicial object in a preadditive category. In a future PR, by taking the "limit" of this sequence, we shall get the projection on the normalized subcomplex, parallel to the degenerate sucomplex.
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…phisms (#15616) This PR introduces a sequence of idempotent endomorphisms of the alternating face map complex of a simplicial object in a preadditive category. In a future PR, by taking the "limit" of this sequence, we shall get the projection on the normalized subcomplex, parallel to the degenerate sucomplex.
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…phisms (#15616) This PR introduces a sequence of idempotent endomorphisms of the alternating face map complex of a simplicial object in a preadditive category. In a future PR, by taking the "limit" of this sequence, we shall get the projection on the normalized subcomplex, parallel to the degenerate sucomplex.
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This PR introduces a sequence of idempotent endomorphisms of the alternating face map complex of a simplicial object in a preadditive category. In a future PR, by taking the "limit" of this sequence, we shall get the projection on the normalized subcomplex, parallel to the degenerate sucomplex.