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Mathlib.AlgebraicTopology.DoldKan.EquivalenceAdditive

The Dold-Kan equivalence for additive categories. #

This file defines Preadditive.DoldKan.equivalence which is the equivalence of categories Karoubi (SimplicialObject C) ≌ Karoubi (ChainComplex C ℕ).

(See Equivalence.lean for the general strategy of proof of the Dold-Kan equivalence.)

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noncomputable def CategoryTheory.Preadditive.DoldKan.N {C : Type u_1} [Category.{v_1, u_1} C] [Preadditive C] :
Functor (Idempotents.Karoubi (SimplicialObject C)) (Idempotents.Karoubi (ChainComplex C ℕ))

The functor Karoubi (SimplicialObject C) ⥤ Karoubi (ChainComplex C ℕ) of the Dold-Kan equivalence for additive categories.

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    noncomputable def CategoryTheory.Preadditive.DoldKan.Γ {C : Type u_1} [Category.{v_1, u_1} C] [Preadditive C] [Limits.HasFiniteCoproducts C] :
    Functor (Idempotents.Karoubi (ChainComplex C ℕ)) (Idempotents.Karoubi (SimplicialObject C))

    The inverse functor Karoubi (ChainComplex C ℕ) ⥤ Karoubi (SimplicialObject C) of the Dold-Kan equivalence for additive categories.

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      noncomputable def CategoryTheory.Preadditive.DoldKan.equivalence {C : Type u_1} [Category.{v_1, u_1} C] [Preadditive C] [Limits.HasFiniteCoproducts C] :
      Idempotents.Karoubi (SimplicialObject C) ≌ Idempotents.Karoubi (ChainComplex C ℕ)

      The Dold-Kan equivalence Karoubi (SimplicialObject C) ≌ Karoubi (ChainComplex C ℕ) for additive categories.

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        @[simp]
        theorem CategoryTheory.Preadditive.DoldKan.equivalence_inverse {C : Type u_1} [Category.{v_1, u_1} C] [Preadditive C] [Limits.HasFiniteCoproducts C] :
        equivalence.inverse = Γ
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        @[simp]
        theorem CategoryTheory.Preadditive.DoldKan.equivalence_unitIso {C : Type u_1} [Category.{v_1, u_1} C] [Preadditive C] [Limits.HasFiniteCoproducts C] :
        equivalence.unitIso = AlgebraicTopology.DoldKan.Γ₂N₂
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        @[simp]
        theorem CategoryTheory.Preadditive.DoldKan.equivalence_functor {C : Type u_1} [Category.{v_1, u_1} C] [Preadditive C] [Limits.HasFiniteCoproducts C] :
        equivalence.functor = N
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        @[simp]
        theorem CategoryTheory.Preadditive.DoldKan.equivalence_counitIso {C : Type u_1} [Category.{v_1, u_1} C] [Preadditive C] [Limits.HasFiniteCoproducts C] :
        equivalence.counitIso = AlgebraicTopology.DoldKan.N₂Γ₂