Construction of functors N for the Dold-Kan correspondence #

In this file, we construct functors N₁ : SimplicialObject C ⥤ Karoubi (ChainComplex C ℕ) and N₂ : Karoubi (SimplicialObject C) ⥤ Karoubi (ChainComplex C ℕ) for any preadditive category C. (The indices of these functors are the number of occurrences of Karoubi at the source or the target.)

In the case C is additive, the functor N₂ shall be the functor of the equivalence CategoryTheory.Preadditive.DoldKan.equivalence defined in EquivalenceAdditive.lean.

In the case the category C is pseudoabelian, the composition of N₁ with the inverse of the equivalence ChainComplex C ℕ ⥤ Karoubi (ChainComplex C ℕ) will be the functor CategoryTheory.Idempotents.DoldKan.N of the equivalence of categories CategoryTheory.Idempotents.DoldKan.equivalence : SimplicialObject C ≌ ChainComplex C ℕ defined in EquivalencePseudoabelian.lean.

When the category C is abelian, a relation between N₁ and the normalized Moore complex functor shall be obtained in Normalized.lean.

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

theorem AlgebraicTopology.DoldKan.N₁_obj_p {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] (X : CategoryTheory.SimplicialObject C) :
(AlgebraicTopology.DoldKan.N₁.obj X).p = AlgebraicTopology.DoldKan.PInfty

The functor SimplicialObject C ⥤ Karoubi (ChainComplex C ℕ) which maps X to the formal direct factor of K[X] defined by PInfty.

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