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Mathlib.AlgebraicTopology.DoldKan.Decomposition

Decomposition of the Q endomorphisms #

In this file, we obtain a lemma decomposition_Q which expresses explicitly the projection (Q q).f (n+1) : X _[n+1] ⟶ X _[n+1] (X : SimplicialObject C with C a preadditive category) as a sum of terms which are postcompositions with degeneracies.

(TODO @joelriou: when C is abelian, define the degenerate subcomplex of the alternating face map complex of X and show that it is a complement to the normalized Moore complex.)

Then, we introduce an ad hoc structure MorphComponents X n Z which can be used in order to define morphisms X _[n+1] ⟶ Z using the decomposition provided by decomposition_Q. This shall play a critical role in the proof that the functor N₁ : SimplicialObject C ⥤ Karoubi (ChainComplex C ℕ)) reflects isomorphisms.

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

theorem AlgebraicTopology.DoldKan.decomposition_Q {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] {X : CategoryTheory.SimplicialObject C} (n : ) (q : ) :
(AlgebraicTopology.DoldKan.Q q).f (n + 1) = (Finset.filter (fun (i : Fin (n + 1)) => i < q) Finset.univ).sum fun (i : Fin (n + 1)) => CategoryTheory.CategoryStruct.comp ((AlgebraicTopology.DoldKan.P i).f (n + 1)) (CategoryTheory.CategoryStruct.comp (X i.rev.succ) (X i.rev))

In each positive degree, this lemma decomposes the idempotent endomorphism Q q as a sum of morphisms which are postcompositions with suitable degeneracies. As Q q is the complement projection to P q, this implies that in the case of simplicial abelian groups, any $(n+1)$-simplex $x$ can be decomposed as $x = x' + \sum (i=0}^{q-1} σ_{n-i}(y_i)$ where $x'$ is in the image of P q and the $y_i$ are in degree $n$.

The structure MorphComponents is an ad hoc structure that is used in the proof that N₁ : SimplicialObject C ⥤ Karoubi (ChainComplex C ℕ)) reflects isomorphisms. The fields are the data that are needed in order to construct a morphism X _[n+1] ⟶ Z (see φ) using the decomposition of the identity given by decomposition_Q n (n+1).

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    The morphism X _[n+1] ⟶ Z associated to f : MorphComponents X n Z.

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      the canonical MorphComponents whose associated morphism is the identity (see F_id) thanks to decomposition_Q n (n+1)

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        A MorphComponents can be postcomposed with a morphism.

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          A MorphComponents can be precomposed with a morphism of simplicial objects.

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