Construction of homotopies for the Dold-Kan correspondence #
(The general strategy of proof of the Dold-Kan correspondence is explained
in Equivalence.lean.)
The purpose of the files Homotopies.lean, Faces.lean, Projections.lean
and PInfty.lean is to construct an idempotent endomorphism
PInfty : K[X] ⟶ K[X] of the alternating face map complex
for each X : SimplicialObject C when C is a preadditive category.
In the case C is abelian, this PInfty shall be the projection on the
normalized Moore subcomplex of K[X] associated to the decomposition of the
complex K[X] as a direct sum of this normalized subcomplex and of the
degenerate subcomplex.
In PInfty.lean, this endomorphism PInfty shall be obtained by
passing to the limit idempotent endomorphisms P q for all (q : ℕ).
These endomorphisms P q are defined by induction. The idea is to
start from the identity endomorphism P 0 of K[X] and to ensure by
induction that the q higher face maps (except $d_0$) vanish on the
image of P q. Then, in a certain degree n, the image of P q for
a big enough q will be contained in the normalized subcomplex. This
construction is done in Projections.lean.
It would be easy to define the P q degreewise (similarly as it is done
in Simplicial Homotopy Theory by Goerrs-Jardine p. 149), but then we would
have to prove that they are compatible with the differential (i.e. they
are chain complex maps), and also that they are homotopic to the identity.
These two verifications are quite technical. In order to reduce the number
of such technical lemmas, the strategy that is followed here is to define
a series of null homotopic maps Hσ q (attached to families of maps hσ)
and use these in order to construct P q : the endomorphisms P q
shall basically be obtained by altering the identity endomorphism by adding
null homotopic maps, so that we get for free that they are morphisms
of chain complexes and that they are homotopic to the identity. The most
technical verifications that are needed about the null homotopic maps Hσ
are obtained in Faces.lean.
In this file Homotopies.lean, we define the null homotopic maps
Hσ q : K[X] ⟶ K[X], show that they are natural (see natTransHσ) and
compatible the application of additive functors (see map_Hσ).
References #
- [Albrecht Dold, Homology of Symmetric Products and Other Functors of Complexes][dold1958]
- [Paul G. Goerss, John F. Jardine, Simplicial Homotopy Theory][goerss-jardine-2009]
Helper when we need some c.rel i j (i.e. ComplexShape.down ℕ),
e.g. c_mk n (n+1) rfl
This lemma is meant to be used with nullHomotopicMap'_f_of_not_rel_left
The sequence of maps which gives the null homotopic maps Hσ that shall be in
the inductive construction of the projections P q : K[X] ⟶ K[X]
Equations
- AlgebraicTopology.DoldKan.hσ q n = if n < q then 0 else (-1) ^ (n - q) • CategoryTheory.SimplicialObject.σ X { val := n - q, isLt := ⋯ }
Instances For
We can turn hσ into a datum that can be passed to nullHomotopicMap'.
Equations
Instances For
The null homotopic map $(hσ q) ∘ d + d ∘ (hσ q)$
Equations
Instances For
Hσ is null homotopic
Equations
Instances For
In degree 0, the null homotopic map Hσ is zero.
The maps hσ' q n m hnm are natural on the simplicial object
For each q, Hσ q is a natural transformation.
Equations
- AlgebraicTopology.DoldKan.natTransHσ q = { app := fun (X : CategoryTheory.SimplicialObject C) => AlgebraicTopology.DoldKan.Hσ q, naturality := ⋯ }
Instances For
The maps hσ' q n m hnm are compatible with the application of additive functors.
The null homotopic maps Hσ are compatible with the application of additive functors.