Decomposition of the Q endomorphisms #
THIS FILE IS SYNCHRONIZED WITH MATHLIB4. Any changes to this file require a corresponding PR to mathlib4.
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 : simplicial_object 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 morph_components 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 : simplicial_object C ⥤ karoubi (chain_complex C ℕ))
reflects isomorphisms.
(See equivalence.lean for the general strategy of proof of the Dold-Kan equivalence.)
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 $n$.
- a : X.obj (opposite.op (simplex_category.mk (n + 1))) ⟶ Z
- b : fin (n + 1) → (X.obj (opposite.op (simplex_category.mk n)) ⟶ Z)
The structure morph_components is an ad hoc structure that is used in
the proof that N₁ : simplicial_object C ⥤ karoubi (chain_complex 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).
Instances for algebraic_topology.dold_kan.morph_components
- algebraic_topology.dold_kan.morph_components.has_sizeof_inst
The morphism X _[n+1] ⟶ Z associated to f : morph_components X n Z.
the canonical morph_components whose associated morphism is the identity
(see F_id) thanks to decomposition_Q n (n+1)
A morph_components can be postcomposed with a morphism.
A morph_components can be precomposed with a morphism of simplicial objects.
Equations
- f.pre_comp g = {a := g.app (opposite.op (simplex_category.mk (n + 1))) ≫ f.a, b := λ (i : fin (n + 1)), g.app (opposite.op (simplex_category.mk n)) ≫ f.b i}