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namespace OAI
noncomputable section
open CategoryTheory
namespace Problem340.ChainMapRangeLift
variable {R : Type*} [Ring R]
section Linear
variable {M N P : Type*} [AddCommGroup M] [AddCommGroup N] [AddCommGroup P]
[Module R M] [Module R N] [Module R P]
def linearLift (i : M →ₗ[R] N) (hi : Function.Injective i)
(f : P →ₗ[R] N) (hf : ∀ x, f x ∈ LinearMap.range i) : P →ₗ[R] M :=
(LinearEquiv.ofInjective i hi).symm.toLinearMap.comp
(f.codRestrict (LinearMap.range i) hf)
@[simp] theorem linearLift_apply (i : M →ₗ[R] N) (hi : Function.Injective i)
(f : P →ₗ[R] N) (hf : ∀ x, f x ∈ LinearMap.range i) (x : P) :
i (linearLift i hi f hf x) = f x := by
exact congrArg Subtype.val
((LinearEquiv.ofInjective i hi).apply_symm_apply ⟨f x, hf x⟩)
end Linear
section Complexes
variable {ι : Type*} {c : ComplexShape ι}
{C D E : HomologicalComplex (ModuleCat R) c}
theorem comm_apply (f : C ⟶ D) (i j : ι) (x : C.X i) :
(D.d i j).hom ((f.f i).hom x) = (f.f j).hom ((C.d i j).hom x) :=
congrArg (fun k : C.X i ⟶ D.X j => k.hom x) (f.comm i j)
def liftChainMap (φ : C ⟶ D) (hinj : ∀ n, Function.Injective (φ.f n).hom)
(f : E ⟶ D) (hf : ∀ n (x : E.X n), (f.f n).hom x ∈ LinearMap.range (φ.f n).hom) :
E ⟶ C where
f n := ModuleCat.ofHom (linearLift (φ.f n).hom (hinj n) (f.f n).hom (hf n))
comm' i j _ := by
ext x
apply hinj j
change (φ.f j).hom ((C.d i j).hom
(linearLift (φ.f i).hom (hinj i) (f.f i).hom (hf i) x)) =
(φ.f j).hom (linearLift (φ.f j).hom (hinj j) (f.f j).hom (hf j)
((E.d i j).hom x))
rw [← comm_apply φ i j, linearLift_apply, linearLift_apply, comm_apply f i j]
@[simp] theorem liftChainMap_comp (φ : C ⟶ D)
(hinj : ∀ n, Function.Injective (φ.f n).hom)
(f : E ⟶ D) (hf : ∀ n (x : E.X n), (f.f n).hom x ∈ LinearMap.range (φ.f n).hom) :
liftChainMap φ hinj f hf ≫ φ = f := by
ext n x
exact linearLift_apply (φ.f n).hom (hinj n) (f.f n).hom (hf n) x
theorem liftChainMap_eq (φ : C ⟶ D) (hinj : ∀ n, Function.Injective (φ.f n).hom)
(f : E ⟶ D) (hf : ∀ n (x : E.X n), (f.f n).hom x ∈ LinearMap.range (φ.f n).hom)
(g : E ⟶ C) (hg : g ≫ φ = f) : liftChainMap φ hinj f hf = g := by
ext n x
apply hinj n
have h := congrArg (fun k : E ⟶ D => (k.f n).hom x) hg
change (φ.f n).hom ((g.f n).hom x) = (f.f n).hom x at h
exact (linearLift_apply (φ.f n).hom (hinj n) (f.f n).hom (hf n) x).trans h.symm
def liftChainHomotopy (φ : C ⟶ D) (hinj : ∀ n, Function.Injective (φ.f n).hom)
(f g : E ⟶ D)
(hf : ∀ n (x : E.X n), (f.f n).hom x ∈ LinearMap.range (φ.f n).hom)
(hg : ∀ n (x : E.X n), (g.f n).hom x ∈ LinearMap.range (φ.f n).hom)
(H : Homotopy f g)
(hH : ∀ i j (x : E.X i), (H.hom i j).hom x ∈ LinearMap.range (φ.f j).hom) :
Homotopy (liftChainMap φ hinj f hf) (liftChainMap φ hinj g hg) where
hom i j := ModuleCat.ofHom (linearLift (φ.f j).hom (hinj j) (H.hom i j).hom (hH i j))
zero i j hij := by
ext x
apply hinj j
change (φ.f j).hom (linearLift (φ.f j).hom (hinj j)
(H.hom i j).hom (hH i j) x) = (φ.f j).hom 0
rw [map_zero, linearLift_apply, H.zero i j hij]
rfl
comm i := by
ext x
apply hinj i
have h := congrArg (fun k : E.X i ⟶ D.X i => k.hom x) (H.comm i)
change (f.f i).hom x =
(H.hom (c.next i) i).hom ((E.d i (c.next i)).hom x) +
(D.d (c.prev i) i).hom ((H.hom i (c.prev i)).hom x) + (g.f i).hom x at h
change (φ.f i).hom (linearLift (φ.f i).hom (hinj i) (f.f i).hom (hf i) x) =
(φ.f i).hom (linearLift (φ.f i).hom (hinj i)
(H.hom (c.next i) i).hom (hH (c.next i) i) ((E.d i (c.next i)).hom x) +
(C.d (c.prev i) i).hom (linearLift (φ.f (c.prev i)).hom (hinj (c.prev i))
(H.hom i (c.prev i)).hom (hH i (c.prev i)) x) +
linearLift (φ.f i).hom (hinj i) (g.f i).hom (hg i) x)
rw [map_add, map_add, linearLift_apply, linearLift_apply, linearLift_apply,
← comm_apply φ (c.prev i) i, linearLift_apply]
exact h
end Complexes
end Problem340.ChainMapRangeLift
end
end OAI
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