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2c7d975 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 | import OAI.Algebra.AuslanderReiten.HomComplex.FullHomFunctor
namespace OAI
noncomputable section
namespace ArExplicit
open CategoryTheory CochainComplex.HomComplex
variable {C D : Type*} [Category* C] [Category* D] [Preadditive C] [Preadditive D]
variable (Φ : C ⥤ D) [Φ.Additive]
variable (K L : CochainComplex C ℤ) (n : ℤ)
def mapFullHom : CohomologyClass K L n →+
CohomologyClass ((Φ.mapHomologicalComplex _).obj K)
((Φ.mapHomologicalComplex _).obj L) n :=
CohomologyClass.descAddMonoidHom
((CohomologyClass.mkAddMonoidHom _ _ _).comp (fullHomCocycleMap Φ K L n)) (by
intro x hx
apply (CohomologyClass.mk_eq_zero_iff _).mpr
obtain ⟨m, hm, y, hy⟩ := hx
refine ⟨m, hm, y.map Φ, ?_⟩
rw [δ_map, hy]
rfl)
@[simp] theorem mapFullHom_mk (x : Cocycle K L n) :
mapFullHom Φ K L n (CohomologyClass.mk x) =
CohomologyClass.mk (fullHomCocycleMap Φ K L n x) := rfl
theorem mapFullHom_eq [Φ.Full] [Φ.Faithful] :
mapFullHom Φ K L n = fullHomCohomologyMap Φ K L n := rfl
variable {K L n}
theorem mapFullHom_product {M : CochainComplex C ℤ} {m : ℤ}
(x : CohomologyClass K L n) (y : CohomologyClass L M m) :
mapFullHom Φ K M (n+m) (CompleteComparison.fullHomProduct x y) =
CompleteComparison.fullHomProduct (mapFullHom Φ K L n x)
(mapFullHom Φ L M m y) := by
obtain ⟨x, rfl⟩ := CohomologyClass.mk_surjective x
obtain ⟨y, rfl⟩ := CohomologyClass.mk_surjective y
simp only [CompleteComparison.fullHomProduct_mk, mapFullHom_mk]
congr 1
apply Cocycle.ext
apply Cochain.map_comp
rfl
end ArExplicit
end
end OAI
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