English
If a localization L: C → D is triangulated and essentially surjective on arrows, and L⋙F ≅ G with G homological, then F is homological.
Русский
Если локализация L: C → D треугольная и эасвно сюръективна на стрелы, и L⋙F ≅ G, причём G гомологичен, то F гомологичен.
LaTeX
$$$\text{If }L: C\to D\text{ is a localization with }L\text{ triangulated and }L\text{ map-arrow surjective, and }F:D\to A, G:C\to A, e:L\circ F\cong G,\ G\text{ is Homological }\Rightarrow F\text{ is Homological}$$$
Lean4
noncomputable instance (priority := 100) [F.IsHomological] : PreservesLimitsOfShape (Discrete WalkingPair) F :=
by
suffices ∀ (X₁ X₂ : C), PreservesLimit (pair X₁ X₂) F from
⟨fun {X} => preservesLimit_of_iso_diagram F (diagramIsoPair X).symm⟩
intro X₁ X₂
have : HasBinaryBiproduct (F.obj X₁) (F.obj X₂) := HasBinaryBiproducts.has_binary_biproduct _ _
have : Mono (F.biprodComparison X₁ X₂) := by
rw [mono_iff_cancel_zero]
intro Z f hf
let S := (ShortComplex.mk _ _ (biprod.inl_snd (X := X₁) (Y := X₂))).map F
have : Mono S.f := by dsimp [S]; infer_instance
have ex : S.Exact := F.map_distinguished_exact _ (binaryBiproductTriangle_distinguished X₁ X₂)
obtain ⟨g, rfl⟩ := ex.lift' f (by simpa using hf =≫ biprod.snd)
dsimp [S] at hf ⊢
replace hf := hf =≫ biprod.fst
simp only [assoc, biprodComparison_fst, zero_comp, ← F.map_comp, biprod.inl_fst, F.map_id, comp_id] at hf
rw [hf, zero_comp]
have : PreservesBinaryBiproduct X₁ X₂ F := preservesBinaryBiproduct_of_mono_biprodComparison _
apply Limits.preservesBinaryProduct_of_preservesBinaryBiproduct
