English
If t is triangulated and P.IsTriangulated, then P.trW HasLeftCalculusOfFractions with explicit exists_leftFraction and ext lemmas.
Русский
Если t треугольная структура и P IsTriangulated, то P.trW имеет левый вычислительный целевой дроби с явными существованиями.
LaTeX
$$$[IsTriangulated C] [P.IsTriangulated] : P.trW.HasLeftCalculusOfFractions$$$
Lean4
instance [IsTriangulated C] [P.IsTriangulated] : P.trW.HasLeftCalculusOfFractions
where
exists_leftFraction X Y
φ := by
obtain ⟨Z, f, g, H, mem⟩ := φ.hs
obtain ⟨Y', s', f', mem'⟩ := distinguished_cocone_triangle₂ (g ≫ φ.f⟦1⟧')
obtain ⟨b, ⟨hb₁, _⟩⟩ := complete_distinguished_triangle_morphism₂ _ _ H mem' φ.f (𝟙 Z) (by simp)
exact ⟨MorphismProperty.LeftFraction.mk b s' ⟨_, _, _, mem', mem⟩, hb₁.symm⟩
ext := by
rintro X' X Y f₁ f₂ s ⟨Z, g, h, H, mem⟩ hf₁
have hf₂ : s ≫ (f₁ - f₂) = 0 := by rw [comp_sub, hf₁, sub_self]
obtain ⟨q, hq⟩ := Triangle.yoneda_exact₂ _ H _ hf₂
obtain ⟨Y', r, t, mem'⟩ := distinguished_cocone_triangle q
refine ⟨Y', r, ?_, ?_⟩
· exact ⟨_, _, _, rot_of_distTriang _ mem', P.le_shift _ _ mem⟩
· have eq := comp_distTriang_mor_zero₁₂ _ mem'
dsimp at eq
rw [← sub_eq_zero, ← sub_comp, hq, assoc, eq, comp_zero]
