Theorems · Theorem · category theory

CategoryTheory.Triangulated.someOctahedron.congr_simp

∀ {C : Type u_1} [inst : CategoryTheory.Category.{v_1, u_1} C] [inst_1 : CategoryTheory.Preadditive C]
  [inst_2 : CategoryTheory.Limits.HasZeroObject C] [inst_3 : CategoryTheory.HasShift C ℤ]
  [inst_4 : ∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive] [inst_5 : CategoryTheory.Pretriangulated C]
  [inst_6 : CategoryTheory.IsTriangulated C] {X₁ X₂ X₃ Z₁₂ Z₂₃ Z₁₃ : C} {u₁₂ : X₁ ⟶ X₂} {u₂₃ : X₂ ⟶ X₃} {u₁₃ : X₁ ⟶ X₃}
  (comm : CategoryTheory.CategoryStruct.comp u₁₂ u₂₃ = u₁₃) {v₁₂ : X₂ ⟶ Z₁₂}
  {w₁₂ : Z₁₂ ⟶ (CategoryTheory.shiftFunctor C 1).obj X₁}
  (h₁₂ : CategoryTheory.Pretriangulated.Triangle.mk u₁₂ v₁₂ w₁₂ ∈ CategoryTheory.Pretriangulated.distinguishedTriangles)
  {v₂₃ : X₃ ⟶ Z₂₃} {w₂₃ : Z₂₃ ⟶ (CategoryTheory.shiftFunctor C 1).obj X₂}
  (h₂₃ : CategoryTheory.Pretriangulated.Triangle.mk u₂₃ v₂₃ w₂₃ ∈ CategoryTheory.Pretriangulated.distinguishedTriangles)
  {v₁₃ : X₃ ⟶ Z₁₃} {w₁₃ : Z₁₃ ⟶ (CategoryTheory.shiftFunctor C 1).obj X₁}
  (h₁₃ :
    CategoryTheory.Pretriangulated.Triangle.mk u₁₃ v₁₃ w₁₃ ∈ CategoryTheory.Pretriangulated.distinguishedTriangles),
  CategoryTheory.Triangulated.someOctahedron comm h₁₂ h₂₃ h₁₃ =
    CategoryTheory.Triangulated.someOctahedron comm h₁₂ h₂₃ h₁₃

Defined in

Mathlib.CategoryTheory.Triangulated.Triangulated

Cited by

0 results in Mathlib

Foundations

Depth 37 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.PreadditiveCategoryTheory.Limits.HasZeroObjectCategoryTheory.HasShiftCategoryTheory.Functor.AdditiveCategoryTheory.PretriangulatedCategoryTheory.IsTriangulated