Theorems · Definition · category theory

CategoryTheory.Pretriangulated.Opposite.contractibleTriangleIso

{C : Type u_1} →
  [inst : CategoryTheory.Category.{v_1, u_1} C] →
    [inst_1 : CategoryTheory.HasShift C ℤ] →
      [inst_2 : CategoryTheory.Limits.HasZeroObject C] →
        [inst_3 : CategoryTheory.Preadditive C] →
          [∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive] →
            (X : Cᵒᵖ) →
              CategoryTheory.Pretriangulated.contractibleTriangle X ≅
                (CategoryTheory.Pretriangulated.triangleOpEquivalence C).functor.obj
                  (Opposite.op (CategoryTheory.Pretriangulated.contractibleTriangle (Opposite.unop X)).invRotate)

Up to rotation, the contractible triangle X ⟶ X ⟶ 0 ⟶ X⟦1⟧ for X : Cᵒᵖ corresponds to the contractible triangle for X.unop in C.

Defined in

Mathlib.CategoryTheory.Triangulated.Opposite.Pretriangulated

Cited by

7 results in Mathlib

Foundations

Depth 68 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.HasShiftCategoryTheory.Limits.HasZeroObjectCategoryTheory.PreadditiveCategoryTheory.Functor.Additive