Theorems · Definition · category theory

CategoryTheory.Triangulated.Localization.pretriangulated

{C : Type u_1} →
  {D : Type u_2} →
    [inst : CategoryTheory.Category.{v_1, u_1} C] →
      [inst_1 : CategoryTheory.Category.{v_2, u_2} D] →
        (L : CategoryTheory.Functor C D) →
          [inst_2 : CategoryTheory.HasShift C ℤ] →
            [inst_3 : CategoryTheory.Preadditive C] →
              [inst_4 : CategoryTheory.Limits.HasZeroObject C] →
                [inst_5 : ∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive] →
                  [inst_6 : CategoryTheory.Pretriangulated C] →
                    [inst_7 : CategoryTheory.HasShift D ℤ] →
                      [L.CommShift ℤ] →
                        (W : CategoryTheory.MorphismProperty C) →
                          [L.IsLocalization W] →
                            [W.HasLeftCalculusOfFractions] →
                              [W.IsCompatibleWithTriangulation] →
                                [inst_12 : CategoryTheory.Limits.HasZeroObject D] →
                                  [inst_13 : CategoryTheory.Preadditive D] →
                                    [inst_14 : ∀ (n : ℤ), (CategoryTheory.shiftFunctor D n).Additive] →
                                      [L.Additive] → CategoryTheory.Pretriangulated D

The pretriangulated structure on the localized category.

Defined in

Mathlib.CategoryTheory.Localization.Triangulated

Cited by

0 results in Mathlib

Foundations

Depth 50 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.HasShiftCategoryTheory.PreadditiveCategoryTheory.Limits.HasZeroObjectCategoryTheory.Functor.AdditiveCategoryTheory.PretriangulatedCategoryTheory.HasShiftCategoryTheory.Functor.CommShiftCategoryTheory.Functor.IsLocalizationCategoryTheory.MorphismProperty.HasLeftCalculusOfFractionsCategoryTheory.MorphismProperty.IsCompatibleWithTriangulationCategoryTheory.Limits.HasZeroObjectCategoryTheory.PreadditiveCategoryTheory.Functor.AdditiveCategoryTheory.Functor.Additive