Theorems · Inductive type · category theory

CategoryTheory.Triangulated.SpectralObject

(C : Type u_1) →
  (ι : Type u_2) →
    [inst : CategoryTheory.Category.{v_1, u_1} C] →
      [CategoryTheory.Category.{v_2, u_2} ι] →
        [inst_2 : CategoryTheory.Limits.HasZeroObject C] →
          [inst_3 : CategoryTheory.HasShift C ℤ] →
            [inst_4 : CategoryTheory.Preadditive C] →
              [inst_5 : ∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive] →
                [CategoryTheory.Pretriangulated C] → Type (max (max (max u_1 u_2) v_1) v_2)

A spectral object in a pretriangulated category C indexed by a category ι consists of a functor ω₁ : ComposableArrows ι 1 ⥤ C, and a functorial distinguished triangle from the category ComposableArrows ι 2, which must be of the form ω₁.obj (mk₁ f) ⟶ ω₁.obj (mk₁ (f ≫ g)) ⟶ ω₁.obj (mk₁ g) ⟶ ... when evaluated on mk₂ f g : ComposableArrows ι 2.

Defined in

Mathlib.CategoryTheory.Triangulated.SpectralObject

Cited by

39 results in Mathlib

Foundations

Depth 32 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.Limits.HasZeroObjectCategoryTheory.HasShiftCategoryTheory.PreadditiveCategoryTheory.Functor.AdditiveCategoryTheory.Pretriangulated