Theorems · Theorem · category theory

CategoryTheory.Triangulated.TStructure.eTruncLTGEIsoGELT_naturality_app

∀ {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]
  (t : CategoryTheory.Triangulated.TStructure C) [inst_6 : CategoryTheory.IsTriangulated C] (a b : EInt) (hab : a ≤ b)
  (a' b' : EInt) (hab' : a' ≤ b')
  (φ :
    CategoryTheory.ComposableArrows.mk₁ (CategoryTheory.homOfLE hab) ⟶
      CategoryTheory.ComposableArrows.mk₁ (CategoryTheory.homOfLE hab'))
  (X : C),
  CategoryTheory.CategoryStruct.comp ((t.eTruncLT.map (φ.app 1)).app ((t.eTruncGE.obj a).obj X))
      (CategoryTheory.CategoryStruct.comp ((t.eTruncLT.obj b').map ((t.eTruncGE.map (φ.app 0)).app X))
        ((t.eTruncLTGEIsoGELT a' b').hom.app X)) =
    CategoryTheory.CategoryStruct.comp ((t.eTruncLTGEIsoGELT a b).hom.app X)
      (CategoryTheory.CategoryStruct.comp ((t.eTruncGE.map (φ.app 0)).app ((t.eTruncLT.obj b).obj X))
        ((t.eTruncGE.obj a').map ((t.eTruncLT.map (φ.app 1)).app X)))

Defined in

Mathlib.CategoryTheory.Triangulated.TStructure.ETrunc

Cited by

2 results in Mathlib

Foundations

Depth 86 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

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