Theorems · Definition · category theory

CategoryTheory.Abelian.SpectralObject.kernelSequenceOpcyclesEIso

{C : Type u_1} →
  {ι : Type u_2} →
    [inst : CategoryTheory.Category.{v_1, u_1} C] →
      [inst_1 : CategoryTheory.Category.{v_2, u_2} ι] →
        [inst_2 : CategoryTheory.Abelian C] →
          (X : CategoryTheory.Abelian.SpectralObject C ι) →
            {i j k l : ι} →
              (f₁ : i ⟶ j) →
                (f₂ : j ⟶ k) →
                  (f₃ : k ⟶ l) →
                    (n₀ n₁ n₂ : ℤ) →
                      (hn₁ :
                          autoParam (n₀ + 1 = n₁)
                            CategoryTheory.Abelian.SpectralObject.kernelSequenceOpcyclesEIso._auto_1) →
                        (hn₂ :
                            autoParam (n₁ + 1 = n₂)
                              CategoryTheory.Abelian.SpectralObject.kernelSequenceOpcyclesEIso._auto_3) →
                          X.kernelSequenceOpcyclesE f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂ ≅
                            { X₁ := (X.shortComplex f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).homology,
                              X₂ := (X.shortComplex f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).opcycles,
                              X₃ := (X.shortComplex f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).X₃,
                              f := (X.shortComplex f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).homologyι,
                              g := (X.shortComplex f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).fromOpcycles, zero := ⋯ }

The short complex E^n(f₁, f₂, f₃) ⟶ opZ^n(f₂, f₃) ⟶ H^{n+1}(f₁) identifies to the kernel sequence of the definition of the homology of the short complex shortComplex as a kernel of ShortComplex.fromOpcycles.

Defined in

Mathlib.Algebra.Homology.SpectralObject.Page

Cited by

7 results in Mathlib

Foundations

Depth 107 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.Abelian