Theorems · Theorem · category theory

HomologicalComplex.extend.d_comp_eq_zero_iff

∀ {ι : Type u_1} {ι' : Type u_2} {c : ComplexShape ι} {c' : ComplexShape ι'} {C : Type u_3}
  [inst : CategoryTheory.Category.{v_1, u_3} C] [inst_1 : CategoryTheory.Limits.HasZeroMorphisms C]
  [inst_2 : CategoryTheory.Limits.HasZeroObject C] (K : HomologicalComplex C c) (e : c.Embedding c') {i j : ι}
  {i' j' : ι'} (hj' : e.f j = j'),
  c.prev j = i →
    c'.prev j' = i' →
      ∀ ⦃W : C⦄ (φ : K.X j ⟶ W),
        CategoryTheory.CategoryStruct.comp (K.d i j) φ = 0 ↔
          CategoryTheory.CategoryStruct.comp ((K.extend e).d i' j')
              (CategoryTheory.CategoryStruct.comp (K.extendXIso e hj').hom φ) =
            0

Defined in

Mathlib.Algebra.Homology.Embedding.ExtendHomology

Cited by

0 results in Mathlib

Foundations

Depth 22 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.Limits.HasZeroMorphismsCategoryTheory.Limits.HasZeroObject