Theorems · Theorem · category theory

CategoryTheory.ShortComplex.isIso_homologyMap_of_isIso_opcyclesMap_of_mono

∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] [inst_1 : CategoryTheory.Limits.HasZeroMorphisms C]
  {S₁ S₂ : CategoryTheory.ShortComplex C} {φ : S₁ ⟶ S₂} [inst_2 : S₁.HasHomology] [inst_3 : S₂.HasHomology],
  CategoryTheory.IsIso (CategoryTheory.ShortComplex.opcyclesMap φ) →
    CategoryTheory.Mono φ.τ₃ → CategoryTheory.IsIso (CategoryTheory.ShortComplex.homologyMap φ)

Defined in: Mathlib.Algebra.Homology.ShortComplex.Homology
Cited by: 0 results in Mathlib
Foundations: Depth 49 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: CategoryTheory.Category CategoryTheory.Limits.HasZeroMorphisms CategoryTheory.ShortComplex.HasHomology CategoryTheory.ShortComplex.HasHomology

Mathlib docs Source Loogle LeanSearch

Around this declaration

Dashed lines are statement dependencies; solid lines are citations in proofs.

Cites34

Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.

CategoryTheory.Categorystatement and proof · cited by 32,673
Quiver.Homstatement and proof · cited by 32,603
CategoryTheory.CategoryStruct.compproof · cited by 17,999
CategoryTheory.Category.assocproof · cited by 6,433
CategoryTheory.CategoryStruct.idproof · cited by 6,235
CategoryTheory.Limits.HasZeroMorphismsstatement and proof · cited by 3,275
CategoryTheory.Category.comp_idproof · cited by 2,119
CategoryTheory.Category.id_compproof · cited by 1,998
CategoryTheory.ShortComplexstatement and proof · cited by 1,850
CategoryTheory.Limits.Cone.ptproof · cited by 1,298
CategoryTheory.IsIsostatement and proof · cited by 1,156
CategoryTheory.Monostatement and proof · cited by 893

Cited by0

Results whose statement or proof uses this declaration.

Nothing cites this yet.