Theorems · Theorem · category theory

CategoryTheory.ShortComplex.exact_iff_homology_iso_zero

∀ {C : Type u_1} [inst : CategoryTheory.Category.{v_1, u_1} C] [inst_1 : CategoryTheory.Limits.HasZeroMorphisms C]
  (S : CategoryTheory.ShortComplex C) [inst_2 : S.HasHomology] [inst_3 : CategoryTheory.Limits.HasZeroObject C],
  S.Exact ↔ Nonempty (S.homology ≅ 0)

Defined in: Mathlib.Algebra.Homology.ShortComplex.Exact
Cited by: 0 results in Mathlib
Foundations: Depth 41 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: CategoryTheory.Category CategoryTheory.Limits.HasZeroMorphisms CategoryTheory.ShortComplex.HasHomology CategoryTheory.Limits.HasZeroObject

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CategoryTheory.Categorystatement and proof · cited by 32,673
CategoryTheory.Isostatement and proof · cited by 3,963
CategoryTheory.Limits.HasZeroMorphismsstatement and proof · cited by 3,275
CategoryTheory.ShortComplexstatement and proof · cited by 1,850
CategoryTheory.Limits.HasZeroObjectstatement and proof · cited by 1,298
CategoryTheory.Limits.IsZeroproof · cited by 306
CategoryTheory.ShortComplex.Exactstatement · cited by 292
CategoryTheory.ShortComplex.HasHomologystatement and proof · cited by 253
CategoryTheory.ShortComplex.homologystatement and proof · cited by 216
CategoryTheory.Limits.HasZeroObject.zero'statement · cited by 115
CategoryTheory.Limits.IsZero.of_isoproof · cited by 35
CategoryTheory.Limits.isZero_zeroproof · cited by 34

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