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Theorems · Theorem · category theory

HomotopyCategory.quotient_obj_mem_subcategoryAcyclic_iff_acyclic

∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] [inst_1 : CategoryTheory.Abelian C] (K : CochainComplex C ℤ),
  HomotopyCategory.subcategoryAcyclic C ((HomotopyCategory.quotient C (ComplexShape.up ℤ)).obj K) ↔
    HomologicalComplex.Acyclic K
Defined in
Mathlib.Algebra.Homology.HomotopyCategory.Acyclic
Cited by
2 results in Mathlib
Foundations
Depth 97 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes
CategoryTheory.CategoryCategoryTheory.Abelian
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CochainComplex.isKProjective_iff_leftOrthogonal · cited by 1CochainComplex.isKProject…CochainComplex.isKInjective_iff_rightOrthogonal · cited by 1CochainComplex.isKInjecti…CategoryTheory.Category · cited by 32673CategoryTheory.CategoryCategoryTheory.Functor.obj · cited by 19642Functor.objCategoryTheory.Abelian · cited by 1753CategoryTheory.AbelianHomologicalComplex · cited by 1691HomologicalComplexComplexShape.up · cited by 1123ComplexShape.upCochainComplex · cited by 1016CochainComplexHomotopyCategory · cited by 132HomotopyCategoryHomotopyCategory.quotient · cited by 109HomotopyCategory.quotientHomologicalComplex.Acyclic · cited by 28HomologicalComplex.AcyclicHomotopyCategory.subcategoryAcyclic · cited by 10HomotopyCategory.subcateg…HomotopyCategory.quotient_obj_mem_subcategoryAcyclic_iff_exactAt · cited by 1HomotopyCategory.quotient…HomotopyCategory.quotient_obj…CITED BYCITES

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