Theorems · Inductive type · category theory

CategoryTheory.Functor.PreservesHomology

{C : Type u_1} →
  {D : Type u_2} →
    [inst : CategoryTheory.Category.{v_1, u_1} C] →
      [inst_1 : CategoryTheory.Category.{v_2, u_2} D] →
        [inst_2 : CategoryTheory.Limits.HasZeroMorphisms C] →
          [inst_3 : CategoryTheory.Limits.HasZeroMorphisms D] →
            (F : CategoryTheory.Functor C D) → [F.PreservesZeroMorphisms] → Prop

A functor preserves homology when it preserves both kernels and cokernels.

Defined in: Mathlib.Algebra.Homology.ShortComplex.PreservesHomology
Cited by: 42 results in Mathlib
Foundations: Depth 3 from the axioms · uses no axioms
Assumes: CategoryTheory.Category CategoryTheory.Category CategoryTheory.Limits.HasZeroMorphisms CategoryTheory.Limits.HasZeroMorphisms CategoryTheory.Functor.PreservesZeroMorphisms

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Cites4

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

CategoryTheory.Categorystatement · cited by 32,673
CategoryTheory.Functorstatement · cited by 16,252
CategoryTheory.Limits.HasZeroMorphismsstatement · cited by 3,275
CategoryTheory.Functor.PreservesZeroMorphismsstatement · cited by 458

Cited by56

Results whose statement or proof uses this declaration.

CategoryTheory.Functor.mapHomologicalComplexUpToQuasiIsoLocalizerMorphismstatement and proof · cited by 5
HomologicalComplex.i_cyclesMkstatement and proof · cited by 5
CategoryTheory.Functor.preservesFiniteLimits_of_preservesHomologystatement and proof · cited by 4
CategoryTheory.Functor.exact_tfaestatement and proof · cited by 3
CategoryTheory.Functor.mapHomologicalComplexUpToQuasiIsostatement and proof · cited by 3
CategoryTheory.Functor.mapHomologicalComplexUpToQuasiIsoFactorshstatement and proof · cited by 3
CategoryTheory.Preadditive.mono_iff_injectivestatement and proof · cited by 3
CategoryTheory.Functor.preservesHomology_of_map_exactstatement · cited by 2
CategoryTheory.Functor.mapHomologicalComplexUpToQuasiIsoFactorsstatement and proof · cited by 2
CategoryTheory.Functor.mapHomologicalComplexUpToQuasiIsoFactorsh_hom_appstatement and proof · cited by 2
HomologicalComplex.cyclesMkstatement and proof · cited by 2
CategoryTheory.ShortComplex.homologyFunctorIsostatement and proof · cited by 2