Theorems · Inductive type · category theory

CategoryTheory.ProjectiveResolution.Hom

{C : Type u} →
  [inst : CategoryTheory.Category.{v, u} C] →
    [inst_1 : CategoryTheory.Limits.HasZeroObject C] →
      [inst_2 : CategoryTheory.Limits.HasZeroMorphisms C] →
        {Z : C} →
          CategoryTheory.ProjectiveResolution Z → {Z' : C} → CategoryTheory.ProjectiveResolution Z' → (Z ⟶ Z') → Type v

Given projective resolutions P and P' of two objects Z and Z', and a morphism f : Z ⟶ Z', this structure contains the data of a morphism P.complex ⟶ P'.complex which is compatible with f

Defined in: Mathlib.CategoryTheory.Preadditive.Projective.Resolution
Cited by: 12 results in Mathlib
Foundations: Depth 3 from the axioms · uses no axioms
Assumes: CategoryTheory.Category CategoryTheory.Limits.HasZeroObject CategoryTheory.Limits.HasZeroMorphisms

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

Cites5

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

CategoryTheory.Categorystatement · cited by 32,673
Quiver.Homstatement · cited by 32,603
CategoryTheory.Limits.HasZeroMorphismsstatement · cited by 3,275
CategoryTheory.Limits.HasZeroObjectstatement · cited by 1,298
CategoryTheory.ProjectiveResolutionstatement · cited by 92

Cited by20

Results whose statement or proof uses this declaration.

CategoryTheory.ProjectiveResolution.Hom.homstatement and proof · cited by 8
CategoryTheory.ProjectiveResolution.Hom.hom'statement and proof · cited by 5
CategoryTheory.ProjectiveResolution.Hom.hom_f_zero_comp_π_f_zerostatement and proof · cited by 3
CategoryTheory.ProjectiveResolution.Hom.hom'_fstatement and proof · cited by 3
CategoryTheory.ProjectiveResolution.Hom.hom'_comp_π'statement and proof · cited by 2
CategoryTheory.ProjectiveResolution.Hom.hom_comp_πstatement and proof · cited by 1
CategoryTheory.ProjectiveResolution.Hom.mk.injstatement · cited by 1
CategoryTheory.ProjectiveResolution.Hom.mk.noConfusionstatement · cited by 1
CategoryTheory.ProjectiveResolution.Hom.hom'_f_assocstatement and proof · cited by 0
CategoryTheory.ProjectiveResolution.Hom.hom_comp_π_assocstatement and proof · cited by 0
CategoryTheory.ProjectiveResolution.Hom.hom_f_zero_comp_π_f_zero_assocstatement and proof · cited by 0
CategoryTheory.ProjectiveResolution.Hom.noConfusionstatement and proof · cited by 0