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Theorems · Definition · category theory

Homotopy.refl

{ι : Type u_1} →
  {V : Type u} →
    [inst : CategoryTheory.Category.{v, u} V] →
      [inst_1 : CategoryTheory.Preadditive V] →
        {c : ComplexShape ι} → {C D : HomologicalComplex V c} → (f : C ⟶ D) → Homotopy f f

Every chain map is homotopic to itself.

Defined in
Mathlib.Algebra.Homology.Homotopy
Cited by
7 results in Mathlib
Foundations
Depth 22 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes
CategoryTheory.CategoryCategoryTheory.Preadditive
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Around this declaration

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

HomologicalComplex.cylinder.π · cited by 16cylinder.πHomologicalComplex.cylinder.ι₀_π · cited by 3cylinder.ι₀_πHomologicalComplex.cylinder.ι₁_π · cited by 3cylinder.ι₁_πHomologicalComplex.cylinder.inlX_π · cited by 1cylinder.inlX_πHomologicalComplex.cylinder.inrX_π · cited by 1cylinder.inrX_πCochainComplex.HomComplex.Cochain.ofHomotopy_refl · cited by 0Cochain.ofHomotopy_reflHomotopy.refl_hom · cited by 0Homotopy.refl_homCochainComplex.cm5b.homotopyEquiv · cited by 0cm5b.homotopyEquivAlgebraicTopology.DoldKan.homotopyPToId.eq_def · cited by 0homotopyPToId.eq_defCategoryTheory.Category · cited by 32673CategoryTheory.CategoryQuiver.Hom · cited by 32603Quiver.HomCategoryTheory.Preadditive · cited by 3309CategoryTheory.PreadditiveHomologicalComplex · cited by 1691HomologicalComplexComplexShape · cited by 1684ComplexShapeHomotopy · cited by 106HomotopyHomotopy.ofEq · cited by 25Homotopy.ofEqHomotopy.reflCITED BYCITES

Cites7

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Cited by9

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