Theorems · Definition · category theory

HomotopicalAlgebra.leftHomotopyClassEquivRightHomotopyClass

{C : Type u} →
  [inst : CategoryTheory.Category.{v, u} C] →
    [inst_1 : HomotopicalAlgebra.ModelCategory C] →
      {X Y : C} →
        [HomotopicalAlgebra.IsCofibrant X] →
          [HomotopicalAlgebra.IsFibrant Y] →
            HomotopicalAlgebra.LeftHomotopyClass X Y ≃ HomotopicalAlgebra.RightHomotopyClass X Y

Left homotopy classes of maps X ⟶ Y identify to right homotopy classes when X is cofibrant and Y is fibrant.

Defined in: Mathlib.AlgebraicTopology.ModelCategory.Homotopy
Cited by: 4 results in Mathlib
Foundations: Depth 84 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: CategoryTheory.Category HomotopicalAlgebra.ModelCategory HomotopicalAlgebra.IsCofibrant HomotopicalAlgebra.IsFibrant

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Cites12

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

CategoryTheory.Categorystatement and proof · cited by 32,673
Quiver.Homproof · cited by 32,603
Equivstatement · cited by 8,337
HomotopicalAlgebra.ModelCategorystatement and proof · cited by 141
HomotopicalAlgebra.IsCofibrantstatement and proof · cited by 56
HomotopicalAlgebra.IsFibrantstatement and proof · cited by 56
HomotopicalAlgebra.LeftHomotopyRelproof · cited by 22
HomotopicalAlgebra.RightHomotopyRelproof · cited by 22
HomotopicalAlgebra.RightHomotopyClass.mkproof · cited by 17
HomotopicalAlgebra.RightHomotopyClassstatement · cited by 16
HomotopicalAlgebra.LeftHomotopyClassstatement · cited by 15
HomotopicalAlgebra.LeftHomotopyClass.mkproof · cited by 14

Cited by5

Results whose statement or proof uses this declaration.

HomotopicalAlgebra.bijective_leftHomotopyClassToHom_iff_bijective_rightHomotopyClassToHomproof · cited by 1
HomotopicalAlgebra.BifibrantObject.HoCat.homEquivLeftproof · cited by 1
HomotopicalAlgebra.leftHomotopyClassEquivRightHomotopyClass_mkstatement · cited by 0
HomotopicalAlgebra.leftHomotopyClassEquivRightHomotopyClass_symm_mkstatement · cited by 0
HomotopicalAlgebra.leftHomotopyClassEquivRightHomotopyClass.congr_simpstatement and proof · cited by 0