Theorems · Definition · category theory

HomotopicalAlgebra.BifibrantObject.HoCat.homEquivRight

{C : Type u} →
  [inst : CategoryTheory.Category.{v, u} C] →
    [inst_1 : HomotopicalAlgebra.ModelCategory C] →
      {X Y : C} →
        [inst_2 : HomotopicalAlgebra.IsCofibrant X] →
          [inst_3 : HomotopicalAlgebra.IsCofibrant Y] →
            [inst_4 : HomotopicalAlgebra.IsFibrant X] →
              [inst_5 : HomotopicalAlgebra.IsFibrant Y] →
                HomotopicalAlgebra.RightHomotopyClass X Y ≃
                  (HomotopicalAlgebra.BifibrantObject.toHoCat.obj (HomotopicalAlgebra.BifibrantObject.mk X) ⟶
                    HomotopicalAlgebra.BifibrantObject.toHoCat.obj (HomotopicalAlgebra.BifibrantObject.mk Y))

Right homotopy classes of maps between bifibrant objects identify to morphisms in the homotopy category BifibrantObject.HoCat.

Defined in

Mathlib.AlgebraicTopology.ModelCategory.BifibrantObjectHomotopy

Cited by

4 results in Mathlib

Foundations

Depth 85 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryHomotopicalAlgebra.ModelCategoryHomotopicalAlgebra.IsCofibrantHomotopicalAlgebra.IsCofibrantHomotopicalAlgebra.IsFibrantHomotopicalAlgebra.IsFibrant