Theorems · Theorem · category theory

HomotopicalAlgebra.CofibrantObject.weakEquivalence_homMk_iff

∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] [inst_1 : HomotopicalAlgebra.CategoryWithCofibrations C]
  [inst_2 : CategoryTheory.Limits.HasInitial C] [inst_3 : HomotopicalAlgebra.CategoryWithWeakEquivalences C] {X Y : C}
  [inst_4 : HomotopicalAlgebra.IsCofibrant X] [inst_5 : HomotopicalAlgebra.IsCofibrant Y] (f : X ⟶ Y),
  HomotopicalAlgebra.WeakEquivalence (HomotopicalAlgebra.CofibrantObject.homMk f) ↔ HomotopicalAlgebra.WeakEquivalence f

Defined in: Mathlib.AlgebraicTopology.ModelCategory.Bifibrant
Cited by: 0 results in Mathlib
Foundations: Depth 30 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: CategoryTheory.Category HomotopicalAlgebra.CategoryWithCofibrations CategoryTheory.Limits.HasInitial HomotopicalAlgebra.CategoryWithWeakEquivalences HomotopicalAlgebra.IsCofibrant HomotopicalAlgebra.IsCofibrant

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CategoryTheory.Categorystatement and proof · cited by 32,673
Quiver.Homstatement and proof · cited by 32,603
CategoryTheory.Limits.HasInitialstatement and proof · cited by 185
HomotopicalAlgebra.CategoryWithWeakEquivalencesstatement and proof · cited by 77
HomotopicalAlgebra.IsCofibrantstatement and proof · cited by 56
HomotopicalAlgebra.CategoryWithCofibrationsstatement and proof · cited by 46
HomotopicalAlgebra.WeakEquivalencestatement · cited by 45
HomotopicalAlgebra.CofibrantObjectstatement · cited by 35
HomotopicalAlgebra.cofibrantObjectsstatement · cited by 34
HomotopicalAlgebra.CofibrantObject.mkstatement · cited by 16
HomotopicalAlgebra.CofibrantObject.homMkstatement and proof · cited by 13

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