Theorems · Inductive type · category theory

HomotopicalAlgebra.PathObject.IsVeryGood

{C : Type u} →
  [inst : CategoryTheory.Category.{v, u} C] →
    {A : C} →
      [inst_1 : HomotopicalAlgebra.CategoryWithWeakEquivalences C] →
        HomotopicalAlgebra.PathObject A →
          [CategoryTheory.Limits.HasBinaryProduct A A] →
            [HomotopicalAlgebra.CategoryWithFibrations C] → [HomotopicalAlgebra.CategoryWithCofibrations C] → Prop

A good path object P is very good if P.ι is a (trivial) cofibration.

Defined in: Mathlib.AlgebraicTopology.ModelCategory.PathObject
Cited by: 7 results in Mathlib
Foundations: Depth 15 from the axioms · uses propext
Assumes: CategoryTheory.Category HomotopicalAlgebra.CategoryWithWeakEquivalences CategoryTheory.Limits.HasBinaryProduct HomotopicalAlgebra.CategoryWithFibrations HomotopicalAlgebra.CategoryWithCofibrations

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Cites6

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

CategoryTheory.Categorystatement · cited by 32,673
CategoryTheory.Limits.HasBinaryProductstatement · cited by 169
HomotopicalAlgebra.CategoryWithWeakEquivalencesstatement · cited by 77
HomotopicalAlgebra.CategoryWithFibrationsstatement · cited by 46
HomotopicalAlgebra.CategoryWithCofibrationsstatement · cited by 46
HomotopicalAlgebra.PathObjectstatement · cited by 32

Cited by9

Results whose statement or proof uses this declaration.

HomotopicalAlgebra.RightHomotopyRel.exists_very_good_pathObjectstatement · cited by 3
HomotopicalAlgebra.LeftHomotopyRel.rightHomotopyRelproof · cited by 2
HomotopicalAlgebra.PathObject.exists_very_goodstatement · cited by 2
HomotopicalAlgebra.RightHomotopyRel.postcompproof · cited by 0
HomotopicalAlgebra.CofibrantObject.factorsThroughLocalizationproof · cited by 0
HomotopicalAlgebra.BifibrantObject.inverts_iff_factorsproof · cited by 0
HomotopicalAlgebra.PathObject.IsVeryGood.casesOnstatement and proof · cited by 0
HomotopicalAlgebra.PathObject.IsVeryGood.congr_simpstatement and proof · cited by 0
HomotopicalAlgebra.PathObject.IsVeryGood.recOnstatement and proof · cited by 0