Theorems · Theorem · category theory

CategoryTheory.MorphismProperty.relative_map

∀ {C : Type u₁} [inst : CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [inst_1 : CategoryTheory.Category.{v₂, u₂} D]
  {F : CategoryTheory.Functor C D} {P : CategoryTheory.MorphismProperty C} [F.Faithful] [F.Full]
  [CategoryTheory.Limits.HasPullbacks C] [P.IsStableUnderBaseChange] {a b : C} {f : a ⟶ b}
  [∀ (c : C) (g : c ⟶ b), CategoryTheory.Limits.PreservesLimit (CategoryTheory.Limits.cospan f g) F],
  P f → CategoryTheory.MorphismProperty.relative F P (F.map f)

If P : MorphismProperty C is stable under base change, F is fully faithful and preserves pullbacks, and C has all pullbacks, then for any f : a ⟶ b in C, F.map f satisfies P.relative if f satisfies P.

Defined in

Mathlib.CategoryTheory.MorphismProperty.Representable

Cited by

1 results in Mathlib

Foundations

Depth 38 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.Functor.FaithfulCategoryTheory.Functor.FullCategoryTheory.Limits.HasPullbacksCategoryTheory.MorphismProperty.IsStableUnderBaseChangeCategoryTheory.Limits.PreservesLimit