Theorems · Theorem · category theory

CategoryTheory.Functor.restrictedTopology_eq_inducedTopology

∀ {C : Type u₁} {D : Type u₂} [inst : CategoryTheory.Category.{v₁, u₁} C] [inst_1 : CategoryTheory.Category.{v₂, u₂} D]
  {F : CategoryTheory.Functor C D} {K : CategoryTheory.GrothendieckTopology D}
  [F.IsContinuous (F.restrictedTopology K) K], F.restrictedTopology K = F.inducedTopology K

If F is continuous with the restricted topology, the restricted topology agrees with the induced topology. This holds for example if G is locally faithful, locally full and cover dense.

Defined in

Mathlib.CategoryTheory.Sites.InducedTopology

Cited by

1 results in Mathlib

Foundations

Depth 74 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.Functor.IsContinuous