Theorems · Theorem · category theory

CategoryTheory.ran_isSheaf_of_isCocontinuous

∀ {C : Type u_1} {D : Type u_2} [inst : CategoryTheory.Category.{v_1, u_1} C]
  [inst_1 : CategoryTheory.Category.{v_2, u_2} D] (G : CategoryTheory.Functor C D) {A : Type w}
  [inst_2 : CategoryTheory.Category.{w', w} A] {J : CategoryTheory.GrothendieckTopology C}
  (K : CategoryTheory.GrothendieckTopology D) [G.IsCocontinuous J K]
  [inst_4 : ∀ (F : CategoryTheory.Functor Cᵒᵖ A), G.op.HasPointwiseRightKanExtension F] (ℱ : CategoryTheory.Sheaf J A),
  CategoryTheory.Presheaf.IsSheaf K (G.op.ran.obj ℱ.obj)

If G is cocontinuous, then G.op.ran pushes sheaves to sheaves. This is SGA 4 III 2.2.

Defined in

Mathlib.CategoryTheory.Sites.CoverLifting

Cited by

0 results in Mathlib

Foundations

Depth 54 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.Functor.IsCocontinuousCategoryTheory.Functor.HasPointwiseRightKanExtension