Theorems · Theorem · category theory
CategoryTheory.le_topology_of_closedSieves_isSheaf
∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] {J₁ J₂ : CategoryTheory.GrothendieckTopology C},
CategoryTheory.Presieve.IsSheaf J₁ (CategoryTheory.Functor.closedSieves J₂).toFunctor → J₁ ≤ J₂If presheaf of J₁-closed sieves is a J₂-sheaf then J₁ ≤ J₂. Note the converse is true by
classifier_isSheaf and isSheaf_of_le.
- Defined in
- Mathlib.CategoryTheory.Sites.Closed
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 40 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- CategoryTheory.Category
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites10
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coeproof · cited by 62,936
- CategoryTheory.Categorystatement and proof · cited by 32,673
- Oppositestatement · cited by 8,081
- CategoryTheory.GrothendieckTopologystatement and proof · cited by 1,415
- CategoryTheory.Sieveproof · cited by 552
- CategoryTheory.Subfunctor.toFunctorstatement and proof · cited by 90
- CategoryTheory.Presieve.IsSheafstatement and proof · cited by 66
- CategoryTheory.Functor.sievesstatement · cited by 20
- CategoryTheory.Functor.closedSievesstatement and proof · cited by 11
- CategoryTheory.GrothendieckTopology.mem_iff_isSheafFor_closedSievesproof · cited by 2
Cited by2
Results whose statement or proof uses this declaration.
- CategoryTheory.Precoverage.Generates.toGrothendieck_eqproof · cited by 1
- CategoryTheory.topology_eq_iff_same_sheavesproof · cited by 0