Theorems · Theorem · category theory
CategoryTheory.Sieve.downward_closed
∀ {C : Type u₁} [inst : CategoryTheory.Category.{v₁, u₁} C] {X : C} (self : CategoryTheory.Sieve X) {Y Z : C}
{f : Y ⟶ X}, self.arrows f → ∀ (g : Z ⟶ Y), self.arrows (CategoryTheory.CategoryStruct.comp g f)stability by precomposition
- Defined in
- Mathlib.CategoryTheory.Sites.Sieves
- Cited by
- 39 results in Mathlib
- Foundations
- Depth 4 from the axioms · uses no axioms
- Assumes
- CategoryTheory.Category
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites5
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- CategoryTheory.Categorystatement and proof · cited by 32,673
- Quiver.Homstatement · cited by 32,603
- CategoryTheory.CategoryStruct.compstatement · cited by 17,999
- CategoryTheory.Sievestatement and proof · cited by 552
- CategoryTheory.Sieve.arrowsstatement · cited by 446
Cited by40
Results whose statement or proof uses this declaration.
- CategoryTheory.Sieve.generate_le_iffproof · cited by 16
- CategoryTheory.Sieve.functor_galoisConnectionproof · cited by 7
- CategoryTheory.Presieve.compatible_iff_sieveCompatibleproof · cited by 6
- CategoryTheory.Sieve.pullbackArrows_commproof · cited by 6
- CategoryTheory.Presieve.FamilyOfElements.SieveCompatibleproof · cited by 5
- CategoryTheory.classifier_isSheafproof · cited by 4
- CategoryTheory.Functor.coverPreserving_restrictedTopologyproof · cited by 3
- CategoryTheory.Sieve.id_mem_iff_eq_topproof · cited by 3
- CategoryTheory.Adjunction.isCocontinuous_iff_coverPreservingproof · cited by 2
- CategoryTheory.Sieve.mem_ofArrows_iffproof · cited by 2
- CategoryTheory.Presieve.extend_restrictproof · cited by 2
- CategoryTheory.RanIsSheafOfIsCocontinuous.liftAux_mapproof · cited by 2