Theorems · Theorem · category theory
CategoryTheory.Sieve.forallYonedaIsSheaf_iff_colimit
∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] {X : C} (S : CategoryTheory.Sieve X),
(∀ (W : C), CategoryTheory.Presieve.IsSheafFor (CategoryTheory.yoneda.obj W) S.arrows) ↔
Nonempty (CategoryTheory.Limits.IsColimit S.arrows.cocone)The base of a sieve S is a colimit of S iff all Yoneda-presheaves satisfy
the sheaf condition for S.
- Cited by
- 4 results in Mathlib
- Foundations
- Depth 32 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.
Cites42
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
- Quiver.Homproof · cited by 32,603
- CategoryTheory.Functor.objstatement and proof · cited by 19,642
- CategoryTheory.CategoryStruct.compproof · cited by 17,999
- CategoryTheory.Functorstatement · cited by 16,252
- CategoryTheory.Functor.mapproof · cited by 8,698
- Oppositestatement · cited by 8,081
- CategoryTheory.NatTrans.appproof · cited by 7,406
- CategoryTheory.ConcreteCategory.homproof · cited by 4,022
- CategoryTheory.Functor.idproof · cited by 3,333
- CategoryTheory.Discreteproof · cited by 2,447
Cited by4
Results whose statement or proof uses this declaration.
- CategoryTheory.EffectiveEpiFamily.transitive_of_finiteproof · cited by 1
- CategoryTheory.coherentTopology.isSheaf_yoneda_objproof · cited by 1
- CategoryTheory.Sieve.EffectiveEpimorphic.iff_forall_isSheafFor_yonedaproof · cited by 0
- CategoryTheory.regularTopology.isSheaf_yoneda_objproof · cited by 0