Theorems · Theorem · category theory

CategoryTheory.regularTopology.equalizerCondition_iff_isSheaf

∀ {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] (F : CategoryTheory.Functor Cᵒᵖ D)
  [inst_2 : CategoryTheory.Preregular C]
  [∀ {Y X : C} (f : Y ⟶ X) [CategoryTheory.EffectiveEpi f], CategoryTheory.Limits.HasPullback f f],
  CategoryTheory.regularTopology.EqualizerCondition F ↔
    CategoryTheory.Presheaf.IsSheaf (CategoryTheory.regularTopology C) F

Defined in: Mathlib.CategoryTheory.Sites.Coherent.RegularSheaves
Cited by: 1 results in Mathlib
Foundations: Depth 46 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: CategoryTheory.Category CategoryTheory.Category CategoryTheory.Preregular CategoryTheory.Limits.HasPullback

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Cites23

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 and proof · cited by 32,603
CategoryTheory.Functorstatement and proof · cited by 16,252
Oppositestatement and proof · cited by 8,081
CategoryTheory.Presheaf.IsSheafstatement · cited by 991
CategoryTheory.Limits.IsLimitproof · cited by 664
CategoryTheory.Presieveproof · cited by 449
CategoryTheory.Sieve.arrowsproof · cited by 446
CategoryTheory.Limits.HasPullbackstatement and proof · cited by 434
CategoryTheory.Precoverage.coveringsproof · cited by 194
CategoryTheory.Presieve.ofArrowsproof · cited by 150
CategoryTheory.Functor.mapConeproof · cited by 147

Cited by1

Results whose statement or proof uses this declaration.

CategoryTheory.Presheaf.isSheaf_iff_preservesFiniteProducts_and_equalizerConditionproof · cited by 5