Theorems · Theorem · category theory

CategoryTheory.Equalizer.Presieve.isSheafFor_singleton_iff

∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] {F : CategoryTheory.Functor Cᵒᵖ (Type u_1)} {X Y : C}
  {f : X ⟶ Y} (c : CategoryTheory.Limits.PullbackCone f f) (hc : CategoryTheory.Limits.IsLimit c),
  CategoryTheory.Presieve.IsSheafFor F (CategoryTheory.Presieve.singleton f) ↔
    Nonempty (CategoryTheory.Limits.IsLimit (CategoryTheory.Limits.Fork.ofι (F.map f.op) ⋯))

The sheaf condition for a single morphism is the same as the canonical fork diagram being limiting.

Defined in

Mathlib.CategoryTheory.Sites.EqualizerSheafCondition

Cited by

1 results in Mathlib

Foundations

Depth 32 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.Category