Theorems · Theorem · category theory

CategoryTheory.Equalizer.Presieve.Arrows.sheaf_condition

∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] (P : CategoryTheory.Functor Cᵒᵖ (Type w)) {B : C} {I : Type t}
  [inst_1 : Small.{w, t} I] (X : I → C) (π : (i : I) → X i ⟶ B)
  [inst_2 : (CategoryTheory.Presieve.ofArrows X π).HasPairwisePullbacks],
  CategoryTheory.Presieve.IsSheafFor P (CategoryTheory.Presieve.ofArrows X π) ↔
    Nonempty
      (CategoryTheory.Limits.IsLimit
        (CategoryTheory.Limits.Fork.ofι (CategoryTheory.Equalizer.Presieve.Arrows.forkMap P X π) ⋯))

P is a sheaf for Presieve.ofArrows X π, iff the fork given by w is an equalizer.

Defined in

Mathlib.CategoryTheory.Sites.EqualizerSheafCondition

Cited by

1 results in Mathlib

Foundations

Depth 38 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategorySmallCategoryTheory.Presieve.HasPairwisePullbacks