Theorems · Theorem · category theory

CategoryTheory.Equalizer.Presieve.compatible_iff

∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] (P : CategoryTheory.Functor Cᵒᵖ (Type (max v u))) {X : C}
  (R : CategoryTheory.Presieve X) [inst_1 : R.HasPairwisePullbacks] (x : CategoryTheory.Equalizer.FirstObj P R),
  CategoryTheory.Presieve.FamilyOfElements.Compatible
      ((CategoryTheory.ConcreteCategory.hom (CategoryTheory.Equalizer.firstObjEqFamily P R).hom) x) ↔
    (CategoryTheory.ConcreteCategory.hom (CategoryTheory.Equalizer.Presieve.firstMap P R)) x =
      (CategoryTheory.ConcreteCategory.hom (CategoryTheory.Equalizer.Presieve.secondMap P R)) x

The family of elements given by x : FirstObj P S is compatible iff firstMap and secondMap map it to the same point.

Defined in

Mathlib.CategoryTheory.Sites.EqualizerSheafCondition

Cited by

0 results in Mathlib

Foundations

Depth 35 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.Presieve.HasPairwisePullbacks