Theorems · Theorem · category theory

CategoryTheory.Equalizer.Presieve.Arrows.compatible_iff_of_small

∀ {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]
  (x : CategoryTheory.Equalizer.Presieve.Arrows.FirstObj P X),
  CategoryTheory.Presieve.Arrows.Compatible P π
      ((equivShrink ((i : I) → P.obj (Opposite.op (X i)))).symm
        ((CategoryTheory.ConcreteCategory.hom
            (CategoryTheory.Limits.Types.Small.productIso fun i => P.obj (Opposite.op (X i))).hom)
          x)) ↔
    (CategoryTheory.ConcreteCategory.hom (CategoryTheory.Equalizer.Presieve.Arrows.firstMap P X π)) x =
      (CategoryTheory.ConcreteCategory.hom (CategoryTheory.Equalizer.Presieve.Arrows.secondMap P X π)) x

Version of CategoryTheory.Equalizer.Presieve.Arrows.compatible_iff for a small indexing type.

Defined in

Mathlib.CategoryTheory.Sites.EqualizerSheafCondition

Cited by

0 results in Mathlib

Foundations

Depth 36 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategorySmallCategoryTheory.Presieve.HasPairwisePullbacks