Theorems · Theorem · category theory

CategoryTheory.Equivalence.preregular_isSheaf_iff_of_essentiallySmall

∀ {C : Type u_1} [inst : CategoryTheory.Category.{v_1, u_1} C] [inst_1 : CategoryTheory.Preregular C] (A : Type u_3)
  [inst_2 : CategoryTheory.Category.{v_3, u_3} A] [inst_3 : CategoryTheory.EssentiallySmall.{u_4, v_1, u_1} C]
  (F : CategoryTheory.Functor Cᵒᵖ A),
  CategoryTheory.Presheaf.IsSheaf (CategoryTheory.regularTopology C) F ↔
    CategoryTheory.Presheaf.IsSheaf (CategoryTheory.regularTopology (CategoryTheory.SmallModel.{u_4, v_1, u_1} C))
      ((CategoryTheory.equivSmallModel C).inverse.op.comp F)

The regular sheaf condition on an essentially small site can be checked after precomposing with the equivalence with a small category.

Defined in

Mathlib.CategoryTheory.Sites.Coherent.Equivalence

Cited by

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Foundations

Depth 88 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.PreregularCategoryTheory.CategoryCategoryTheory.EssentiallySmall