Theorems · Theorem · category theory

CategoryTheory.Sheaf.isConstant_iff_of_equivalence

∀ {C : Type u_1} [inst : CategoryTheory.Category.{v_1, u_1} C] (J : CategoryTheory.GrothendieckTopology C)
  {D : Type u_2} [inst_1 : CategoryTheory.Category.{v_2, u_2} D] [inst_2 : CategoryTheory.HasWeakSheafify J D]
  {C' : Type u_3} [inst_3 : CategoryTheory.Category.{v_3, u_3} C'] (K : CategoryTheory.GrothendieckTopology C')
  [inst_4 : CategoryTheory.HasWeakSheafify K D] (G : CategoryTheory.Functor C C')
  [inst_5 : ∀ (X : C'ᵒᵖ), CategoryTheory.Limits.HasLimitsOfShape (CategoryTheory.StructuredArrow X G.op) D]
  [inst_6 : CategoryTheory.Functor.IsDenseSubsite J K G] {T : C} (hT : CategoryTheory.Limits.IsTerminal T)
  (hT' : CategoryTheory.Limits.IsTerminal (G.obj T)) (F : CategoryTheory.Sheaf K D),
  CategoryTheory.Sheaf.IsConstant J ((CategoryTheory.Functor.IsDenseSubsite.sheafEquiv J K G D).inverse.obj F) ↔
    CategoryTheory.Sheaf.IsConstant K F

The property of a sheaf of being constant is invariant under equivalence of sheaf categories.

Defined in

Mathlib.CategoryTheory.Sites.ConstantSheaf

Cited by

2 results in Mathlib

Foundations

Depth 78 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.HasWeakSheafifyCategoryTheory.CategoryCategoryTheory.HasWeakSheafifyCategoryTheory.Limits.HasLimitsOfShapeCategoryTheory.Functor.IsDenseSubsite