Theorems · Theorem · category theory

CategoryTheory.Sheaf.isConstant_iff_isIso_counit_app

∀ {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]
  [(CategoryTheory.constantSheaf J D).Faithful] [(CategoryTheory.constantSheaf J D).Full] (F : CategoryTheory.Sheaf J D)
  {T : C} (hT : CategoryTheory.Limits.IsTerminal T),
  CategoryTheory.Sheaf.IsConstant J F ↔ CategoryTheory.IsIso ((CategoryTheory.constantSheafAdj J D hT).counit.app F)

If the constant sheaf functor is fully faithful, then a sheaf is constant if and only if the counit of the constant sheaf adjunction applied to it is an isomorphism.

Defined in

Mathlib.CategoryTheory.Sites.ConstantSheaf

Cited by

5 results in Mathlib

Foundations

Depth 42 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.HasWeakSheafifyCategoryTheory.Functor.FaithfulCategoryTheory.Functor.Full