isConstant_iff_isIso_counit_app — Mathlib

English

A sheaf F is constant if and only if the counit of the constant-sheaf adjunction at F is an isomorphism.

Русский

У касательной F константности равносильно тому, что у контргруппового сопоставления counit является изоморфизмом.

LaTeX

$$$\mathrm{IsConstant}(J,F) \;\iff\; \mathrm{IsIso}\big((\text{constantSheafAdj } J D h_T).\text{counit.app }F\big)$$$

Lean4

/-- 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.
-/
theorem isConstant_iff_isIso_counit_app [(constantSheaf J D).Faithful] [(constantSheaf J D).Full] (F : Sheaf J D)
    {T : C} (hT : IsTerminal T) : IsConstant J F ↔ (IsIso <| (constantSheafAdj J D hT).counit.app F) :=
  ⟨fun _ ↦ inferInstance, fun _ ↦ ⟨_, ⟨asIso <| (constantSheafAdj J D hT).counit.app F⟩⟩⟩

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