isSeparated_iff_subsingleton — Mathlib

English

The amalgamate operation in a sheaf yields an E → P(X) morphism satisfying the gluing axioms.

Русский

Операция амальгамирования в шейфе дает морфизм E → P(X), удовлетворяющий axioms glueing.

LaTeX

$$$\\text{IsSheaf.amalgamate}$$$

Lean4

/-- A presheaf `P` is separated for the Grothendieck topology `J` iff for every covering sieve
    `S` of `J`, the natural cone associated to `P` and `S` admits at most one morphism from every
    cone in the same category. -/
theorem isSeparated_iff_subsingleton :
    (∀ E : A, Presieve.IsSeparated J (P ⋙ coyoneda.obj (op E))) ↔
      ∀ ⦃X : C⦄ (S : Sieve X), S ∈ J X → ∀ c, Subsingleton (c ⟶ P.mapCone S.arrows.cocone.op) :=
  ⟨fun h _ S hS => (subsingleton_iff_isSeparatedFor P S).2 fun E => h E.unop S hS, fun h E _ S hS =>
    (subsingleton_iff_isSeparatedFor P S).1 (h S hS) (op E)⟩

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