English
Under a sheaf condition, amalgamation commutes with postcomposition by P.map I.f.op: (hP.amalgamate S x hx) ≫ P.map I.f.op = x I.
Русский
При условииshеaf, амальгамация сохраняет композицию с P.map I.f.op: (hP.amalgamate S x hx) ≫ P.map I.f.op = x I.
LaTeX
$$$hP.amalgamate S x hx \\;\\; \\Rightarrow\\; hP.amalgamate S x hx \\mapsto P.map I.f.op = x I$$$
Lean4
/-- A presheaf `P` is a sheaf for the Grothendieck topology `J` iff for every covering sieve
`S` of `J`, the natural cone associated to `P` and `S` is a limit cone. -/
theorem isSheaf_iff_isLimit :
IsSheaf J P ↔ ∀ ⦃X : C⦄ (S : Sieve X), S ∈ J X → Nonempty (IsLimit (P.mapCone S.arrows.cocone.op)) :=
⟨fun h _ S hS => (isLimit_iff_isSheafFor P S).2 fun E => h E.unop S hS, fun h E _ S hS =>
(isLimit_iff_isSheafFor P S).1 (h S hS) (op E)⟩
