isSheaf_iff_isLimit_pretopology — Mathlib

English

There is an instance making the hom component of OppositeShift.natIsoComp A F G into a NatTrans.CommShift between F.CommShift A and G.CommShift A, i.e., the shift-commutativity condition holds.

Русский

Существует инстанс, превращающий гом-компонент OppositeShift.natIsoComp A F G в NatTrans.CommShift между F.CommShift A и G.CommShift A; условие shift-comm выполняется.

LaTeX

$$$[F.CommShift A][G.CommShift A]\; :\; \mathrm{NatTrans.CommShift}((\mathrm{OppositeShift.natIsoComp}\ a\ F\ G).\mathrm{hom} A)$$$

Lean4

/-- A presheaf `P` is a sheaf for the Grothendieck topology generated by a pretopology `K`
    iff for every covering presieve `R` of `K`, the natural cone associated to `P` and
    `Sieve.generate R` is a limit cone. -/
theorem isSheaf_iff_isLimit_pretopology [HasPullbacks C] (K : Pretopology C) :
    IsSheaf K.toGrothendieck P ↔
      ∀ ⦃X : C⦄ (R : Presieve X), R ∈ K X → Nonempty (IsLimit (P.mapCone (generate R).arrows.cocone.op)) :=
  by
  dsimp [IsSheaf]
  simp_rw [isSheaf_pretopology]
  exact
    ⟨fun h X R hR => (isLimit_iff_isSheafFor_presieve P R).2 fun E => h E.unop R hR, fun h E X R hR =>
      (isLimit_iff_isSheafFor_presieve P R).1 (h R hR) (op E)⟩

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