sheafificationWhiskerLeftIso_hom_app

English

There is a canonical isomorphism, natural in P and F, relating the right whiskering of P with the sheafification of F to the whiskering of F with the sheafification of P.

Русский

Существует каноническое естественное изоморфизмo, естественное в P и F, связывающее правой взмах P с шейфификацией F и взмах F с шейфификацией P.

LaTeX

$$$J.sheafification\_Whisker\_Right(P) \cong P \text{ whiskered by } F \;\text{then sheafified}$$$

Lean4

@[simp]
theorem sheafificationWhiskerLeftIso_hom_app (P : Cᵒᵖ ⥤ D) (F : D ⥤ E)
    [∀ (F : D ⥤ E) (X : C), PreservesColimitsOfShape (J.Cover X)ᵒᵖ F]
    [∀ (F : D ⥤ E) (X : C) (W : J.Cover X) (P : Cᵒᵖ ⥤ D), PreservesLimit (W.index P).multicospan F] :
    (sheafificationWhiskerLeftIso J P).hom.app F = (J.sheafifyCompIso F P).hom :=
  by
  dsimp [sheafificationWhiskerLeftIso, sheafifyCompIso]
  rw [Category.comp_id]

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