toPushforwardStalk_comp — Mathlib

English

The canonical relation between pushforward stalk and stalk maps composes correctly with stalk functors and sheaf morphisms.

Русский

Каноническое соответствие между pushforward stalk и stalk-map корректно компонуется с феноменами стока и морфизмами sheaf.

LaTeX

$$$f \\;\\gg\\; \\mathrm{StructureSheaf.toPushforwardStalk}\, f\\; p = \\mathrm{StructureSheaf.toStalk}\, R\\, p \\;\\gg\\; (\\mathrm{TopCat.Presheaf.stalkFunctor}_.map (\\mathrm{Spec.sheafedSpaceMap}\, f).c).$$$

Lean4

@[reassoc]
theorem toPushforwardStalk_comp :
    f ≫ StructureSheaf.toPushforwardStalk f p =
      StructureSheaf.toStalk R p ≫ (TopCat.Presheaf.stalkFunctor _ _).map (Spec.sheafedSpaceMap f).c :=
  by
  rw [StructureSheaf.toStalk, Category.assoc, TopCat.Presheaf.stalkFunctor_map_germ]
  exact Spec_Γ_naturality_assoc f _

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