to_skyscraper_fromStalk — Mathlib

English

For any f: F stalk p0 ⟶ c, the induced natural transformation fromSkyscraperPresheaf p0 (fromStalk p0 f) equals f.

Русский

Для любого f: F_STALK_p0 ⟶ c индуцированное преобразование fromStalk к skyscraperPresheaf p0 f равно f.

LaTeX

$$$toSkyscraperPresheaf p_0 (fromStalk _ f) = f$$$

Lean4

theorem to_skyscraper_fromStalk {𝓕 : Presheaf C X} {c : C} (f : 𝓕 ⟶ skyscraperPresheaf p₀ c) :
    toSkyscraperPresheaf p₀ (fromStalk _ f) = f :=
  by
  apply NatTrans.ext
  ext U
  dsimp
  split_ifs with h
  · rw [← Category.assoc, germ_fromStalk, Category.assoc, eqToHom_trans, eqToHom_refl, Category.comp_id]
  · exact ((if_neg h).symm.ndrec terminalIsTerminal).hom_ext ..

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