ofRestrict_top_c — Mathlib

English

Restriction morphism along an open embedding is mono when the embedding is open.

Русский

Ограничение морфизма вдоль открытого вложения является моно, когда вложение открыто.

LaTeX

$$$\text{ofRestrict_mono}(X) \text{ holds for top embedding}$$$

Lean4

theorem ofRestrict_top_c (X : PresheafedSpace C) :
    (X.ofRestrict (Opens.isOpenEmbedding ⊤)).c =
      eqToHom
        (by
          rw [restrict_top_presheaf, ← Presheaf.Pushforward.comp_eq]
          tauto) :=
  by
  /- another approach would be to prove the left-hand side
         is a natural isomorphism, but I encountered a universe
         issue when `apply NatIso.isIso_of_isIso_app`. -/

  ext
  dsimp [ofRestrict]
  erw [eqToHom_map, eqToHom_app]
  simp

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