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