English
There is a canonical unit morphism from the domain to the constant functor object in the forgetful setting.
Русский
Существует каноническая единичная морфизм от области к константному объекту в забывающем контексте.
LaTeX
$$$1_{X} : \mathrm{id} \Rightarrow \mathrm{Hom}$$$
Lean4
/-- Given a submonoid at each of the stalks, we may define a submonoid presheaf consisting of
sections whose restriction onto each stalk falls in the given submonoid. -/
@[simps]
noncomputable def submonoidPresheafOfStalk (S : ∀ x : X, Submonoid (F.stalk x)) : F.SubmonoidPresheaf
where
obj U := ⨅ x : U.unop, Submonoid.comap (F.germ U.unop x.1 x.2).hom (S x)
map {U V}
i := by
intro s hs
simp only [Submonoid.mem_comap, Submonoid.mem_iInf] at hs ⊢
intro x
change (F.map i.unop.op ≫ F.germ V.unop x.1 x.2) s ∈ _
rw [F.germ_res]
exact hs ⟨_, i.unop.le x.2⟩
