English
There is a canonical monoid hom from a profinite group P to the limit of its quotients by open normal subgroups (filtered by inclusion). It sends p to the family of images in each quotient, consistently.
Русский
Существует канонический моноид-хомоморфизм from P to предел его частичных частичных фактор-групп; он отправляет p в семейство образов в каждом фактор-группе.
LaTeX
$$$P \to \mathrm{limit}(P\!\to\mathrm{FiniteQuotients}\!\cdot \mathrm{forget}_{\mathrm{FiniteGrp}\to \mathrm{ProfiniteGrp}})$$$
Lean4
/-- The `MonoidHom` from a profinite group `P` to the projective limit of its quotients by
open normal subgroups ordered by inclusion -/
def toLimit_fun (P : ProfiniteGrp.{u}) : P →* limit (toFiniteQuotientFunctor P ⋙ forget₂ FiniteGrp ProfiniteGrp)
where
toFun p := ⟨fun _ => QuotientGroup.mk p, fun _ ↦ fun _ _ ↦ rfl⟩
map_one' := Subtype.val_inj.mp rfl
map_mul' _ _ := Subtype.val_inj.mp rfl
