English
For a full and faithful functor F: C → D between cartesian-monoidal categories and G a group object of D, the essential image of F.mapGrp at G is equivalent to the essential image of F at the underlying object G.X.
Русский
Для полного и верного функторa F: C → D между карриентно-монойной категориями и G — групповым объектом D, экзистентная образность F.mapGrp на G эквивалентна образности F на базовом объекте G.X.
LaTeX
$$$ F.mapGrp.essImage G \iff F.essImage G.X $$$
Lean4
/-- The essential image of a full and faithful functor between cartesian-monoidal categories is the
same on group objects as on objects. -/
@[simp]
theorem essImage_mapGrp [F.Full] [F.Faithful] {G : Grp_ D} : F.mapGrp.essImage G ↔ F.essImage G.X
where
mp := by rintro ⟨H, ⟨e⟩⟩; exact ⟨H.X, ⟨(Grp_.forget _).mapIso e⟩⟩
mpr := by
rintro ⟨H, ⟨e⟩⟩
let : GrpObj (F.obj H) := .ofIso e.symm
let : GrpObj H := (FullyFaithful.ofFullyFaithful F).grpObj H
refine ⟨⟨H⟩, ⟨Grp_.mkIso e ?_ ?_⟩⟩ <;> simp
