English
Let F : C ⥤ FintypeCat be a fiber functor. Then the induced functor to finite Aut F-sets, functorToAction F, is full; i.e., for any objects X,Y in C and any morphism between the finite Aut F-sets F.obj X and F.obj Y, there exists a morphism g : X ⟶ Y in C whose image under functorToAction F equals that morphism.
Русский
Пусть F: C ⥤ FintypeCat — фибер-функтор. Тогда индуцированный функтор в конечные множества с действием группы Aut F, названный functorToAction F, полно кого: для любых объектов X,Y в C и любого гомоморфизма между Aut F-множестами F.obj X и F.obj Y существует морфизм g : X ⟶ Y в C, Such that его образ через functorToAction F совпадает с данным гомоморфизмом.
LaTeX
$$$\forall X,Y$ in $\mathcal{C}$,\quad \mathrm{Hom}_{\mathrm{Aut}\,F\text{-Set}}(F(X),F(Y)) = \mathrm{Im}\Bigl( \mathrm{Hom}_{\mathcal{C}}(X,Y) \to \mathrm{Hom}_{\mathrm{Aut}\,F\text{-Set}}(F(X),F(Y))\Bigr).$$$
Lean4
/-- The by a fiber functor `F : C ⥤ FintypeCat` induced functor `functorToAction F` to
finite `Aut F`-sets is full. -/
instance functorToAction_full : Functor.Full (functorToAction F) where
map_surjective {X Y}
f :=
by
let u : (functorToAction F).obj X ⟶ (functorToAction F).obj X ⨯ (functorToAction F).obj Y := prod.lift (𝟙 _) f
let i : (functorToAction F).obj X ⟶ (functorToAction F).obj (X ⨯ Y) :=
u ≫ (PreservesLimitPair.iso (functorToAction F) X Y).inv
have : Mono i := by
have : Mono (u ≫ prod.fst) := prod.lift_fst (𝟙 _) f ▸ inferInstance
have : Mono u := mono_of_mono u prod.fst
apply mono_comp u _
obtain ⟨Z, g, v, _, hvgi⟩ := exists_lift_of_mono F (Limits.prod X Y) ((functorToAction F).obj X) i
let ψ : Z ⟶ X := g ≫ prod.fst
have hgvi : (functorToAction F).map g = v.inv ≫ i := by simp [← hvgi]
have : IsIso ((functorToAction F).map ψ) :=
by
simp only [map_comp, hgvi, Category.assoc, ψ]
have : IsIso (i ≫ (functorToAction F).map prod.fst) :=
by
suffices h : IsIso (𝟙 ((functorToAction F).obj X)) by simpa [i, u]
infer_instance
apply IsIso.comp_isIso
have : IsIso ψ := isIso_of_reflects_iso ψ (functorToAction F)
use inv ψ ≫ g ≫ prod.snd
rw [← cancel_epi ((functorToAction F).map ψ)]
ext (z : F.obj Z)
simp [-FintypeCat.comp_apply, -Action.comp_hom, i, u, ψ, hgvi]
