English
If f: X → Y has pullbacks along f and epi-compatibility for pullback projections, then the pullback along f is a faithful functor.
Русский
Если у f: X → Y существуют указанные обратные каналы (pullbacks) и для каждого Z→Y карта pullback.fst гомоциклически эпимортизируется, то pullback f — это верная (faithful) функционор.
LaTeX
$$ (pullback f).Faithful $$
Lean4
/-- The pullback along an epi that's preserved under pullbacks is faithful.
This "preserved under pullbacks" condition is automatically satisfied in abelian categories:
```
example [Abelian C] [Epi f] : (pullback f).Faithful := inferInstance
```
-/
instance faithful_pullback {X Y : C} (f : X ⟶ Y) [HasPullbacksAlong f] [∀ Z (g : Z ⟶ Y), Epi (pullback.fst g f)] :
(pullback f).Faithful :=
by
have (Z : Over Y) : Epi ((mapPullbackAdj f).counit.app Z) := by
simp only [Functor.comp_obj, Functor.id_obj, mapPullbackAdj_counit_app]; infer_instance
exact (mapPullbackAdj f).faithful_R_of_epi_counit_app
