English
The presheaf of monomorphisms on C maps into monomorphisms on Type; in particular, presheaf of monomorphisms on C is contained in monomorphisms on opposite Category.
Русский
Пресхраф монооморфизмов C подсказывает монооморфизмы в Type; в частности, presheaf монооморфизмов в Opposite C лежит в монооморфизмах.
LaTeX
$$$\text{presheaf}(\text{monomorphisms } C) \le \text{monomorphisms}_{Type}$$$
Lean4
/-- Morphisms satisfying `(monomorphism C).presheaf` are in particular monomorphisms. -/
theorem presheaf_monomorphisms_le_monomorphisms : (monomorphisms C).presheaf ≤ monomorphisms _ := fun F G f hf ↦
by
suffices ∀ {X : C} {a b : yoneda.obj X ⟶ F}, a ≫ f = b ≫ f → a = b from
⟨fun _ _ h ↦ hom_ext_yoneda (fun _ _ ↦ this (by simp only [assoc, h]))⟩
intro X a b h
suffices hf.rep.lift (g := a ≫ f) a (𝟙 X) (by simp) = hf.rep.lift b (𝟙 X) (by simp [← h]) by
simpa using
yoneda.congr_map this =≫
(hf.rep.fst (a ≫ f))
-- This follows from the fact that the induced maps `hf.rep.pullback g ⟶ X` are mono.
have : Mono (hf.rep.snd (a ≫ f)) := hf.property_snd (a ≫ f)
simp only [← cancel_mono (hf.rep.snd (a ≫ f)), lift_snd]
