English
In a category with a projective generator G, and IsSeparator, the induced map from morphisms G → X under the preadditive Yoneda construction is surjective for all X.
Русский
В категории с проектором-генератором G и разделителем, отображение, полученное через прегруппу предадит Yoneda, образно сюръектно на все X.
LaTeX
$$$\text{preadditiveCoyonedaObj } G\text{ map_surjective}$$$
Lean4
theorem preadditiveCoyonedaObj_map_surjective {G : C} [Projective G] (hG : IsSeparator G) {X : C} (p : G ⟶ X) [Epi p]
{Y : C} : Function.Surjective ((preadditiveCoyonedaObj G).map : (X ⟶ Y) → _) :=
by
rw [← Functor.coe_mapAddHom, ← AddCommGrpCat.hom_ofHom (preadditiveCoyonedaObj G).mapAddHom, ←
AddCommGrpCat.epi_iff_surjective]
let cm : ShortComplex C := ⟨kernel.ι p, p, by simp⟩
have exact : cm.Exact := ShortComplex.exact_of_f_is_kernel _ (kernelIsKernel _)
have mono : Mono cm.op.f := by dsimp [cm]; infer_instance
let φ := preadditiveCoyonedaObj G
have faithful : φ.Faithful := by rwa [← isSeparator_iff_faithful_preadditiveCoyonedaObj]
apply ShortComplex.epi_of_mono_of_epi_of_mono (cm.op.mapNatTrans (preadditiveYonedaMap _ _))
· exact exact.op.map_of_mono_of_preservesKernel _ mono inferInstance
· simp only [ShortComplex.map_f]
infer_instance
· suffices φ.map.Surjective by simpa [AddCommGrpCat.epi_iff_surjective, Functor.coe_mapAddHom]
exact fun f => ⟨f (𝟙 G), by cat_disch⟩
· simp [AddCommGrpCat.mono_iff_injective, Functor.coe_mapAddHom, Functor.map_injective]
