English
Every Yoneda presheaf is a sheaf for the regular topology on C.
Русский
Каждый фаворит Йонада является шервисом для регулярной топологии на C.
LaTeX
$$isSheaf_yoneda_obj W$$
Lean4
/-- Every Yoneda-presheaf is a sheaf for the regular topology. -/
theorem isSheaf_yoneda_obj [Preregular C] (W : C) : Presieve.IsSheaf (regularTopology C) (yoneda.obj W) :=
by
rw [regularTopology, isSheaf_coverage]
intro X S ⟨_, hS⟩
have : S.regular := ⟨_, hS⟩
obtain ⟨Y, f, rfl, hf⟩ := Presieve.regular.single_epi (R := S)
have h_colim := isColimitOfEffectiveEpiStruct f hf.effectiveEpi.some
rw [← Sieve.generateSingleton_eq, ← Presieve.ofArrows_pUnit] at h_colim
intro x hx
let x_ext := Presieve.FamilyOfElements.sieveExtend x
have hx_ext := Presieve.FamilyOfElements.Compatible.sieveExtend hx
let S := Sieve.generate (Presieve.ofArrows (fun () ↦ Y) (fun () ↦ f))
obtain ⟨t, t_amalg, t_uniq⟩ := (Sieve.forallYonedaIsSheaf_iff_colimit S).mpr ⟨h_colim⟩ W x_ext hx_ext
refine ⟨t, ?_, ?_⟩
· convert
Presieve.isAmalgamation_restrict (Sieve.le_generate (Presieve.ofArrows (fun () ↦ Y) (fun () ↦ f))) _ _ t_amalg
exact (Presieve.restrict_extend hx).symm
· exact fun y hy ↦ t_uniq y <| Presieve.isAmalgamation_sieveExtend x y hy
