English
There is a natural isomorphism of presheaves between the commutative-structure presheaf and the Type-valued presheaf after forgetting the ring structure. Concretely, the presheaf structurePresheafInCommRing 𝒜, composed with the forgetful functor to sets, is naturally isomorphic to the Type-valued structure presheaf (structureSheafInType 𝒜).1.
Русский
Существет естественное изоморфизмопредsheaf между структурной предsheaf в CommRing и предsheaf типов после забывания структуры кольца; т.е. структураPresheafInCommRing 𝒜, композиция с забыванием равна естественному изоморфизму к структуре, осуществляемой полем (structureSheafInType 𝒜).1.
LaTeX
$$$\text{structurePresheafInCommRing } 𝒜 \circ \text{forget} \cong (\text{structureSheafInType } 𝒜).1$$$
Lean4
/-- Some glue, verifying that the structure presheaf valued in `CommRing` agrees with the
`Type`-valued structure presheaf. -/
def structurePresheafCompForget : structurePresheafInCommRing 𝒜 ⋙ forget CommRingCat ≅ (structureSheafInType 𝒜).1 :=
NatIso.ofComponents (fun _ => Iso.refl _) (by cat_disch)
