smoothPresheafCommRing — Mathlib

English

The presheaf of smooth functions to R forms a sheaf of rings with the gluing condition inherited from the smooth functions.

Русский

Прешеф гладких функций в R образует sheaf колец, удовлетворяющий условию сцепления.

LaTeX

$$smoothSheafRing IM I M R$$

Lean4

/-- The presheaf of smooth functions from `M` to `R`, for `R` a smooth commutative ring, as a
presheaf of commutative rings. -/
def smoothPresheafCommRing : TopCat.Presheaf CommRingCat.{u} (TopCat.of M) :=
  { obj := fun U ↦ CommRingCat.of ((smoothSheaf IM I M R).presheaf.obj U)
    map := fun h ↦ CommRingCat.ofHom <| ContMDiffMap.restrictRingHom IM I R <| CategoryTheory.leOfHom h.unop
    map_id := fun _ ↦ rfl
    map_comp := fun _ _ ↦ rfl }

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