opensImagePreimageMap — Mathlib

English

The associativity of opensImagePreimageMap_app' ensures the commutativity holds after composing with further maps; the naturality is preserved.

Русский

Ассоциативность opensImagePreimageMap_app' гарантирует сохранение коммутативности после дополнительной композиции; сохраняется натуральность.

LaTeX

$$$ opensImagePreimageMap\\_app' \\;\\text{associativity} \\;$$

Lean4

/-- (Implementation). The map `Γ(𝒪_{U_i}, U) ⟶ Γ(𝒪_{U_j}, 𝖣.ι j ⁻¹' (𝖣.ι i '' U))` -/
def opensImagePreimageMap (i j : D.J) (U : Opens (D.U i).carrier) :
    (D.U i).presheaf.obj (op U) ⟶
      (D.U j).presheaf.obj (op <| (Opens.map (𝖣.ι j).base).obj ((D.ι_isOpenEmbedding i).isOpenMap.functor.obj U)) :=
  (D.f i j).c.app (op U) ≫
    (D.t j i).c.app _ ≫
      (D.f_open j i).invApp _ (unop _) ≫ (𝖣.U j).presheaf.map (eqToHom (D.ι_image_preimage_eq i j U)).op

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