comp_toGlued_eq — Mathlib

English

If a : U → X_i and b : U → X_j satisfy a certain compatibility with f_i and f_j, then they glue to the same morphism into the glued object.

Русский

Если a : U → X_i и b : U → X_j удовлетворяют совместимости с f_i и f_j, то они склейваются в одну морфизму в склейку.

LaTeX

$$If $a: U\\to X_i$, $b: U\\to X_j$, and $h: \\yoneda.map a = \\yoneda.map b$ after composing with $f_i$ and $f_j$, then $a\\,\\toGlued hf i = b\\,\\toGlued hf j$.$$

Lean4

theorem comp_toGlued_eq {U : Scheme} {i j : ι} (a : U ⟶ X i) (b : U ⟶ X j)
    (h : yoneda.map a ≫ f i = yoneda.map b ≫ f j) : a ≫ toGlued hf i = b ≫ toGlued hf j :=
  by
  rw [← (hf i).rep.lift'_fst a b h, assoc]
  conv_rhs => rw [← (hf i).rep.lift'_snd a b h, assoc]
  congr 1
  exact ((glueData hf).glue_condition i j).symm.trans (by simp [toGlued])

Похожие утверждения