iso_of_restrict_iso — Mathlib

English

Pushforward_mem_iff asserts that pushforward memory of K along G matches the membership criterion in J: S.functorPushforward G ∈ K on (G.obj X) is equivalent to S ∈ J on X.

Русский

Утверждение PushforwardMem: S.functorPushforward G ∈ K на (G.obj X) эквивалентно S ∈ J на X.

LaTeX

$$$S.\mathrm{functorPushforward}\ G \in K\; (G.obj X) \ \Leftrightarrow\; S \in J\; X.$$$

Lean4

/-- Given a locally-full and cover-dense functor `G` and a natural transformation of sheaves
`α : ℱ ⟶ ℱ'`, if the pullback of `α` along `G` is iso, then `α` is also iso.
-/
theorem iso_of_restrict_iso {ℱ ℱ' : Sheaf K A} (α : ℱ ⟶ ℱ') (i : IsIso (whiskerLeft G.op α.val)) : IsIso α :=
  by
  convert (sheafIso (asIso (whiskerLeft G.op α.val))).isIso_hom using 1
  ext1
  apply (sheafHom_eq _ _).symm

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