pullback_cone_of_left_condition — Mathlib

English

The stalk map is an isomorphism for open immersions in presheafed spaces under colimits assumptions.

Русский

Сталкерное отображение является изоморфизмом для открытых вложений в прешефед пространства при наличии колимитов.

LaTeX

$$IsIso (AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.stalkMap f x)$$

Lean4

theorem pullback_cone_of_left_condition : pullbackConeOfLeftFst f g ≫ f = Y.ofRestrict _ ≫ g := by
  -- Porting note (https://github.com/leanprover-community/mathlib4/issues/11041): `ext` did not pick up `NatTrans.ext`

  refine PresheafedSpace.Hom.ext _ _ ?_ <| NatTrans.ext <| funext fun U => ?_
  · simpa using pullback.condition
  · induction U
    rw [NatTrans.comp_app]
    dsimp only [comp_c_app, unop_op, Functor.whiskerRight_app, pullbackConeOfLeftFst]
      -- simp only [ofRestrict_c_app, NatTrans.comp_app]

    simp only [app_invApp_assoc, eqToHom_app, Category.assoc, NatTrans.naturality_assoc]
    erw [← Y.presheaf.map_comp, ← Y.presheaf.map_comp]
    congr 1

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