pushforwardPullbackAdjunction_unit_pullback_map_germToPullbackStalk

English

A simplified version showing that germToPullbackStalk acts as a morphism from the pullback to the stalk with no extraneous data.

Русский

Упрощённая версия: germToPullbackStalk действует как морфизм из pullback в stalk без лишних данных.

LaTeX

$$germToPullbackStalk is a universal morphism from pullback to stalk.$$

Lean4

@[reassoc (attr := simp)]
theorem pushforwardPullbackAdjunction_unit_pullback_map_germToPullbackStalk (f : X ⟶ Y) (F : Y.Presheaf C) (U : Opens X)
    (x : X) (hx : x ∈ U) (V : Opens Y) (hV : U ≤ (Opens.map f).obj V) :
    ((pushforwardPullbackAdjunction C f).unit.app F).app (op V) ≫
        ((pullback C f).obj F).map (homOfLE hV).op ≫ germToPullbackStalk C f F U x hx =
      F.germ _ (f x) (hV hx) :=
  by
  simpa [pushforwardPullbackAdjunction] using
    ((Opens.map f).op.isPointwiseLeftKanExtensionLeftKanExtensionUnit F (op U)).fac _
      (CostructuredArrow.mk (homOfLE hV).op)

Похожие утверждения