File: Mathlib/Topology/Sheaves/Operations.lean

full_name: TopCat.Presheaf.SubmonoidPresheaf.obj
code: /-- A subpresheaf with a submonoid structure on each of the components. -/
structure SubmonoidPresheaf [∀ X : C, MulOneClass X] [∀ X Y : C, MonoidHomClass (X ⟶ Y) X Y]
    (F : X.Presheaf C) where
  obj : ∀ U, Submonoid (F.obj U)
  map : ∀ {U V : (Opens X)ᵒᵖ} (i : U ⟶ V), obj U ≤ (obj V).comap (F.map i)
goal: Submonoid ((CategoryTheory.forget C).obj (F.obj U))

full_name: TopCat.Presheaf.SubmonoidPresheaf
code: /-- A subpresheaf with a submonoid structure on each of the components. -/
structure SubmonoidPresheaf [∀ X : C, MulOneClass X] [∀ X Y : C, MonoidHomClass (X ⟶ Y) X Y]
    (F : X.Presheaf C) where
  obj : ∀ U, Submonoid (F.obj U)
  map : ∀ {U V : (Opens X)ᵒᵖ} (i : U ⟶ V), obj U ≤ (obj V).comap (F.map i)
goal: Type (max u_1 w)

full_name: TopCat.Presheaf.SubmonoidPresheaf.map
code: /-- A subpresheaf with a submonoid structure on each of the components. -/
structure SubmonoidPresheaf [∀ X : C, MulOneClass X] [∀ X Y : C, MonoidHomClass (X ⟶ Y) X Y]
    (F : X.Presheaf C) where
  obj : ∀ U, Submonoid (F.obj U)
  map : ∀ {U V : (Opens X)ᵒᵖ} (i : U ⟶ V), obj U ≤ (obj V).comap (F.map i)
goal: self.obj U ≤ Submonoid.comap (F.map i) (self.obj V)

full_name: TopCat.Presheaf.SubmonoidPresheaf.localizationPresheaf
code: /-- The localization of a presheaf of `CommRing`s with respect to a `SubmonoidPresheaf`. -/
protected noncomputable def SubmonoidPresheaf.localizationPresheaf : X.Presheaf CommRingCat 
goal: TopCat.Presheaf CommRingCat X

full_name: TopCat.Presheaf.instAlgebraObjCommRingCatForgetOppositeOpensαTopologicalSpaceCommRingLocalizationPresheaf
code: instance (U) : Algebra ((forget CommRingCat).obj (F.obj U)) (G.localizationPresheaf.obj U) 
goal: Algebra ((CategoryTheory.forget CommRingCat).obj (F.obj U)) ↑(G.localizationPresheaf.obj U)

full_name: TopCat.Presheaf.instIsLocalizationObjCommRingCatForgetOppositeOpensαTopologicalSpaceObjCommRingLocalizationPresheaf
code: instance (U) : IsLocalization (G.obj U) (G.localizationPresheaf.obj U) 
goal: IsLocalization (G.obj U) ↑(G.localizationPresheaf.obj U)

full_name: TopCat.Presheaf.SubmonoidPresheaf.toLocalizationPresheaf
code: /-- The map into the localization presheaf. -/
@[simps app]
def SubmonoidPresheaf.toLocalizationPresheaf : F ⟶ G.localizationPresheaf 
goal: F ⟶ G.localizationPresheaf

full_name: TopCat.Presheaf.SubmonoidPresheaf.toLocalizationPresheaf_app
code: /-- The map into the localization presheaf. -/
@[simps app]
def SubmonoidPresheaf.toLocalizationPresheaf : F ⟶ G.localizationPresheaf 
goal: G.toLocalizationPresheaf.app U = CommRingCat.ofHom (algebraMap (↑(F.obj U)) (Localization (G.obj U)))

full_name: TopCat.Presheaf.epi_toLocalizationPresheaf
code: instance epi_toLocalizationPresheaf : Epi G.toLocalizationPresheaf 
goal: CategoryTheory.Epi G.toLocalizationPresheaf

full_name: TopCat.Presheaf.submonoidPresheafOfStalk
code: /-- Given a submonoid at each of the stalks, we may define a submonoid presheaf consisting of
sections whose restriction onto each stalk falls in the given submonoid. -/
@[simps]
noncomputable def submonoidPresheafOfStalk (S : ∀ x : X, Submonoid (F.stalk x)) :
    F.SubmonoidPresheaf 
goal: F.SubmonoidPresheaf

full_name: TopCat.Presheaf.submonoidPresheafOfStalk_obj
code: /-- Given a submonoid at each of the stalks, we may define a submonoid presheaf consisting of
sections whose restriction onto each stalk falls in the given submonoid. -/
@[simps]
noncomputable def submonoidPresheafOfStalk (S : ∀ x : X, Submonoid (F.stalk x)) :
    F.SubmonoidPresheaf 
goal: (F.submonoidPresheafOfStalk S).obj U = ⨅ x, Submonoid.comap (F.germ x) (S ↑x)

full_name: TopCat.Presheaf.instInhabitedSubmonoidPresheafCommRingCat
code: noncomputable instance : Inhabited F.SubmonoidPresheaf 
goal: Inhabited F.SubmonoidPresheaf

full_name: TopCat.Presheaf.totalQuotientPresheaf
code: /-- The localization of a presheaf of `CommRing`s at locally non-zero-divisor sections. -/
noncomputable def totalQuotientPresheaf : X.Presheaf CommRingCat.{w} 
goal: TopCat.Presheaf CommRingCat X

full_name: TopCat.Presheaf.toTotalQuotientPresheaf
code: /-- The map into the presheaf of total quotient rings -/
noncomputable def toTotalQuotientPresheaf : F ⟶ F.totalQuotientPresheaf 
goal: F ⟶ F.totalQuotientPresheaf

full_name: TopCat.Presheaf.instEpiCommRingCatToTotalQuotientPresheaf
code: instance : Epi (toTotalQuotientPresheaf F) 
goal: CategoryTheory.Epi F.toTotalQuotientPresheaf

full_name: TopCat.Presheaf.instMonoCommRingCatToTotalQuotientPresheafPresheaf
code: instance (F : X.Sheaf CommRingCat.{w}) : Mono F.presheaf.toTotalQuotientPresheaf 
goal: CategoryTheory.Mono F.presheaf.toTotalQuotientPresheaf