Open immersions of structured spaces #

We say that a morphism of presheafed spaces f : X ⟶ Y is an open immersion if the underlying map of spaces is an open embedding f : X ⟶ U ⊆ Y, and the sheaf map Y(V) ⟶ f _* X(V) is an iso for each V ⊆ U.

Abbreviations are also provided for SheafedSpace, LocallyRingedSpace and Scheme.

Main definitions #

AlgebraicGeometry.PresheafedSpace.IsOpenImmersion: the Prop-valued typeclass asserting that a PresheafedSpace hom f is an open_immersion.
AlgebraicGeometry.IsOpenImmersion: the Prop-valued typeclass asserting that a Scheme morphism f is an open_immersion.
AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.isoRestrict: The source of an open immersion is isomorphic to the restriction of the target onto the image.
AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.lift: Any morphism whose range is contained in an open immersion factors though the open immersion.
AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.toSheafedSpace: If f : X ⟶ Y is an open immersion of presheafed spaces, and Y is a sheafed space, then X is also a sheafed space. The morphism as morphisms of sheafed spaces is given by toSheafedSpaceHom.
AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.toLocallyRingedSpace: If f : X ⟶ Y is an open immersion of presheafed spaces, and Y is a locally ringed space, then X is also a locally ringed space. The morphism as morphisms of locally ringed spaces is given by toLocallyRingedSpaceHom.

Main results #

AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.comp: The composition of two open immersions is an open immersion.
AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.ofIso: An iso is an open immersion.
AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.to_iso: A surjective open immersion is an isomorphism.
AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.stalk_iso: An open immersion induces an isomorphism on stalks.
AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.hasPullback_of_left: If f is an open immersion, then the pullback (f, g) exists (and the forgetful functor to TopCat preserves it).
AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.pullbackSndOfLeft: Open immersions are stable under pullbacks.
AlgebraicGeometry.SheafedSpace.IsOpenImmersion.of_stalk_iso An (topological) open embedding between two sheafed spaces is an open immersion if all the stalk maps are isomorphisms.

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class AlgebraicGeometry.PresheafedSpace.IsOpenImmersion {C : Type u} [CategoryTheory.Category.{v, u} C] {X Y : PresheafedSpace C} (f : X ⟶ Y) :

Prop

An open immersion of PresheafedSpaces is an open embedding f : X ⟶ U ⊆ Y of the underlying spaces, such that the sheaf map Y(V) ⟶ f _* X(V) is an iso for each V ⊆ U.

base_open : Topology.IsOpenEmbedding ⇑f.base
the underlying continuous map of underlying spaces from the source to an open subset of the target.
c_iso (U : TopologicalSpace.Opens ↑↑X) : CategoryTheory.IsIso (f.c.app (Opposite.op (⋯.functor.obj U)))
the underlying sheaf morphism is an isomorphism on each open subset

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@[reducible, inline]

abbrev AlgebraicGeometry.SheafedSpace.IsOpenImmersion {C : Type u} [CategoryTheory.Category.{v, u} C] {X Y : SheafedSpace C} (f : X ⟶ Y) :

Prop

A morphism of SheafedSpaces is an open immersion if it is an open immersion as a morphism of PresheafedSpaces

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AlgebraicGeometry.SheafedSpace.IsOpenImmersion f = AlgebraicGeometry.PresheafedSpace.IsOpenImmersion f

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@[reducible, inline]

abbrev AlgebraicGeometry.LocallyRingedSpace.IsOpenImmersion {X Y : LocallyRingedSpace} (f : X ⟶ Y) :

Prop

A morphism of LocallyRingedSpaces is an open immersion if it is an open immersion as a morphism of SheafedSpaces

Equations

AlgebraicGeometry.LocallyRingedSpace.IsOpenImmersion f = AlgebraicGeometry.SheafedSpace.IsOpenImmersion f.toHom

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@[reducible, inline]

abbrev AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.opensFunctor {C : Type u} [CategoryTheory.Category.{v, u} C] {X Y : PresheafedSpace C} (f : X ⟶ Y) [H : IsOpenImmersion f] :

CategoryTheory.Functor (TopologicalSpace.Opens ↑↑X) (TopologicalSpace.Opens ↑↑Y)

The functor Opens X ⥤ Opens Y associated with an open immersion f : X ⟶ Y.

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AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.opensFunctor f = ⋯.functor

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noncomputable def AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.isoRestrict {C : Type u} [CategoryTheory.Category.{v, u} C] {X Y : PresheafedSpace C} (f : X ⟶ Y) [H : IsOpenImmersion f] :

X ≅ Y.restrict ⋯

An open immersion f : X ⟶ Y induces an isomorphism X ≅ Y|_{f(X)}.

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@[simp]

theorem AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.isoRestrict_hom_c_app {C : Type u} [CategoryTheory.Category.{v, u} C] {X Y : PresheafedSpace C} (f : X ⟶ Y) [H : IsOpenImmersion f] (X✝ : (TopologicalSpace.Opens ↑↑(Y.restrict ⋯))ᵒᵖ) :

(isoRestrict f).hom.c.app X✝ = CategoryTheory.CategoryStruct.comp (f.c.app (Opposite.op ((opensFunctor f).obj (Opposite.unop X✝)))) (X.presheaf.map (CategoryTheory.eqToHom ⋯))

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@[simp]

theorem AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.isoRestrict_hom_ofRestrict {C : Type u} [CategoryTheory.Category.{v, u} C] {X Y : PresheafedSpace C} (f : X ⟶ Y) [H : IsOpenImmersion f] :

CategoryTheory.CategoryStruct.comp (isoRestrict f).hom (Y.ofRestrict ⋯) = f

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@[simp]

theorem AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.isoRestrict_hom_ofRestrict_assoc {C : Type u} [CategoryTheory.Category.{v, u} C] {X Y : PresheafedSpace C} (f : X ⟶ Y) [H : IsOpenImmersion f] {Z : PresheafedSpace C} (h : Y ⟶ Z) :

CategoryTheory.CategoryStruct.comp (isoRestrict f).hom (CategoryTheory.CategoryStruct.comp (Y.ofRestrict ⋯) h) = CategoryTheory.CategoryStruct.comp f h

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@[simp]

theorem AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.isoRestrict_inv_ofRestrict {C : Type u} [CategoryTheory.Category.{v, u} C] {X Y : PresheafedSpace C} (f : X ⟶ Y) [H : IsOpenImmersion f] :

CategoryTheory.CategoryStruct.comp (isoRestrict f).inv f = Y.ofRestrict ⋯

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@[simp]

theorem AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.isoRestrict_inv_ofRestrict_assoc {C : Type u} [CategoryTheory.Category.{v, u} C] {X Y : PresheafedSpace C} (f : X ⟶ Y) [H : IsOpenImmersion f] {Z : PresheafedSpace C} (h : Y ⟶ Z) :

CategoryTheory.CategoryStruct.comp (isoRestrict f).inv (CategoryTheory.CategoryStruct.comp f h) = CategoryTheory.CategoryStruct.comp (Y.ofRestrict ⋯) h

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instance AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.mono {C : Type u} [CategoryTheory.Category.{v, u} C] {X Y : PresheafedSpace C} (f : X ⟶ Y) [H : IsOpenImmersion f] :

CategoryTheory.Mono f

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theorem AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.c_iso' {C : Type u} [CategoryTheory.Category.{v, u} C] {X Y : PresheafedSpace C} (f : X ⟶ Y) [H : IsOpenImmersion f] {V : TopologicalSpace.Opens ↑↑Y} (U : TopologicalSpace.Opens ↑↑X) (h : V = (opensFunctor f).obj U) :

CategoryTheory.IsIso (f.c.app (Opposite.op V))

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instance AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.comp {C : Type u} [CategoryTheory.Category.{v, u} C] {X Y : PresheafedSpace C} (f : X ⟶ Y) [H : IsOpenImmersion f] {Z : PresheafedSpace C} (g : Y ⟶ Z) [hg : IsOpenImmersion g] :

IsOpenImmersion (CategoryTheory.CategoryStruct.comp f g)

The composition of two open immersions is an open immersion.

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noncomputable def AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.invApp {C : Type u} [CategoryTheory.Category.{v, u} C] {X Y : PresheafedSpace C} (f : X ⟶ Y) [H : IsOpenImmersion f] (U : TopologicalSpace.Opens ↑↑X) :

X.presheaf.obj (Opposite.op U) ⟶ Y.presheaf.obj (Opposite.op ((opensFunctor f).obj U))

For an open immersion f : X ⟶ Y and an open set U ⊆ X, we have the map X(U) ⟶ Y(U).

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@[simp]

theorem AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.inv_naturality {C : Type u} [CategoryTheory.Category.{v, u} C] {X Y : PresheafedSpace C} (f : X ⟶ Y) [H : IsOpenImmersion f] {U V : (TopologicalSpace.Opens ↑↑X)ᵒᵖ} (i : U ⟶ V) :

CategoryTheory.CategoryStruct.comp (X.presheaf.map i) (invApp f (Opposite.unop V)) = CategoryTheory.CategoryStruct.comp (invApp f (Opposite.unop U)) (Y.presheaf.map ((opensFunctor f).op.map i))

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theorem AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.inv_naturality_assoc {C : Type u} [CategoryTheory.Category.{v, u} C] {X Y : PresheafedSpace C} (f : X ⟶ Y) [H : IsOpenImmersion f] {U V : (TopologicalSpace.Opens ↑↑X)ᵒᵖ} (i : U ⟶ V) {Z : C} (h : Y.presheaf.obj (Opposite.op ((opensFunctor f).obj (Opposite.unop V))) ⟶ Z) :

CategoryTheory.CategoryStruct.comp (X.presheaf.map i) (CategoryTheory.CategoryStruct.comp (invApp f (Opposite.unop V)) h) = CategoryTheory.CategoryStruct.comp (invApp f (Opposite.unop U)) (CategoryTheory.CategoryStruct.comp (Y.presheaf.map ((opensFunctor f).op.map i)) h)

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instance AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.instIsIsoInvApp {C : Type u} [CategoryTheory.Category.{v, u} C] {X Y : PresheafedSpace C} (f : X ⟶ Y) [H : IsOpenImmersion f] (U : TopologicalSpace.Opens ↑↑X) :

CategoryTheory.IsIso (invApp f U)

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theorem AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.inv_invApp {C : Type u} [CategoryTheory.Category.{v, u} C] {X Y : PresheafedSpace C} (f : X ⟶ Y) [H : IsOpenImmersion f] (U : TopologicalSpace.Opens ↑↑X) :

CategoryTheory.inv (invApp f U) = CategoryTheory.CategoryStruct.comp (f.c.app (Opposite.op ((opensFunctor f).obj U))) (X.presheaf.map (CategoryTheory.eqToHom ⋯))

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@[simp]

theorem AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.invApp_app {C : Type u} [CategoryTheory.Category.{v, u} C] {X Y : PresheafedSpace C} (f : X ⟶ Y) [H : IsOpenImmersion f] (U : TopologicalSpace.Opens ↑↑X) :

CategoryTheory.CategoryStruct.comp (invApp f U) (f.c.app (Opposite.op ((opensFunctor f).obj U))) = X.presheaf.map (CategoryTheory.eqToHom ⋯)

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theorem AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.invApp_app_assoc {C : Type u} [CategoryTheory.Category.{v, u} C] {X Y : PresheafedSpace C} (f : X ⟶ Y) [H : IsOpenImmersion f] (U : TopologicalSpace.Opens ↑↑X) {Z : C} (h : ((TopCat.Presheaf.pushforward C f.base).obj X.presheaf).obj (Opposite.op ((opensFunctor f).obj U)) ⟶ Z) :

CategoryTheory.CategoryStruct.comp (invApp f U) (CategoryTheory.CategoryStruct.comp (f.c.app (Opposite.op ((opensFunctor f).obj U))) h) = CategoryTheory.CategoryStruct.comp (X.presheaf.map (CategoryTheory.eqToHom ⋯)) h

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theorem AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.invApp_app_apply {C : Type u} [CategoryTheory.Category.{v, u} C] {X Y : PresheafedSpace C} (f : X ⟶ Y) [H : IsOpenImmersion f] (U : TopologicalSpace.Opens ↑↑X) [inst : CategoryTheory.HasForget C] (x : (CategoryTheory.forget C).obj (X.presheaf.obj (Opposite.op U))) :

(f.c.app (Opposite.op ((opensFunctor f).obj U))) ((invApp f U) x) = (X.presheaf.map (CategoryTheory.eqToHom ⋯)) x

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@[simp]

theorem AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.app_invApp {C : Type u} [CategoryTheory.Category.{v, u} C] {X Y : PresheafedSpace C} (f : X ⟶ Y) [H : IsOpenImmersion f] (U : TopologicalSpace.Opens ↑↑Y) :

CategoryTheory.CategoryStruct.comp (f.c.app (Opposite.op U)) (invApp f ((TopologicalSpace.Opens.map f.base).obj U)) = Y.presheaf.map (CategoryTheory.homOfLE ⋯).op

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theorem AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.app_invApp_assoc {C : Type u} [CategoryTheory.Category.{v, u} C] {X Y : PresheafedSpace C} (f : X ⟶ Y) [H : IsOpenImmersion f] (U : TopologicalSpace.Opens ↑↑Y) {Z : C} (h : Y.presheaf.obj (Opposite.op ((opensFunctor f).obj ((TopologicalSpace.Opens.map f.base).obj U))) ⟶ Z) :

CategoryTheory.CategoryStruct.comp (f.c.app (Opposite.op U)) (CategoryTheory.CategoryStruct.comp (invApp f ((TopologicalSpace.Opens.map f.base).obj U)) h) = CategoryTheory.CategoryStruct.comp (Y.presheaf.map (CategoryTheory.homOfLE ⋯).op) h

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theorem AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.app_inv_app' {C : Type u} [CategoryTheory.Category.{v, u} C] {X Y : PresheafedSpace C} (f : X ⟶ Y) [H : IsOpenImmersion f] (U : TopologicalSpace.Opens ↑↑Y) (hU : ↑U ⊆ Set.range ⇑f.base) :

CategoryTheory.CategoryStruct.comp (f.c.app (Opposite.op U)) (invApp f ((TopologicalSpace.Opens.map f.base).obj U)) = Y.presheaf.map (CategoryTheory.eqToHom ⋯).op

A variant of app_inv_app that gives an eqToHom instead of homOfLe.

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theorem AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.app_inv_app'_assoc {C : Type u} [CategoryTheory.Category.{v, u} C] {X Y : PresheafedSpace C} (f : X ⟶ Y) [H : IsOpenImmersion f] (U : TopologicalSpace.Opens ↑↑Y) (hU : ↑U ⊆ Set.range ⇑f.base) {Z : C} (h : Y.presheaf.obj (Opposite.op ((opensFunctor f).obj ((TopologicalSpace.Opens.map f.base).obj U))) ⟶ Z) :

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instance AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.ofIso {C : Type u} [CategoryTheory.Category.{v, u} C] {X Y : PresheafedSpace C} (H : X ≅ Y) :

IsOpenImmersion H.hom

An isomorphism is an open immersion.

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@[instance 100]

instance AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.ofIsIso {C : Type u} [CategoryTheory.Category.{v, u} C] {X Y : PresheafedSpace C} (f : X ⟶ Y) [CategoryTheory.IsIso f] :

IsOpenImmersion f

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instance AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.ofRestrict {C : Type u} [CategoryTheory.Category.{v, u} C] {X : TopCat} (Y : PresheafedSpace C) {f : X ⟶ ↑Y} (hf : Topology.IsOpenEmbedding ⇑f) :

IsOpenImmersion (Y.ofRestrict hf)

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@[simp]

theorem AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.ofRestrict_invApp {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] (X : PresheafedSpace C) {Y : TopCat} {f : Y ⟶ TopCat.of ↑↑X} (h : Topology.IsOpenEmbedding ⇑f) (U : TopologicalSpace.Opens ↑↑(X.restrict h)) :

invApp (X.ofRestrict h) U = CategoryTheory.CategoryStruct.id ((X.restrict h).presheaf.obj (Opposite.op U))

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theorem AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.ofRestrict_invApp_apply {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] (X : PresheafedSpace C) {Y : TopCat} {f : Y ⟶ TopCat.of ↑↑X} (h : Topology.IsOpenEmbedding ⇑f) (U : TopologicalSpace.Opens ↑↑(X.restrict h)) [inst : CategoryTheory.HasForget C] (x : (CategoryTheory.forget C).obj ((X.restrict h).presheaf.obj (Opposite.op U))) :

(invApp (X.ofRestrict h) U) x = x

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theorem AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.to_iso {C : Type u} [CategoryTheory.Category.{v, u} C] {X Y : PresheafedSpace C} (f : X ⟶ Y) [H : IsOpenImmersion f] [h' : CategoryTheory.Epi f.base] :

CategoryTheory.IsIso f

An open immersion is an iso if the underlying continuous map is epi.

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instance AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.stalk_iso {C : Type u} [CategoryTheory.Category.{v, u} C] {X Y : PresheafedSpace C} (f : X ⟶ Y) [H : IsOpenImmersion f] [CategoryTheory.Limits.HasColimits C] (x : ↑↑X) :

CategoryTheory.IsIso (Hom.stalkMap f x)

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def AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.pullbackConeOfLeftFst {C : Type u} [CategoryTheory.Category.{v, u} C] {X Y Z : PresheafedSpace C} (f : X ⟶ Z) [hf : IsOpenImmersion f] (g : Y ⟶ Z) :

Y.restrict ⋯ ⟶ X

(Implementation.) The projection map when constructing the pullback along an open immersion.

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theorem AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.pullback_cone_of_left_condition {C : Type u} [CategoryTheory.Category.{v, u} C] {X Y Z : PresheafedSpace C} (f : X ⟶ Z) [hf : IsOpenImmersion f] (g : Y ⟶ Z) :

CategoryTheory.CategoryStruct.comp (pullbackConeOfLeftFst f g) f = CategoryTheory.CategoryStruct.comp (Y.ofRestrict ⋯) g

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def AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.pullbackConeOfLeft {C : Type u} [CategoryTheory.Category.{v, u} C] {X Y Z : PresheafedSpace C} (f : X ⟶ Z) [hf : IsOpenImmersion f] (g : Y ⟶ Z) :

CategoryTheory.Limits.PullbackCone f g

We construct the pullback along an open immersion via restricting along the pullback of the maps of underlying spaces (which is also an open embedding).

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def AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.pullbackConeOfLeftLift {C : Type u} [CategoryTheory.Category.{v, u} C] {X Y Z : PresheafedSpace C} (f : X ⟶ Z) [hf : IsOpenImmersion f] (g : Y ⟶ Z) (s : CategoryTheory.Limits.PullbackCone f g) :

s.pt ⟶ (pullbackConeOfLeft f g).pt

(Implementation.) Any cone over cospan f g indeed factors through the constructed cone.

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theorem AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.pullbackConeOfLeftLift_fst {C : Type u} [CategoryTheory.Category.{v, u} C] {X Y Z : PresheafedSpace C} (f : X ⟶ Z) [hf : IsOpenImmersion f] (g : Y ⟶ Z) (s : CategoryTheory.Limits.PullbackCone f g) :

CategoryTheory.CategoryStruct.comp (pullbackConeOfLeftLift f g s) (pullbackConeOfLeft f g).fst = s.fst

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theorem AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.pullbackConeOfLeftLift_snd {C : Type u} [CategoryTheory.Category.{v, u} C] {X Y Z : PresheafedSpace C} (f : X ⟶ Z) [hf : IsOpenImmersion f] (g : Y ⟶ Z) (s : CategoryTheory.Limits.PullbackCone f g) :

CategoryTheory.CategoryStruct.comp (pullbackConeOfLeftLift f g s) (pullbackConeOfLeft f g).snd = s.snd

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instance AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.pullbackConeSndIsOpenImmersion {C : Type u} [CategoryTheory.Category.{v, u} C] {X Y Z : PresheafedSpace C} (f : X ⟶ Z) [hf : IsOpenImmersion f] (g : Y ⟶ Z) :

IsOpenImmersion (pullbackConeOfLeft f g).snd

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def AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.pullbackConeOfLeftIsLimit {C : Type u} [CategoryTheory.Category.{v, u} C] {X Y Z : PresheafedSpace C} (f : X ⟶ Z) [hf : IsOpenImmersion f] (g : Y ⟶ Z) :

CategoryTheory.Limits.IsLimit (pullbackConeOfLeft f g)

The constructed pullback cone is indeed the pullback.

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instance AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.hasPullback_of_left {C : Type u} [CategoryTheory.Category.{v, u} C] {X Y Z : PresheafedSpace C} (f : X ⟶ Z) [hf : IsOpenImmersion f] (g : Y ⟶ Z) :

CategoryTheory.Limits.HasPullback f g

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instance AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.hasPullback_of_right {C : Type u} [CategoryTheory.Category.{v, u} C] {X Y Z : PresheafedSpace C} (f : X ⟶ Z) [hf : IsOpenImmersion f] (g : Y ⟶ Z) :

CategoryTheory.Limits.HasPullback g f

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instance AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.pullbackSndOfLeft {C : Type u} [CategoryTheory.Category.{v, u} C] {X Y Z : PresheafedSpace C} (f : X ⟶ Z) [hf : IsOpenImmersion f] (g : Y ⟶ Z) :

IsOpenImmersion (CategoryTheory.Limits.pullback.snd f g)

Open immersions are stable under base-change.

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instance AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.pullbackFstOfRight {C : Type u} [CategoryTheory.Category.{v, u} C] {X Y Z : PresheafedSpace C} (f : X ⟶ Z) [hf : IsOpenImmersion f] (g : Y ⟶ Z) :

IsOpenImmersion (CategoryTheory.Limits.pullback.fst g f)

Open immersions are stable under base-change.

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instance AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.pullbackToBaseIsOpenImmersion {C : Type u} [CategoryTheory.Category.{v, u} C] {X Y Z : PresheafedSpace C} (f : X ⟶ Z) [hf : IsOpenImmersion f] (g : Y ⟶ Z) [IsOpenImmersion g] :

IsOpenImmersion (CategoryTheory.Limits.limit.π (CategoryTheory.Limits.cospan f g) CategoryTheory.Limits.WalkingCospan.one)

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instance AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.forget_preservesLimitsOfLeft {C : Type u} [CategoryTheory.Category.{v, u} C] {X Y Z : PresheafedSpace C} (f : X ⟶ Z) [hf : IsOpenImmersion f] (g : Y ⟶ Z) :

CategoryTheory.Limits.PreservesLimit (CategoryTheory.Limits.cospan f g) (forget C)

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instance AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.forget_preservesLimitsOfRight {C : Type u} [CategoryTheory.Category.{v, u} C] {X Y Z : PresheafedSpace C} (f : X ⟶ Z) [hf : IsOpenImmersion f] (g : Y ⟶ Z) :

CategoryTheory.Limits.PreservesLimit (CategoryTheory.Limits.cospan g f) (forget C)

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theorem AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.pullback_snd_isIso_of_range_subset {C : Type u} [CategoryTheory.Category.{v, u} C] {X Y Z : PresheafedSpace C} (f : X ⟶ Z) [hf : IsOpenImmersion f] (g : Y ⟶ Z) (H : Set.range ⇑g.base ⊆ Set.range ⇑f.base) :

CategoryTheory.IsIso (CategoryTheory.Limits.pullback.snd f g)

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def AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.lift {C : Type u} [CategoryTheory.Category.{v, u} C] {X Y Z : PresheafedSpace C} (f : X ⟶ Z) [hf : IsOpenImmersion f] (g : Y ⟶ Z) (H : Set.range ⇑g.base ⊆ Set.range ⇑f.base) :

Y ⟶ X

The universal property of open immersions: For an open immersion f : X ⟶ Z, given any morphism of schemes g : Y ⟶ Z whose topological image is contained in the image of f, we can lift this morphism to a unique Y ⟶ X that commutes with these maps.

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@[simp]

theorem AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.lift_fac {C : Type u} [CategoryTheory.Category.{v, u} C] {X Y Z : PresheafedSpace C} (f : X ⟶ Z) [hf : IsOpenImmersion f] (g : Y ⟶ Z) (H : Set.range ⇑g.base ⊆ Set.range ⇑f.base) :

CategoryTheory.CategoryStruct.comp (lift f g H) f = g

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theorem AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.lift_fac_assoc {C : Type u} [CategoryTheory.Category.{v, u} C] {X Y Z : PresheafedSpace C} (f : X ⟶ Z) [hf : IsOpenImmersion f] (g : Y ⟶ Z) (H : Set.range ⇑g.base ⊆ Set.range ⇑f.base) {Z✝ : PresheafedSpace C} (h : Z ⟶ Z✝) :

CategoryTheory.CategoryStruct.comp (lift f g H) (CategoryTheory.CategoryStruct.comp f h) = CategoryTheory.CategoryStruct.comp g h

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theorem AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.lift_uniq {C : Type u} [CategoryTheory.Category.{v, u} C] {X Y Z : PresheafedSpace C} (f : X ⟶ Z) [hf : IsOpenImmersion f] (g : Y ⟶ Z) (H : Set.range ⇑g.base ⊆ Set.range ⇑f.base) (l : Y ⟶ X) (hl : CategoryTheory.CategoryStruct.comp l f = g) :

l = lift f g H

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def AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.isoOfRangeEq {C : Type u} [CategoryTheory.Category.{v, u} C] {X Y Z : PresheafedSpace C} (f : X ⟶ Z) [hf : IsOpenImmersion f] (g : Y ⟶ Z) [IsOpenImmersion g] (e : Set.range ⇑f.base = Set.range ⇑g.base) :

X ≅ Y

Two open immersions with equal range is isomorphic.

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theorem AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.isoOfRangeEq_inv {C : Type u} [CategoryTheory.Category.{v, u} C] {X Y Z : PresheafedSpace C} (f : X ⟶ Z) [hf : IsOpenImmersion f] (g : Y ⟶ Z) [IsOpenImmersion g] (e : Set.range ⇑f.base = Set.range ⇑g.base) :

(isoOfRangeEq f g e).inv = lift f g ⋯

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@[simp]

theorem AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.isoOfRangeEq_hom {C : Type u} [CategoryTheory.Category.{v, u} C] {X Y Z : PresheafedSpace C} (f : X ⟶ Z) [hf : IsOpenImmersion f] (g : Y ⟶ Z) [IsOpenImmersion g] (e : Set.range ⇑f.base = Set.range ⇑g.base) :

(isoOfRangeEq f g e).hom = lift g f ⋯

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def AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.toSheafedSpace {C : Type u} [CategoryTheory.Category.{v, u} C] {X : PresheafedSpace C} (Y : SheafedSpace C) (f : X ⟶ Y.toPresheafedSpace) [H : IsOpenImmersion f] :

SheafedSpace C

If X ⟶ Y is an open immersion, and Y is a SheafedSpace, then so is X.

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AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.toSheafedSpace Y f = { toPresheafedSpace := X, IsSheaf := ⋯ }

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theorem AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.toSheafedSpace_toPresheafedSpace {C : Type u} [CategoryTheory.Category.{v, u} C] {X : PresheafedSpace C} (Y : SheafedSpace C) (f : X ⟶ Y.toPresheafedSpace) [H : IsOpenImmersion f] :

(toSheafedSpace Y f).toPresheafedSpace = X

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def AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.toSheafedSpaceHom {C : Type u} [CategoryTheory.Category.{v, u} C] {X : PresheafedSpace C} (Y : SheafedSpace C) (f : X ⟶ Y.toPresheafedSpace) [H : IsOpenImmersion f] :

toSheafedSpace Y f ⟶ Y

If X ⟶ Y is an open immersion of PresheafedSpaces, and Y is a SheafedSpace, we can upgrade it into a morphism of SheafedSpaces.

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AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.toSheafedSpaceHom Y f = f

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theorem AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.toSheafedSpaceHom_base {C : Type u} [CategoryTheory.Category.{v, u} C] {X : PresheafedSpace C} (Y : SheafedSpace C) (f : X ⟶ Y.toPresheafedSpace) [H : IsOpenImmersion f] :

(toSheafedSpaceHom Y f).base = f.base

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@[simp]

theorem AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.toSheafedSpaceHom_c {C : Type u} [CategoryTheory.Category.{v, u} C] {X : PresheafedSpace C} (Y : SheafedSpace C) (f : X ⟶ Y.toPresheafedSpace) [H : IsOpenImmersion f] :

(toSheafedSpaceHom Y f).c = f.c

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instance AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.toSheafedSpace_isOpenImmersion {C : Type u} [CategoryTheory.Category.{v, u} C] {X : PresheafedSpace C} (Y : SheafedSpace C) (f : X ⟶ Y.toPresheafedSpace) [H : IsOpenImmersion f] :

SheafedSpace.IsOpenImmersion (toSheafedSpaceHom Y f)

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@[simp]

theorem AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.sheafedSpace_toSheafedSpace {C : Type u} [CategoryTheory.Category.{v, u} C] {X Y : SheafedSpace C} (f : X ⟶ Y) [IsOpenImmersion f] :

toSheafedSpace Y f = X

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def AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.toLocallyRingedSpace {X : PresheafedSpace CommRingCat} (Y : LocallyRingedSpace) (f : X ⟶ Y.toPresheafedSpace) [H : IsOpenImmersion f] :

LocallyRingedSpace

If X ⟶ Y is an open immersion, and Y is a LocallyRingedSpace, then so is X.

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theorem AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.toLocallyRingedSpace_toSheafedSpace {X : PresheafedSpace CommRingCat} (Y : LocallyRingedSpace) (f : X ⟶ Y.toPresheafedSpace) [H : IsOpenImmersion f] :

(toLocallyRingedSpace Y f).toSheafedSpace = toSheafedSpace Y.toSheafedSpace f

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def AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.toLocallyRingedSpaceHom {X : PresheafedSpace CommRingCat} (Y : LocallyRingedSpace) (f : X ⟶ Y.toPresheafedSpace) [H : IsOpenImmersion f] :

toLocallyRingedSpace Y f ⟶ Y

If X ⟶ Y is an open immersion of PresheafedSpaces, and Y is a LocallyRingedSpace, we can upgrade it into a morphism of LocallyRingedSpace.

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AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.toLocallyRingedSpaceHom Y f = { toHom := f, prop := ⋯ }

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@[simp]

theorem AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.toLocallyRingedSpaceHom_val {X : PresheafedSpace CommRingCat} (Y : LocallyRingedSpace) (f : X ⟶ Y.toPresheafedSpace) [H : IsOpenImmersion f] :

LocallyRingedSpace.Hom.toShHom (toLocallyRingedSpaceHom Y f) = f

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instance AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.toLocallyRingedSpace_isOpenImmersion {X : PresheafedSpace CommRingCat} (Y : LocallyRingedSpace) (f : X ⟶ Y.toPresheafedSpace) [H : IsOpenImmersion f] :

LocallyRingedSpace.IsOpenImmersion (toLocallyRingedSpaceHom Y f)

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@[simp]

theorem AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.locallyRingedSpace_toLocallyRingedSpace {X Y : LocallyRingedSpace} (f : X ⟶ Y) [LocallyRingedSpace.IsOpenImmersion f] :

toLocallyRingedSpace Y f.toHom = X

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theorem AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.isIso_of_subset {C : Type u} [CategoryTheory.Category.{v, u} C] {X Y : PresheafedSpace C} (f : X ⟶ Y) [H : IsOpenImmersion f] (U : TopologicalSpace.Opens ↑↑Y) (hU : ↑U ⊆ Set.range ⇑f.base) :

CategoryTheory.IsIso (f.c.app (Opposite.op U))

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@[instance 100]

instance AlgebraicGeometry.SheafedSpace.IsOpenImmersion.of_isIso {C : Type u} [CategoryTheory.Category.{v, u} C] {X Y : SheafedSpace C} (f : X ⟶ Y) [CategoryTheory.IsIso f] :

IsOpenImmersion f

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instance AlgebraicGeometry.SheafedSpace.IsOpenImmersion.comp {C : Type u} [CategoryTheory.Category.{v, u} C] {X Y Z : SheafedSpace C} (f : X ⟶ Y) (g : Y ⟶ Z) [IsOpenImmersion f] [IsOpenImmersion g] :

IsOpenImmersion (CategoryTheory.CategoryStruct.comp f g)

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instance AlgebraicGeometry.SheafedSpace.IsOpenImmersion.instMono {C : Type u} [CategoryTheory.Category.{v, u} C] {X Z : SheafedSpace C} (f : X ⟶ Z) [H : IsOpenImmersion f] :

CategoryTheory.Mono f

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instance AlgebraicGeometry.SheafedSpace.IsOpenImmersion.forgetMapIsOpenImmersion {C : Type u} [CategoryTheory.Category.{v, u} C] {X Z : SheafedSpace C} (f : X ⟶ Z) [H : IsOpenImmersion f] :

PresheafedSpace.IsOpenImmersion (forgetToPresheafedSpace.map f)

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instance AlgebraicGeometry.SheafedSpace.IsOpenImmersion.hasLimit_cospan_forget_of_left {C : Type u} [CategoryTheory.Category.{v, u} C] {X Y Z : SheafedSpace C} (f : X ⟶ Z) (g : Y ⟶ Z) [H : IsOpenImmersion f] :

CategoryTheory.Limits.HasLimit ((CategoryTheory.Limits.cospan f g).comp forgetToPresheafedSpace)

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instance AlgebraicGeometry.SheafedSpace.IsOpenImmersion.hasLimit_cospan_forget_of_left' {C : Type u} [CategoryTheory.Category.{v, u} C] {X Y Z : SheafedSpace C} (f : X ⟶ Z) (g : Y ⟶ Z) [H : IsOpenImmersion f] :

CategoryTheory.Limits.HasLimit (CategoryTheory.Limits.cospan (((CategoryTheory.Limits.cospan f g).comp forgetToPresheafedSpace).map CategoryTheory.Limits.WalkingCospan.Hom.inl) (((CategoryTheory.Limits.cospan f g).comp forgetToPresheafedSpace).map CategoryTheory.Limits.WalkingCospan.Hom.inr))

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instance AlgebraicGeometry.SheafedSpace.IsOpenImmersion.hasLimit_cospan_forget_of_right {C : Type u} [CategoryTheory.Category.{v, u} C] {X Y Z : SheafedSpace C} (f : X ⟶ Z) (g : Y ⟶ Z) [H : IsOpenImmersion f] :

CategoryTheory.Limits.HasLimit ((CategoryTheory.Limits.cospan g f).comp forgetToPresheafedSpace)

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instance AlgebraicGeometry.SheafedSpace.IsOpenImmersion.hasLimit_cospan_forget_of_right' {C : Type u} [CategoryTheory.Category.{v, u} C] {X Y Z : SheafedSpace C} (f : X ⟶ Z) (g : Y ⟶ Z) [H : IsOpenImmersion f] :

CategoryTheory.Limits.HasLimit (CategoryTheory.Limits.cospan (((CategoryTheory.Limits.cospan g f).comp forgetToPresheafedSpace).map CategoryTheory.Limits.WalkingCospan.Hom.inl) (((CategoryTheory.Limits.cospan g f).comp forgetToPresheafedSpace).map CategoryTheory.Limits.WalkingCospan.Hom.inr))

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instance AlgebraicGeometry.SheafedSpace.IsOpenImmersion.forgetCreatesPullbackOfLeft {C : Type u} [CategoryTheory.Category.{v, u} C] {X Y Z : SheafedSpace C} (f : X ⟶ Z) (g : Y ⟶ Z) [H : IsOpenImmersion f] :

CategoryTheory.CreatesLimit (CategoryTheory.Limits.cospan f g) forgetToPresheafedSpace

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instance AlgebraicGeometry.SheafedSpace.IsOpenImmersion.forgetCreatesPullbackOfRight {C : Type u} [CategoryTheory.Category.{v, u} C] {X Y Z : SheafedSpace C} (f : X ⟶ Z) (g : Y ⟶ Z) [H : IsOpenImmersion f] :

CategoryTheory.CreatesLimit (CategoryTheory.Limits.cospan g f) forgetToPresheafedSpace

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One or more equations did not get rendered due to their size.

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instance AlgebraicGeometry.SheafedSpace.IsOpenImmersion.sheafedSpace_forgetPreserves_of_left {C : Type u} [CategoryTheory.Category.{v, u} C] {X Y Z : SheafedSpace C} (f : X ⟶ Z) (g : Y ⟶ Z) [H : IsOpenImmersion f] :

CategoryTheory.Limits.PreservesLimit (CategoryTheory.Limits.cospan f g) (forget C)

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instance AlgebraicGeometry.SheafedSpace.IsOpenImmersion.sheafedSpace_forgetPreserves_of_right {C : Type u} [CategoryTheory.Category.{v, u} C] {X Y Z : SheafedSpace C} (f : X ⟶ Z) (g : Y ⟶ Z) [H : IsOpenImmersion f] :

CategoryTheory.Limits.PreservesLimit (CategoryTheory.Limits.cospan g f) (forget C)

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instance AlgebraicGeometry.SheafedSpace.IsOpenImmersion.sheafedSpace_hasPullback_of_left {C : Type u} [CategoryTheory.Category.{v, u} C] {X Y Z : SheafedSpace C} (f : X ⟶ Z) (g : Y ⟶ Z) [H : IsOpenImmersion f] :

CategoryTheory.Limits.HasPullback f g

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instance AlgebraicGeometry.SheafedSpace.IsOpenImmersion.sheafedSpace_hasPullback_of_right {C : Type u} [CategoryTheory.Category.{v, u} C] {X Y Z : SheafedSpace C} (f : X ⟶ Z) (g : Y ⟶ Z) [H : IsOpenImmersion f] :

CategoryTheory.Limits.HasPullback g f

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instance AlgebraicGeometry.SheafedSpace.IsOpenImmersion.sheafedSpace_pullback_snd_of_left {C : Type u} [CategoryTheory.Category.{v, u} C] {X Y Z : SheafedSpace C} (f : X ⟶ Z) (g : Y ⟶ Z) [H : IsOpenImmersion f] :

IsOpenImmersion (CategoryTheory.Limits.pullback.snd f g)

Open immersions are stable under base-change.

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instance AlgebraicGeometry.SheafedSpace.IsOpenImmersion.sheafedSpace_pullback_fst_of_right {C : Type u} [CategoryTheory.Category.{v, u} C] {X Y Z : SheafedSpace C} (f : X ⟶ Z) (g : Y ⟶ Z) [H : IsOpenImmersion f] :

IsOpenImmersion (CategoryTheory.Limits.pullback.fst g f)

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instance AlgebraicGeometry.SheafedSpace.IsOpenImmersion.sheafedSpace_pullback_to_base_isOpenImmersion {C : Type u} [CategoryTheory.Category.{v, u} C] {X Y Z : SheafedSpace C} (f : X ⟶ Z) (g : Y ⟶ Z) [H : IsOpenImmersion f] [IsOpenImmersion g] :

IsOpenImmersion (CategoryTheory.Limits.limit.π (CategoryTheory.Limits.cospan f g) CategoryTheory.Limits.WalkingCospan.one)

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theorem AlgebraicGeometry.SheafedSpace.IsOpenImmersion.of_stalk_iso {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasLimits C] [CategoryTheory.Limits.HasColimits C] [CategoryTheory.HasForget C] [(CategoryTheory.forget C).ReflectsIsomorphisms] [CategoryTheory.Limits.PreservesLimits (CategoryTheory.forget C)] [CategoryTheory.Limits.PreservesFilteredColimits (CategoryTheory.forget C)] {X Y : SheafedSpace C} (f : X ⟶ Y) (hf : Topology.IsOpenEmbedding ⇑f.base) [H : ∀ (x : ↑↑X.toPresheafedSpace), CategoryTheory.IsIso (PresheafedSpace.Hom.stalkMap f x)] :

IsOpenImmersion f

Suppose X Y : SheafedSpace C, where C is a concrete category, whose forgetful functor reflects isomorphisms, preserves limits and filtered colimits. Then a morphism X ⟶ Y that is a topological open embedding is an open immersion iff every stalk map is an iso.

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@[reducible, inline]