state stringlengths 0 159k | srcUpToTactic stringlengths 387 167k | nextTactic stringlengths 3 9k | declUpToTactic stringlengths 22 11.5k | declId stringlengths 38 95 | decl stringlengths 16 1.89k | file_tag stringlengths 17 73 |
|---|---|---|---|---|---|---|
case convert_2.naturality
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X Y Z : TopCat
f : X ⟶ Y
hf : OpenEmbedding ⇑f
F : Presheaf C X
x : ↑X
this : Functor.Initial (IsOpenMap.functorNhds (_ : IsOpenMap ⇑f) x)
⊢ autoParam
(∀ {X_1 Y_1 : (OpenNhds x)ᵒᵖ} (f_1 : X_1 ⟶ Y_1),
((IsOpenMap.functorNhds ... | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Topology.Category.TopCat.OpenNhds
import Mathlib.Topology.Sheaves.Presheaf
import Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing... | intro U V i | theorem stalkPushforward_iso_of_openEmbedding {f : X ⟶ Y} (hf : OpenEmbedding f) (F : X.Presheaf C)
(x : X) : IsIso (F.stalkPushforward _ f x) := by
haveI := Functor.initial_of_adjunction (hf.isOpenMap.adjunctionNhds x)
convert IsIso.of_iso
((Functor.Final.colimitIso (hf.isOpenMap.functorNhds x).op
... | Mathlib.Topology.Sheaves.Stalks.202_0.hsVUPKIHRY0xmFk | theorem stalkPushforward_iso_of_openEmbedding {f : X ⟶ Y} (hf : OpenEmbedding f) (F : X.Presheaf C)
(x : X) : IsIso (F.stalkPushforward _ f x) | Mathlib_Topology_Sheaves_Stalks |
case convert_2.naturality
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X Y Z : TopCat
f : X ⟶ Y
hf : OpenEmbedding ⇑f
F : Presheaf C X
x : ↑X
this : Functor.Initial (IsOpenMap.functorNhds (_ : IsOpenMap ⇑f) x)
U V : (OpenNhds x)ᵒᵖ
i : U ⟶ V
⊢ ((IsOpenMap.functorNhds (_ : IsOpenMap ⇑f) x).op ⋙ (OpenNhds.i... | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Topology.Category.TopCat.OpenNhds
import Mathlib.Topology.Sheaves.Presheaf
import Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing... | erw [← F.map_comp, ← F.map_comp] | theorem stalkPushforward_iso_of_openEmbedding {f : X ⟶ Y} (hf : OpenEmbedding f) (F : X.Presheaf C)
(x : X) : IsIso (F.stalkPushforward _ f x) := by
haveI := Functor.initial_of_adjunction (hf.isOpenMap.adjunctionNhds x)
convert IsIso.of_iso
((Functor.Final.colimitIso (hf.isOpenMap.functorNhds x).op
... | Mathlib.Topology.Sheaves.Stalks.202_0.hsVUPKIHRY0xmFk | theorem stalkPushforward_iso_of_openEmbedding {f : X ⟶ Y} (hf : OpenEmbedding f) (F : X.Presheaf C)
(x : X) : IsIso (F.stalkPushforward _ f x) | Mathlib_Topology_Sheaves_Stalks |
case convert_2.naturality
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X Y Z : TopCat
f : X ⟶ Y
hf : OpenEmbedding ⇑f
F : Presheaf C X
x : ↑X
this : Functor.Initial (IsOpenMap.functorNhds (_ : IsOpenMap ⇑f) x)
U V : (OpenNhds x)ᵒᵖ
i : U ⟶ V
⊢ F.map
((Opens.map f).op.map
((OpenNhds.inclusi... | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Topology.Category.TopCat.OpenNhds
import Mathlib.Topology.Sheaves.Presheaf
import Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing... | congr 1 | theorem stalkPushforward_iso_of_openEmbedding {f : X ⟶ Y} (hf : OpenEmbedding f) (F : X.Presheaf C)
(x : X) : IsIso (F.stalkPushforward _ f x) := by
haveI := Functor.initial_of_adjunction (hf.isOpenMap.adjunctionNhds x)
convert IsIso.of_iso
((Functor.Final.colimitIso (hf.isOpenMap.functorNhds x).op
... | Mathlib.Topology.Sheaves.Stalks.202_0.hsVUPKIHRY0xmFk | theorem stalkPushforward_iso_of_openEmbedding {f : X ⟶ Y} (hf : OpenEmbedding f) (F : X.Presheaf C)
(x : X) : IsIso (F.stalkPushforward _ f x) | Mathlib_Topology_Sheaves_Stalks |
case h.e'_5.h
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X Y Z : TopCat
f : X ⟶ Y
hf : OpenEmbedding ⇑f
F : Presheaf C X
x : ↑X
this : Functor.Initial (IsOpenMap.functorNhds (_ : IsOpenMap ⇑f) x)
e_3✝ : stalk (f _* F) (f x) = colimit ((OpenNhds.inclusion (f x)).op ⋙ f _* F)
e_4✝ : stalk F x = colim.obj... | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Topology.Category.TopCat.OpenNhds
import Mathlib.Topology.Sheaves.Presheaf
import Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing... | change (_ : colimit _ ⟶ _) = (_ : colimit _ ⟶ _) | theorem stalkPushforward_iso_of_openEmbedding {f : X ⟶ Y} (hf : OpenEmbedding f) (F : X.Presheaf C)
(x : X) : IsIso (F.stalkPushforward _ f x) := by
haveI := Functor.initial_of_adjunction (hf.isOpenMap.adjunctionNhds x)
convert IsIso.of_iso
((Functor.Final.colimitIso (hf.isOpenMap.functorNhds x).op
... | Mathlib.Topology.Sheaves.Stalks.202_0.hsVUPKIHRY0xmFk | theorem stalkPushforward_iso_of_openEmbedding {f : X ⟶ Y} (hf : OpenEmbedding f) (F : X.Presheaf C)
(x : X) : IsIso (F.stalkPushforward _ f x) | Mathlib_Topology_Sheaves_Stalks |
case h.e'_5.h
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X Y Z : TopCat
f : X ⟶ Y
hf : OpenEmbedding ⇑f
F : Presheaf C X
x : ↑X
this : Functor.Initial (IsOpenMap.functorNhds (_ : IsOpenMap ⇑f) x)
e_3✝ : stalk (f _* F) (f x) = colimit ((OpenNhds.inclusion (f x)).op ⋙ f _* F)
e_4✝ : stalk F x = colim.obj... | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Topology.Category.TopCat.OpenNhds
import Mathlib.Topology.Sheaves.Presheaf
import Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing... | ext U | theorem stalkPushforward_iso_of_openEmbedding {f : X ⟶ Y} (hf : OpenEmbedding f) (F : X.Presheaf C)
(x : X) : IsIso (F.stalkPushforward _ f x) := by
haveI := Functor.initial_of_adjunction (hf.isOpenMap.adjunctionNhds x)
convert IsIso.of_iso
((Functor.Final.colimitIso (hf.isOpenMap.functorNhds x).op
... | Mathlib.Topology.Sheaves.Stalks.202_0.hsVUPKIHRY0xmFk | theorem stalkPushforward_iso_of_openEmbedding {f : X ⟶ Y} (hf : OpenEmbedding f) (F : X.Presheaf C)
(x : X) : IsIso (F.stalkPushforward _ f x) | Mathlib_Topology_Sheaves_Stalks |
case h.e'_5.h.w
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X Y Z : TopCat
f : X ⟶ Y
hf : OpenEmbedding ⇑f
F : Presheaf C X
x : ↑X
this : Functor.Initial (IsOpenMap.functorNhds (_ : IsOpenMap ⇑f) x)
e_3✝ : stalk (f _* F) (f x) = colimit ((OpenNhds.inclusion (f x)).op ⋙ f _* F)
e_4✝ : stalk F x = colim.o... | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Topology.Category.TopCat.OpenNhds
import Mathlib.Topology.Sheaves.Presheaf
import Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing... | rw [← Iso.comp_inv_eq] | theorem stalkPushforward_iso_of_openEmbedding {f : X ⟶ Y} (hf : OpenEmbedding f) (F : X.Presheaf C)
(x : X) : IsIso (F.stalkPushforward _ f x) := by
haveI := Functor.initial_of_adjunction (hf.isOpenMap.adjunctionNhds x)
convert IsIso.of_iso
((Functor.Final.colimitIso (hf.isOpenMap.functorNhds x).op
... | Mathlib.Topology.Sheaves.Stalks.202_0.hsVUPKIHRY0xmFk | theorem stalkPushforward_iso_of_openEmbedding {f : X ⟶ Y} (hf : OpenEmbedding f) (F : X.Presheaf C)
(x : X) : IsIso (F.stalkPushforward _ f x) | Mathlib_Topology_Sheaves_Stalks |
case h.e'_5.h.w
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X Y Z : TopCat
f : X ⟶ Y
hf : OpenEmbedding ⇑f
F : Presheaf C X
x : ↑X
this : Functor.Initial (IsOpenMap.functorNhds (_ : IsOpenMap ⇑f) x)
e_3✝ : stalk (f _* F) (f x) = colimit ((OpenNhds.inclusion (f x)).op ⋙ f _* F)
e_4✝ : stalk F x = colim.o... | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Topology.Category.TopCat.OpenNhds
import Mathlib.Topology.Sheaves.Presheaf
import Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing... | erw [colimit.ι_map_assoc] | theorem stalkPushforward_iso_of_openEmbedding {f : X ⟶ Y} (hf : OpenEmbedding f) (F : X.Presheaf C)
(x : X) : IsIso (F.stalkPushforward _ f x) := by
haveI := Functor.initial_of_adjunction (hf.isOpenMap.adjunctionNhds x)
convert IsIso.of_iso
((Functor.Final.colimitIso (hf.isOpenMap.functorNhds x).op
... | Mathlib.Topology.Sheaves.Stalks.202_0.hsVUPKIHRY0xmFk | theorem stalkPushforward_iso_of_openEmbedding {f : X ⟶ Y} (hf : OpenEmbedding f) (F : X.Presheaf C)
(x : X) : IsIso (F.stalkPushforward _ f x) | Mathlib_Topology_Sheaves_Stalks |
case h.e'_5.h.w
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X Y Z : TopCat
f : X ⟶ Y
hf : OpenEmbedding ⇑f
F : Presheaf C X
x : ↑X
this : Functor.Initial (IsOpenMap.functorNhds (_ : IsOpenMap ⇑f) x)
e_3✝ : stalk (f _* F) (f x) = colimit ((OpenNhds.inclusion (f x)).op ⋙ f _* F)
e_4✝ : stalk F x = colim.o... | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Topology.Category.TopCat.OpenNhds
import Mathlib.Topology.Sheaves.Presheaf
import Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing... | rw [colimit.ι_pre, Category.assoc] | theorem stalkPushforward_iso_of_openEmbedding {f : X ⟶ Y} (hf : OpenEmbedding f) (F : X.Presheaf C)
(x : X) : IsIso (F.stalkPushforward _ f x) := by
haveI := Functor.initial_of_adjunction (hf.isOpenMap.adjunctionNhds x)
convert IsIso.of_iso
((Functor.Final.colimitIso (hf.isOpenMap.functorNhds x).op
... | Mathlib.Topology.Sheaves.Stalks.202_0.hsVUPKIHRY0xmFk | theorem stalkPushforward_iso_of_openEmbedding {f : X ⟶ Y} (hf : OpenEmbedding f) (F : X.Presheaf C)
(x : X) : IsIso (F.stalkPushforward _ f x) | Mathlib_Topology_Sheaves_Stalks |
case h.e'_5.h.w
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X Y Z : TopCat
f : X ⟶ Y
hf : OpenEmbedding ⇑f
F : Presheaf C X
x : ↑X
this : Functor.Initial (IsOpenMap.functorNhds (_ : IsOpenMap ⇑f) x)
e_3✝ : stalk (f _* F) (f x) = colimit ((OpenNhds.inclusion (f x)).op ⋙ f _* F)
e_4✝ : stalk F x = colim.o... | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Topology.Category.TopCat.OpenNhds
import Mathlib.Topology.Sheaves.Presheaf
import Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing... | erw [colimit.ι_map_assoc, colimit.ι_pre, ← F.map_comp_assoc] | theorem stalkPushforward_iso_of_openEmbedding {f : X ⟶ Y} (hf : OpenEmbedding f) (F : X.Presheaf C)
(x : X) : IsIso (F.stalkPushforward _ f x) := by
haveI := Functor.initial_of_adjunction (hf.isOpenMap.adjunctionNhds x)
convert IsIso.of_iso
((Functor.Final.colimitIso (hf.isOpenMap.functorNhds x).op
... | Mathlib.Topology.Sheaves.Stalks.202_0.hsVUPKIHRY0xmFk | theorem stalkPushforward_iso_of_openEmbedding {f : X ⟶ Y} (hf : OpenEmbedding f) (F : X.Presheaf C)
(x : X) : IsIso (F.stalkPushforward _ f x) | Mathlib_Topology_Sheaves_Stalks |
case h.e'_5.h.w
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X Y Z : TopCat
f : X ⟶ Y
hf : OpenEmbedding ⇑f
F : Presheaf C X
x : ↑X
this : Functor.Initial (IsOpenMap.functorNhds (_ : IsOpenMap ⇑f) x)
e_3✝ : stalk (f _* F) (f x) = colimit ((OpenNhds.inclusion (f x)).op ⋙ f _* F)
e_4✝ : stalk F x = colim.o... | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Topology.Category.TopCat.OpenNhds
import Mathlib.Topology.Sheaves.Presheaf
import Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing... | apply colimit.w ((OpenNhds.inclusion (f x)).op ⋙ f _* F) _ | theorem stalkPushforward_iso_of_openEmbedding {f : X ⟶ Y} (hf : OpenEmbedding f) (F : X.Presheaf C)
(x : X) : IsIso (F.stalkPushforward _ f x) := by
haveI := Functor.initial_of_adjunction (hf.isOpenMap.adjunctionNhds x)
convert IsIso.of_iso
((Functor.Final.colimitIso (hf.isOpenMap.functorNhds x).op
... | Mathlib.Topology.Sheaves.Stalks.202_0.hsVUPKIHRY0xmFk | theorem stalkPushforward_iso_of_openEmbedding {f : X ⟶ Y} (hf : OpenEmbedding f) (F : X.Presheaf C)
(x : X) : IsIso (F.stalkPushforward _ f x) | Mathlib_Topology_Sheaves_Stalks |
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X Y Z : TopCat
f : X ⟶ Y
hf : OpenEmbedding ⇑f
F : Presheaf C X
x : ↑X
this : Functor.Initial (IsOpenMap.functorNhds (_ : IsOpenMap ⇑f) x)
e_3✝ : stalk (f _* F) (f x) = colimit ((OpenNhds.inclusion (f x)).op ⋙ f _* F)
e_4✝ : stalk F x = colim.obj (((whiskering... | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Topology.Category.TopCat.OpenNhds
import Mathlib.Topology.Sheaves.Presheaf
import Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing... | dsimp only [Functor.op] | theorem stalkPushforward_iso_of_openEmbedding {f : X ⟶ Y} (hf : OpenEmbedding f) (F : X.Presheaf C)
(x : X) : IsIso (F.stalkPushforward _ f x) := by
haveI := Functor.initial_of_adjunction (hf.isOpenMap.adjunctionNhds x)
convert IsIso.of_iso
((Functor.Final.colimitIso (hf.isOpenMap.functorNhds x).op
... | Mathlib.Topology.Sheaves.Stalks.202_0.hsVUPKIHRY0xmFk | theorem stalkPushforward_iso_of_openEmbedding {f : X ⟶ Y} (hf : OpenEmbedding f) (F : X.Presheaf C)
(x : X) : IsIso (F.stalkPushforward _ f x) | Mathlib_Topology_Sheaves_Stalks |
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X Y Z : TopCat
f : X ⟶ Y
hf : OpenEmbedding ⇑f
F : Presheaf C X
x : ↑X
this : Functor.Initial (IsOpenMap.functorNhds (_ : IsOpenMap ⇑f) x)
e_3✝ : stalk (f _* F) (f x) = colimit ((OpenNhds.inclusion (f x)).op ⋙ f _* F)
e_4✝ : stalk F x = colim.obj (((whiskering... | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Topology.Category.TopCat.OpenNhds
import Mathlib.Topology.Sheaves.Presheaf
import Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing... | refine' ((homOfLE _).op : op (unop U) ⟶ _) | theorem stalkPushforward_iso_of_openEmbedding {f : X ⟶ Y} (hf : OpenEmbedding f) (F : X.Presheaf C)
(x : X) : IsIso (F.stalkPushforward _ f x) := by
haveI := Functor.initial_of_adjunction (hf.isOpenMap.adjunctionNhds x)
convert IsIso.of_iso
((Functor.Final.colimitIso (hf.isOpenMap.functorNhds x).op
... | Mathlib.Topology.Sheaves.Stalks.202_0.hsVUPKIHRY0xmFk | theorem stalkPushforward_iso_of_openEmbedding {f : X ⟶ Y} (hf : OpenEmbedding f) (F : X.Presheaf C)
(x : X) : IsIso (F.stalkPushforward _ f x) | Mathlib_Topology_Sheaves_Stalks |
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X Y Z : TopCat
f : X ⟶ Y
hf : OpenEmbedding ⇑f
F : Presheaf C X
x : ↑X
this : Functor.Initial (IsOpenMap.functorNhds (_ : IsOpenMap ⇑f) x)
e_3✝ : stalk (f _* F) (f x) = colimit ((OpenNhds.inclusion (f x)).op ⋙ f _* F)
e_4✝ : stalk F x = colim.obj (((whiskering... | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Topology.Category.TopCat.OpenNhds
import Mathlib.Topology.Sheaves.Presheaf
import Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing... | exact Set.image_preimage_subset _ _ | theorem stalkPushforward_iso_of_openEmbedding {f : X ⟶ Y} (hf : OpenEmbedding f) (F : X.Presheaf C)
(x : X) : IsIso (F.stalkPushforward _ f x) := by
haveI := Functor.initial_of_adjunction (hf.isOpenMap.adjunctionNhds x)
convert IsIso.of_iso
((Functor.Final.colimitIso (hf.isOpenMap.functorNhds x).op
... | Mathlib.Topology.Sheaves.Stalks.202_0.hsVUPKIHRY0xmFk | theorem stalkPushforward_iso_of_openEmbedding {f : X ⟶ Y} (hf : OpenEmbedding f) (F : X.Presheaf C)
(x : X) : IsIso (F.stalkPushforward _ f x) | Mathlib_Topology_Sheaves_Stalks |
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X Y Z : TopCat
f : X ⟶ Y
F : Presheaf C Y
U : Opens ↑X
x : ↥U
x✝¹ x✝ : CostructuredArrow (Opens.map f).op (op U)
i : x✝¹ ⟶ x✝
⊢ (Lan.diagram (Opens.map f).op F (op U)).map i ≫
(fun V => germ F { val := f ↑x, property := (_ : ↑x ∈ ↑((Opens.map f).op.obj V... | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Topology.Category.TopCat.OpenNhds
import Mathlib.Topology.Sheaves.Presheaf
import Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing... | erw [Category.comp_id] | /-- The morphism `(f⁻¹ℱ)(U) ⟶ ℱ_{f(x)}` for some `U ∋ x`. -/
def germToPullbackStalk (f : X ⟶ Y) (F : Y.Presheaf C) (U : Opens X) (x : U) :
(pullbackObj f F).obj (op U) ⟶ F.stalk ((f : X → Y) (x : X)) :=
colimit.desc (Lan.diagram (Opens.map f).op F (op U))
{ pt := F.stalk ((f : X → Y) (x : X))
ι :=
... | Mathlib.Topology.Sheaves.Stalks.242_0.hsVUPKIHRY0xmFk | /-- The morphism `(f⁻¹ℱ)(U) ⟶ ℱ_{f(x)}` for some `U ∋ x`. -/
def germToPullbackStalk (f : X ⟶ Y) (F : Y.Presheaf C) (U : Opens X) (x : U) :
(pullbackObj f F).obj (op U) ⟶ F.stalk ((f : X → Y) (x : X)) | Mathlib_Topology_Sheaves_Stalks |
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X Y Z : TopCat
f : X ⟶ Y
F : Presheaf C Y
U : Opens ↑X
x : ↥U
x✝¹ x✝ : CostructuredArrow (Opens.map f).op (op U)
i : x✝¹ ⟶ x✝
⊢ (Lan.diagram (Opens.map f).op F (op U)).map i ≫
(fun V => germ F { val := f ↑x, property := (_ : ↑x ∈ ↑((Opens.map f).op.obj V... | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Topology.Category.TopCat.OpenNhds
import Mathlib.Topology.Sheaves.Presheaf
import Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing... | exact F.germ_res i.left.unop _ | /-- The morphism `(f⁻¹ℱ)(U) ⟶ ℱ_{f(x)}` for some `U ∋ x`. -/
def germToPullbackStalk (f : X ⟶ Y) (F : Y.Presheaf C) (U : Opens X) (x : U) :
(pullbackObj f F).obj (op U) ⟶ F.stalk ((f : X → Y) (x : X)) :=
colimit.desc (Lan.diagram (Opens.map f).op F (op U))
{ pt := F.stalk ((f : X → Y) (x : X))
ι :=
... | Mathlib.Topology.Sheaves.Stalks.242_0.hsVUPKIHRY0xmFk | /-- The morphism `(f⁻¹ℱ)(U) ⟶ ℱ_{f(x)}` for some `U ∋ x`. -/
def germToPullbackStalk (f : X ⟶ Y) (F : Y.Presheaf C) (U : Opens X) (x : U) :
(pullbackObj f F).obj (op U) ⟶ F.stalk ((f : X → Y) (x : X)) | Mathlib_Topology_Sheaves_Stalks |
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X Y Z : TopCat
f : X ⟶ Y
F : Presheaf C Y
x : ↑X
x✝² x✝¹ : (OpenNhds x)ᵒᵖ
x✝ : x✝² ⟶ x✝¹
⊢ ((OpenNhds.inclusion x).op ⋙ pullbackObj f F).map x✝ ≫
(fun U => germToPullbackStalk C f F U.unop.obj { val := x, property := (_ : x ∈ U.unop.obj) }) x✝¹ =
(fu... | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Topology.Category.TopCat.OpenNhds
import Mathlib.Topology.Sheaves.Presheaf
import Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing... | erw [colimit.pre_desc, Category.comp_id] | /-- The morphism `(f⁻¹ℱ)ₓ ⟶ ℱ_{f(x)}`. -/
def stalkPullbackInv (f : X ⟶ Y) (F : Y.Presheaf C) (x : X) :
(pullbackObj f F).stalk x ⟶ F.stalk (f x) :=
colimit.desc ((OpenNhds.inclusion x).op ⋙ Presheaf.pullbackObj f F)
{ pt := F.stalk (f x)
ι :=
{ app := fun U => F.germToPullbackStalk _ f (unop U)... | Mathlib.Topology.Sheaves.Stalks.253_0.hsVUPKIHRY0xmFk | /-- The morphism `(f⁻¹ℱ)ₓ ⟶ ℱ_{f(x)}`. -/
def stalkPullbackInv (f : X ⟶ Y) (F : Y.Presheaf C) (x : X) :
(pullbackObj f F).stalk x ⟶ F.stalk (f x) | Mathlib_Topology_Sheaves_Stalks |
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X Y Z : TopCat
f : X ⟶ Y
F : Presheaf C Y
x : ↑X
x✝² x✝¹ : (OpenNhds x)ᵒᵖ
x✝ : x✝² ⟶ x✝¹
⊢ colimit.desc
(CostructuredArrow.map ((OpenNhds.inclusion x).op.map x✝) ⋙
Lan.diagram (Opens.map f).op F ((OpenNhds.inclusion x).op.obj x✝¹))
(Cocone.... | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Topology.Category.TopCat.OpenNhds
import Mathlib.Topology.Sheaves.Presheaf
import Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing... | congr | /-- The morphism `(f⁻¹ℱ)ₓ ⟶ ℱ_{f(x)}`. -/
def stalkPullbackInv (f : X ⟶ Y) (F : Y.Presheaf C) (x : X) :
(pullbackObj f F).stalk x ⟶ F.stalk (f x) :=
colimit.desc ((OpenNhds.inclusion x).op ⋙ Presheaf.pullbackObj f F)
{ pt := F.stalk (f x)
ι :=
{ app := fun U => F.germToPullbackStalk _ f (unop U)... | Mathlib.Topology.Sheaves.Stalks.253_0.hsVUPKIHRY0xmFk | /-- The morphism `(f⁻¹ℱ)ₓ ⟶ ℱ_{f(x)}`. -/
def stalkPullbackInv (f : X ⟶ Y) (F : Y.Presheaf C) (x : X) :
(pullbackObj f F).stalk x ⟶ F.stalk (f x) | Mathlib_Topology_Sheaves_Stalks |
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X Y Z : TopCat
f : X ⟶ Y
F : Presheaf C Y
x : ↑X
⊢ stalkPullbackHom C f F x ≫ stalkPullbackInv C f F x = 𝟙 (stalk F (f x)) | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Topology.Category.TopCat.OpenNhds
import Mathlib.Topology.Sheaves.Presheaf
import Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing... | delta
stalkPullbackHom stalkPullbackInv stalkFunctor Presheaf.pullback stalkPushforward
germToPullbackStalk germ | /-- The isomorphism `ℱ_{f(x)} ≅ (f⁻¹ℱ)ₓ`. -/
def stalkPullbackIso (f : X ⟶ Y) (F : Y.Presheaf C) (x : X) :
F.stalk (f x) ≅ (pullbackObj f F).stalk x where
hom := stalkPullbackHom _ _ _ _
inv := stalkPullbackInv _ _ _ _
hom_inv_id := by
| Mathlib.Topology.Sheaves.Stalks.264_0.hsVUPKIHRY0xmFk | /-- The isomorphism `ℱ_{f(x)} ≅ (f⁻¹ℱ)ₓ`. -/
def stalkPullbackIso (f : X ⟶ Y) (F : Y.Presheaf C) (x : X) :
F.stalk (f x) ≅ (pullbackObj f F).stalk x where
hom | Mathlib_Topology_Sheaves_Stalks |
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X Y Z : TopCat
f : X ⟶ Y
F : Presheaf C Y
x : ↑X
⊢ (((whiskeringLeft (OpenNhds (f x))ᵒᵖ (Opens ↑Y)ᵒᵖ C).obj (OpenNhds.inclusion (f x)).op ⋙ colim).map
((pushforwardPullbackAdjunction C f).unit.app F) ≫
colim.map (whiskerRight (NatTrans.op (Op... | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Topology.Category.TopCat.OpenNhds
import Mathlib.Topology.Sheaves.Presheaf
import Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing... | change (_ : colimit _ ⟶ _) = (_ : colimit _ ⟶ _) | /-- The isomorphism `ℱ_{f(x)} ≅ (f⁻¹ℱ)ₓ`. -/
def stalkPullbackIso (f : X ⟶ Y) (F : Y.Presheaf C) (x : X) :
F.stalk (f x) ≅ (pullbackObj f F).stalk x where
hom := stalkPullbackHom _ _ _ _
inv := stalkPullbackInv _ _ _ _
hom_inv_id := by
delta
stalkPullbackHom stalkPullbackInv stalkFunctor Presheaf.pu... | Mathlib.Topology.Sheaves.Stalks.264_0.hsVUPKIHRY0xmFk | /-- The isomorphism `ℱ_{f(x)} ≅ (f⁻¹ℱ)ₓ`. -/
def stalkPullbackIso (f : X ⟶ Y) (F : Y.Presheaf C) (x : X) :
F.stalk (f x) ≅ (pullbackObj f F).stalk x where
hom | Mathlib_Topology_Sheaves_Stalks |
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X Y Z : TopCat
f : X ⟶ Y
F : Presheaf C Y
x : ↑X
⊢ (((whiskeringLeft (OpenNhds (f x))ᵒᵖ (Opens ↑Y)ᵒᵖ C).obj (OpenNhds.inclusion (f x)).op ⋙ colim).map
((pushforwardPullbackAdjunction C f).unit.app F) ≫
colim.map (whiskerRight (NatTrans.op (Op... | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Topology.Category.TopCat.OpenNhds
import Mathlib.Topology.Sheaves.Presheaf
import Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing... | ext j | /-- The isomorphism `ℱ_{f(x)} ≅ (f⁻¹ℱ)ₓ`. -/
def stalkPullbackIso (f : X ⟶ Y) (F : Y.Presheaf C) (x : X) :
F.stalk (f x) ≅ (pullbackObj f F).stalk x where
hom := stalkPullbackHom _ _ _ _
inv := stalkPullbackInv _ _ _ _
hom_inv_id := by
delta
stalkPullbackHom stalkPullbackInv stalkFunctor Presheaf.pu... | Mathlib.Topology.Sheaves.Stalks.264_0.hsVUPKIHRY0xmFk | /-- The isomorphism `ℱ_{f(x)} ≅ (f⁻¹ℱ)ₓ`. -/
def stalkPullbackIso (f : X ⟶ Y) (F : Y.Presheaf C) (x : X) :
F.stalk (f x) ≅ (pullbackObj f F).stalk x where
hom | Mathlib_Topology_Sheaves_Stalks |
case w
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X Y Z : TopCat
f : X ⟶ Y
F : Presheaf C Y
x : ↑X
j : (OpenNhds (f x))ᵒᵖ
⊢ colimit.ι (((whiskeringLeft (OpenNhds (f x))ᵒᵖ (Opens ↑Y)ᵒᵖ C).obj (OpenNhds.inclusion (f x)).op).obj F) j ≫
(((whiskeringLeft (OpenNhds (f x))ᵒᵖ (Opens ↑Y)ᵒᵖ C).obj (OpenNh... | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Topology.Category.TopCat.OpenNhds
import Mathlib.Topology.Sheaves.Presheaf
import Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing... | induction' j with j | /-- The isomorphism `ℱ_{f(x)} ≅ (f⁻¹ℱ)ₓ`. -/
def stalkPullbackIso (f : X ⟶ Y) (F : Y.Presheaf C) (x : X) :
F.stalk (f x) ≅ (pullbackObj f F).stalk x where
hom := stalkPullbackHom _ _ _ _
inv := stalkPullbackInv _ _ _ _
hom_inv_id := by
delta
stalkPullbackHom stalkPullbackInv stalkFunctor Presheaf.pu... | Mathlib.Topology.Sheaves.Stalks.264_0.hsVUPKIHRY0xmFk | /-- The isomorphism `ℱ_{f(x)} ≅ (f⁻¹ℱ)ₓ`. -/
def stalkPullbackIso (f : X ⟶ Y) (F : Y.Presheaf C) (x : X) :
F.stalk (f x) ≅ (pullbackObj f F).stalk x where
hom | Mathlib_Topology_Sheaves_Stalks |
case w.h
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X Y Z : TopCat
f : X ⟶ Y
F : Presheaf C Y
x : ↑X
j : OpenNhds (f x)
⊢ colimit.ι (((whiskeringLeft (OpenNhds (f x))ᵒᵖ (Opens ↑Y)ᵒᵖ C).obj (OpenNhds.inclusion (f x)).op).obj F) (op j) ≫
(((whiskeringLeft (OpenNhds (f x))ᵒᵖ (Opens ↑Y)ᵒᵖ C).obj (Ope... | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Topology.Category.TopCat.OpenNhds
import Mathlib.Topology.Sheaves.Presheaf
import Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing... | cases j | /-- The isomorphism `ℱ_{f(x)} ≅ (f⁻¹ℱ)ₓ`. -/
def stalkPullbackIso (f : X ⟶ Y) (F : Y.Presheaf C) (x : X) :
F.stalk (f x) ≅ (pullbackObj f F).stalk x where
hom := stalkPullbackHom _ _ _ _
inv := stalkPullbackInv _ _ _ _
hom_inv_id := by
delta
stalkPullbackHom stalkPullbackInv stalkFunctor Presheaf.pu... | Mathlib.Topology.Sheaves.Stalks.264_0.hsVUPKIHRY0xmFk | /-- The isomorphism `ℱ_{f(x)} ≅ (f⁻¹ℱ)ₓ`. -/
def stalkPullbackIso (f : X ⟶ Y) (F : Y.Presheaf C) (x : X) :
F.stalk (f x) ≅ (pullbackObj f F).stalk x where
hom | Mathlib_Topology_Sheaves_Stalks |
case w.h.mk
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X Y Z : TopCat
f : X ⟶ Y
F : Presheaf C Y
x : ↑X
obj✝ : Opens ↑Y
property✝ : f x ∈ obj✝
⊢ colimit.ι (((whiskeringLeft (OpenNhds (f x))ᵒᵖ (Opens ↑Y)ᵒᵖ C).obj (OpenNhds.inclusion (f x)).op).obj F)
(op { obj := obj✝, property := property✝ }) ≫... | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Topology.Category.TopCat.OpenNhds
import Mathlib.Topology.Sheaves.Presheaf
import Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing... | simp only [TopologicalSpace.OpenNhds.inclusionMapIso_inv, whiskerRight_app, whiskerLeft_app,
whiskeringLeft_obj_map, Functor.comp_map, colimit.ι_map_assoc, NatTrans.op_id, lan_obj_map,
pushforwardPullbackAdjunction_unit_app_app, Category.assoc, colimit.ι_pre_assoc] | /-- The isomorphism `ℱ_{f(x)} ≅ (f⁻¹ℱ)ₓ`. -/
def stalkPullbackIso (f : X ⟶ Y) (F : Y.Presheaf C) (x : X) :
F.stalk (f x) ≅ (pullbackObj f F).stalk x where
hom := stalkPullbackHom _ _ _ _
inv := stalkPullbackInv _ _ _ _
hom_inv_id := by
delta
stalkPullbackHom stalkPullbackInv stalkFunctor Presheaf.pu... | Mathlib.Topology.Sheaves.Stalks.264_0.hsVUPKIHRY0xmFk | /-- The isomorphism `ℱ_{f(x)} ≅ (f⁻¹ℱ)ₓ`. -/
def stalkPullbackIso (f : X ⟶ Y) (F : Y.Presheaf C) (x : X) :
F.stalk (f x) ≅ (pullbackObj f F).stalk x where
hom | Mathlib_Topology_Sheaves_Stalks |
case w.h.mk
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X Y Z : TopCat
f : X ⟶ Y
F : Presheaf C Y
x : ↑X
obj✝ : Opens ↑Y
property✝ : f x ∈ obj✝
⊢ colimit.ι
(Lan.diagram (Opens.map f).op F
(op ((Opens.map f).obj ((OpenNhds.inclusion (f x)).op.obj (op { obj := obj✝, property := property✝... | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Topology.Category.TopCat.OpenNhds
import Mathlib.Topology.Sheaves.Presheaf
import Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing... | erw [colimit.ι_desc, colimit.pre_desc, colimit.ι_desc, Category.comp_id] | /-- The isomorphism `ℱ_{f(x)} ≅ (f⁻¹ℱ)ₓ`. -/
def stalkPullbackIso (f : X ⟶ Y) (F : Y.Presheaf C) (x : X) :
F.stalk (f x) ≅ (pullbackObj f F).stalk x where
hom := stalkPullbackHom _ _ _ _
inv := stalkPullbackInv _ _ _ _
hom_inv_id := by
delta
stalkPullbackHom stalkPullbackInv stalkFunctor Presheaf.pu... | Mathlib.Topology.Sheaves.Stalks.264_0.hsVUPKIHRY0xmFk | /-- The isomorphism `ℱ_{f(x)} ≅ (f⁻¹ℱ)ₓ`. -/
def stalkPullbackIso (f : X ⟶ Y) (F : Y.Presheaf C) (x : X) :
F.stalk (f x) ≅ (pullbackObj f F).stalk x where
hom | Mathlib_Topology_Sheaves_Stalks |
case w.h.mk
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X Y Z : TopCat
f : X ⟶ Y
F : Presheaf C Y
x : ↑X
obj✝ : Opens ↑Y
property✝ : f x ∈ obj✝
⊢ (Cocone.whisker
(CostructuredArrow.map
((𝟙 (OpenNhds.map f x ⋙ OpenNhds.inclusion x).op).app (op { obj := obj✝, property := propert... | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Topology.Category.TopCat.OpenNhds
import Mathlib.Topology.Sheaves.Presheaf
import Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing... | simp | /-- The isomorphism `ℱ_{f(x)} ≅ (f⁻¹ℱ)ₓ`. -/
def stalkPullbackIso (f : X ⟶ Y) (F : Y.Presheaf C) (x : X) :
F.stalk (f x) ≅ (pullbackObj f F).stalk x where
hom := stalkPullbackHom _ _ _ _
inv := stalkPullbackInv _ _ _ _
hom_inv_id := by
delta
stalkPullbackHom stalkPullbackInv stalkFunctor Presheaf.pu... | Mathlib.Topology.Sheaves.Stalks.264_0.hsVUPKIHRY0xmFk | /-- The isomorphism `ℱ_{f(x)} ≅ (f⁻¹ℱ)ₓ`. -/
def stalkPullbackIso (f : X ⟶ Y) (F : Y.Presheaf C) (x : X) :
F.stalk (f x) ≅ (pullbackObj f F).stalk x where
hom | Mathlib_Topology_Sheaves_Stalks |
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X Y Z : TopCat
f : X ⟶ Y
F : Presheaf C Y
x : ↑X
⊢ stalkPullbackInv C f F x ≫ stalkPullbackHom C f F x = 𝟙 (stalk (pullbackObj f F) x) | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Topology.Category.TopCat.OpenNhds
import Mathlib.Topology.Sheaves.Presheaf
import Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing... | delta stalkPullbackHom stalkPullbackInv stalkFunctor Presheaf.pullback stalkPushforward | /-- The isomorphism `ℱ_{f(x)} ≅ (f⁻¹ℱ)ₓ`. -/
def stalkPullbackIso (f : X ⟶ Y) (F : Y.Presheaf C) (x : X) :
F.stalk (f x) ≅ (pullbackObj f F).stalk x where
hom := stalkPullbackHom _ _ _ _
inv := stalkPullbackInv _ _ _ _
hom_inv_id := by
delta
stalkPullbackHom stalkPullbackInv stalkFunctor Presheaf.pu... | Mathlib.Topology.Sheaves.Stalks.264_0.hsVUPKIHRY0xmFk | /-- The isomorphism `ℱ_{f(x)} ≅ (f⁻¹ℱ)ₓ`. -/
def stalkPullbackIso (f : X ⟶ Y) (F : Y.Presheaf C) (x : X) :
F.stalk (f x) ≅ (pullbackObj f F).stalk x where
hom | Mathlib_Topology_Sheaves_Stalks |
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X Y Z : TopCat
f : X ⟶ Y
F : Presheaf C Y
x : ↑X
⊢ colimit.desc ((OpenNhds.inclusion x).op ⋙ pullbackObj f F)
{ pt := stalk F (f x),
ι :=
NatTrans.mk fun U => germToPullbackStalk C f F U.unop.obj { val := x, property := (_ : x ∈ U... | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Topology.Category.TopCat.OpenNhds
import Mathlib.Topology.Sheaves.Presheaf
import Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing... | change (_ : colimit _ ⟶ _) = (_ : colimit _ ⟶ _) | /-- The isomorphism `ℱ_{f(x)} ≅ (f⁻¹ℱ)ₓ`. -/
def stalkPullbackIso (f : X ⟶ Y) (F : Y.Presheaf C) (x : X) :
F.stalk (f x) ≅ (pullbackObj f F).stalk x where
hom := stalkPullbackHom _ _ _ _
inv := stalkPullbackInv _ _ _ _
hom_inv_id := by
delta
stalkPullbackHom stalkPullbackInv stalkFunctor Presheaf.pu... | Mathlib.Topology.Sheaves.Stalks.264_0.hsVUPKIHRY0xmFk | /-- The isomorphism `ℱ_{f(x)} ≅ (f⁻¹ℱ)ₓ`. -/
def stalkPullbackIso (f : X ⟶ Y) (F : Y.Presheaf C) (x : X) :
F.stalk (f x) ≅ (pullbackObj f F).stalk x where
hom | Mathlib_Topology_Sheaves_Stalks |
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X Y Z : TopCat
f : X ⟶ Y
F : Presheaf C Y
x : ↑X
⊢ colimit.desc ((OpenNhds.inclusion x).op ⋙ pullbackObj f F)
{ pt := stalk F (f x),
ι :=
NatTrans.mk fun U => germToPullbackStalk C f F U.unop.obj { val := x, property := (_ : x ∈ U... | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Topology.Category.TopCat.OpenNhds
import Mathlib.Topology.Sheaves.Presheaf
import Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing... | ext ⟨U_obj, U_property⟩ | /-- The isomorphism `ℱ_{f(x)} ≅ (f⁻¹ℱ)ₓ`. -/
def stalkPullbackIso (f : X ⟶ Y) (F : Y.Presheaf C) (x : X) :
F.stalk (f x) ≅ (pullbackObj f F).stalk x where
hom := stalkPullbackHom _ _ _ _
inv := stalkPullbackInv _ _ _ _
hom_inv_id := by
delta
stalkPullbackHom stalkPullbackInv stalkFunctor Presheaf.pu... | Mathlib.Topology.Sheaves.Stalks.264_0.hsVUPKIHRY0xmFk | /-- The isomorphism `ℱ_{f(x)} ≅ (f⁻¹ℱ)ₓ`. -/
def stalkPullbackIso (f : X ⟶ Y) (F : Y.Presheaf C) (x : X) :
F.stalk (f x) ≅ (pullbackObj f F).stalk x where
hom | Mathlib_Topology_Sheaves_Stalks |
case w.mk.mk
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X Y Z : TopCat
f : X ⟶ Y
F : Presheaf C Y
x : ↑X
U_obj : Opens ↑X
U_property : x ∈ U_obj
⊢ colimit.ι (((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion x).op).obj (pullbackObj f F))
{ unop := { obj := U_obj, property ... | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Topology.Category.TopCat.OpenNhds
import Mathlib.Topology.Sheaves.Presheaf
import Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing... | change (_ : colimit _ ⟶ _) = (_ : colimit _ ⟶ _) | /-- The isomorphism `ℱ_{f(x)} ≅ (f⁻¹ℱ)ₓ`. -/
def stalkPullbackIso (f : X ⟶ Y) (F : Y.Presheaf C) (x : X) :
F.stalk (f x) ≅ (pullbackObj f F).stalk x where
hom := stalkPullbackHom _ _ _ _
inv := stalkPullbackInv _ _ _ _
hom_inv_id := by
delta
stalkPullbackHom stalkPullbackInv stalkFunctor Presheaf.pu... | Mathlib.Topology.Sheaves.Stalks.264_0.hsVUPKIHRY0xmFk | /-- The isomorphism `ℱ_{f(x)} ≅ (f⁻¹ℱ)ₓ`. -/
def stalkPullbackIso (f : X ⟶ Y) (F : Y.Presheaf C) (x : X) :
F.stalk (f x) ≅ (pullbackObj f F).stalk x where
hom | Mathlib_Topology_Sheaves_Stalks |
case w.mk.mk
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X Y Z : TopCat
f : X ⟶ Y
F : Presheaf C Y
x : ↑X
U_obj : Opens ↑X
U_property : x ∈ U_obj
⊢ colimit.ι (((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion x).op).obj (pullbackObj f F))
{ unop := { obj := U_obj, property ... | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Topology.Category.TopCat.OpenNhds
import Mathlib.Topology.Sheaves.Presheaf
import Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing... | ext ⟨j_left, ⟨⟨⟩⟩, j_hom⟩ | /-- The isomorphism `ℱ_{f(x)} ≅ (f⁻¹ℱ)ₓ`. -/
def stalkPullbackIso (f : X ⟶ Y) (F : Y.Presheaf C) (x : X) :
F.stalk (f x) ≅ (pullbackObj f F).stalk x where
hom := stalkPullbackHom _ _ _ _
inv := stalkPullbackInv _ _ _ _
hom_inv_id := by
delta
stalkPullbackHom stalkPullbackInv stalkFunctor Presheaf.pu... | Mathlib.Topology.Sheaves.Stalks.264_0.hsVUPKIHRY0xmFk | /-- The isomorphism `ℱ_{f(x)} ≅ (f⁻¹ℱ)ₓ`. -/
def stalkPullbackIso (f : X ⟶ Y) (F : Y.Presheaf C) (x : X) :
F.stalk (f x) ≅ (pullbackObj f F).stalk x where
hom | Mathlib_Topology_Sheaves_Stalks |
case w.mk.mk.w.mk.mk.unit
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X Y Z : TopCat
f : X ⟶ Y
F : Presheaf C Y
x : ↑X
U_obj : Opens ↑X
U_property : x ∈ U_obj
j_left : (Opens ↑Y)ᵒᵖ
j_hom :
(Opens.map f).op.obj j_left ⟶
(Functor.fromPUnit ((OpenNhds.inclusion x).op.obj { unop := { obj := U_obj, pro... | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Topology.Category.TopCat.OpenNhds
import Mathlib.Topology.Sheaves.Presheaf
import Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing... | erw [colimit.map_desc, colimit.map_desc, colimit.ι_desc_assoc, colimit.ι_desc_assoc,
colimit.ι_desc, Category.comp_id] | /-- The isomorphism `ℱ_{f(x)} ≅ (f⁻¹ℱ)ₓ`. -/
def stalkPullbackIso (f : X ⟶ Y) (F : Y.Presheaf C) (x : X) :
F.stalk (f x) ≅ (pullbackObj f F).stalk x where
hom := stalkPullbackHom _ _ _ _
inv := stalkPullbackInv _ _ _ _
hom_inv_id := by
delta
stalkPullbackHom stalkPullbackInv stalkFunctor Presheaf.pu... | Mathlib.Topology.Sheaves.Stalks.264_0.hsVUPKIHRY0xmFk | /-- The isomorphism `ℱ_{f(x)} ≅ (f⁻¹ℱ)ₓ`. -/
def stalkPullbackIso (f : X ⟶ Y) (F : Y.Presheaf C) (x : X) :
F.stalk (f x) ≅ (pullbackObj f F).stalk x where
hom | Mathlib_Topology_Sheaves_Stalks |
case w.mk.mk.w.mk.mk.unit
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X Y Z : TopCat
f : X ⟶ Y
F : Presheaf C Y
x : ↑X
U_obj : Opens ↑X
U_property : x ∈ U_obj
j_left : (Opens ↑Y)ᵒᵖ
j_hom :
(Opens.map f).op.obj j_left ⟶
(Functor.fromPUnit ((OpenNhds.inclusion x).op.obj { unop := { obj := U_obj, pro... | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Topology.Category.TopCat.OpenNhds
import Mathlib.Topology.Sheaves.Presheaf
import Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing... | simp only [Cocone.whisker_ι, colimit.cocone_ι, OpenNhds.inclusionMapIso_inv,
Cocones.precompose_obj_ι, whiskerRight_app, whiskerLeft_app, NatTrans.comp_app,
whiskeringLeft_obj_map, NatTrans.op_id, lan_obj_map,
pushforwardPullbackAdjunction_unit_app_app] | /-- The isomorphism `ℱ_{f(x)} ≅ (f⁻¹ℱ)ₓ`. -/
def stalkPullbackIso (f : X ⟶ Y) (F : Y.Presheaf C) (x : X) :
F.stalk (f x) ≅ (pullbackObj f F).stalk x where
hom := stalkPullbackHom _ _ _ _
inv := stalkPullbackInv _ _ _ _
hom_inv_id := by
delta
stalkPullbackHom stalkPullbackInv stalkFunctor Presheaf.pu... | Mathlib.Topology.Sheaves.Stalks.264_0.hsVUPKIHRY0xmFk | /-- The isomorphism `ℱ_{f(x)} ≅ (f⁻¹ℱ)ₓ`. -/
def stalkPullbackIso (f : X ⟶ Y) (F : Y.Presheaf C) (x : X) :
F.stalk (f x) ≅ (pullbackObj f F).stalk x where
hom | Mathlib_Topology_Sheaves_Stalks |
case w.mk.mk.w.mk.mk.unit
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X Y Z : TopCat
f : X ⟶ Y
F : Presheaf C Y
x : ↑X
U_obj : Opens ↑X
U_property : x ∈ U_obj
j_left : (Opens ↑Y)ᵒᵖ
j_hom :
(Opens.map f).op.obj j_left ⟶
(Functor.fromPUnit ((OpenNhds.inclusion x).op.obj { unop := { obj := U_obj, pro... | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Topology.Category.TopCat.OpenNhds
import Mathlib.Topology.Sheaves.Presheaf
import Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing... | erw [←
colimit.w _
(@homOfLE (OpenNhds x) _ ⟨_, U_property⟩
⟨(Opens.map f).obj (unop j_left), j_hom.unop.le U_property⟩ j_hom.unop.le).op] | /-- The isomorphism `ℱ_{f(x)} ≅ (f⁻¹ℱ)ₓ`. -/
def stalkPullbackIso (f : X ⟶ Y) (F : Y.Presheaf C) (x : X) :
F.stalk (f x) ≅ (pullbackObj f F).stalk x where
hom := stalkPullbackHom _ _ _ _
inv := stalkPullbackInv _ _ _ _
hom_inv_id := by
delta
stalkPullbackHom stalkPullbackInv stalkFunctor Presheaf.pu... | Mathlib.Topology.Sheaves.Stalks.264_0.hsVUPKIHRY0xmFk | /-- The isomorphism `ℱ_{f(x)} ≅ (f⁻¹ℱ)ₓ`. -/
def stalkPullbackIso (f : X ⟶ Y) (F : Y.Presheaf C) (x : X) :
F.stalk (f x) ≅ (pullbackObj f F).stalk x where
hom | Mathlib_Topology_Sheaves_Stalks |
case w.mk.mk.w.mk.mk.unit
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X Y Z : TopCat
f : X ⟶ Y
F : Presheaf C Y
x : ↑X
U_obj : Opens ↑X
U_property : x ∈ U_obj
j_left : (Opens ↑Y)ᵒᵖ
j_hom :
(Opens.map f).op.obj j_left ⟶
(Functor.fromPUnit ((OpenNhds.inclusion x).op.obj { unop := { obj := U_obj, pro... | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Topology.Category.TopCat.OpenNhds
import Mathlib.Topology.Sheaves.Presheaf
import Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing... | erw [colimit.ι_pre_assoc (Lan.diagram _ F _) (CostructuredArrow.map _)] | /-- The isomorphism `ℱ_{f(x)} ≅ (f⁻¹ℱ)ₓ`. -/
def stalkPullbackIso (f : X ⟶ Y) (F : Y.Presheaf C) (x : X) :
F.stalk (f x) ≅ (pullbackObj f F).stalk x where
hom := stalkPullbackHom _ _ _ _
inv := stalkPullbackInv _ _ _ _
hom_inv_id := by
delta
stalkPullbackHom stalkPullbackInv stalkFunctor Presheaf.pu... | Mathlib.Topology.Sheaves.Stalks.264_0.hsVUPKIHRY0xmFk | /-- The isomorphism `ℱ_{f(x)} ≅ (f⁻¹ℱ)ₓ`. -/
def stalkPullbackIso (f : X ⟶ Y) (F : Y.Presheaf C) (x : X) :
F.stalk (f x) ≅ (pullbackObj f F).stalk x where
hom | Mathlib_Topology_Sheaves_Stalks |
case w.mk.mk.w.mk.mk.unit
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X Y Z : TopCat
f : X ⟶ Y
F : Presheaf C Y
x : ↑X
U_obj : Opens ↑X
U_property : x ∈ U_obj
j_left : (Opens ↑Y)ᵒᵖ
j_hom :
(Opens.map f).op.obj j_left ⟶
(Functor.fromPUnit ((OpenNhds.inclusion x).op.obj { unop := { obj := U_obj, pro... | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Topology.Category.TopCat.OpenNhds
import Mathlib.Topology.Sheaves.Presheaf
import Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing... | erw [colimit.ι_pre_assoc (Lan.diagram _ F (op U_obj)) (CostructuredArrow.map _)] | /-- The isomorphism `ℱ_{f(x)} ≅ (f⁻¹ℱ)ₓ`. -/
def stalkPullbackIso (f : X ⟶ Y) (F : Y.Presheaf C) (x : X) :
F.stalk (f x) ≅ (pullbackObj f F).stalk x where
hom := stalkPullbackHom _ _ _ _
inv := stalkPullbackInv _ _ _ _
hom_inv_id := by
delta
stalkPullbackHom stalkPullbackInv stalkFunctor Presheaf.pu... | Mathlib.Topology.Sheaves.Stalks.264_0.hsVUPKIHRY0xmFk | /-- The isomorphism `ℱ_{f(x)} ≅ (f⁻¹ℱ)ₓ`. -/
def stalkPullbackIso (f : X ⟶ Y) (F : Y.Presheaf C) (x : X) :
F.stalk (f x) ≅ (pullbackObj f F).stalk x where
hom | Mathlib_Topology_Sheaves_Stalks |
case w.mk.mk.w.mk.mk.unit
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X Y Z : TopCat
f : X ⟶ Y
F : Presheaf C Y
x : ↑X
U_obj : Opens ↑X
U_property : x ∈ U_obj
j_left : (Opens ↑Y)ᵒᵖ
j_hom :
(Opens.map f).op.obj j_left ⟶
(Functor.fromPUnit ((OpenNhds.inclusion x).op.obj { unop := { obj := U_obj, pro... | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Topology.Category.TopCat.OpenNhds
import Mathlib.Topology.Sheaves.Presheaf
import Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing... | rfl | /-- The isomorphism `ℱ_{f(x)} ≅ (f⁻¹ℱ)ₓ`. -/
def stalkPullbackIso (f : X ⟶ Y) (F : Y.Presheaf C) (x : X) :
F.stalk (f x) ≅ (pullbackObj f F).stalk x where
hom := stalkPullbackHom _ _ _ _
inv := stalkPullbackInv _ _ _ _
hom_inv_id := by
delta
stalkPullbackHom stalkPullbackInv stalkFunctor Presheaf.pu... | Mathlib.Topology.Sheaves.Stalks.264_0.hsVUPKIHRY0xmFk | /-- The isomorphism `ℱ_{f(x)} ≅ (f⁻¹ℱ)ₓ`. -/
def stalkPullbackIso (f : X ⟶ Y) (F : Y.Presheaf C) (x : X) :
F.stalk (f x) ≅ (pullbackObj f F).stalk x where
hom | Mathlib_Topology_Sheaves_Stalks |
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X Y Z : TopCat
F : Presheaf C X
x y : ↑X
h : x ⤳ y
⊢ stalk F y ⟶ stalk F x | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Topology.Category.TopCat.OpenNhds
import Mathlib.Topology.Sheaves.Presheaf
import Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing... | refine' colimit.desc _ ⟨_, fun U => _, _⟩ | /-- If `x` specializes to `y`, then there is a natural map `F.stalk y ⟶ F.stalk x`. -/
noncomputable def stalkSpecializes (F : X.Presheaf C) {x y : X} (h : x ⤳ y) :
F.stalk y ⟶ F.stalk x := by
| Mathlib.Topology.Sheaves.Stalks.310_0.hsVUPKIHRY0xmFk | /-- If `x` specializes to `y`, then there is a natural map `F.stalk y ⟶ F.stalk x`. -/
noncomputable def stalkSpecializes (F : X.Presheaf C) {x y : X} (h : x ⤳ y) :
F.stalk y ⟶ F.stalk x | Mathlib_Topology_Sheaves_Stalks |
case refine'_1
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X Y Z : TopCat
F : Presheaf C X
x y : ↑X
h : x ⤳ y
U : (OpenNhds y)ᵒᵖ
⊢ (((whiskeringLeft (OpenNhds y)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion y).op).obj F).obj U ⟶
((Functor.const (OpenNhds y)ᵒᵖ).obj
(colim.obj (((whiskeringLeft... | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Topology.Category.TopCat.OpenNhds
import Mathlib.Topology.Sheaves.Presheaf
import Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing... | exact
colimit.ι ((OpenNhds.inclusion x).op ⋙ F)
(op ⟨(unop U).1, (specializes_iff_forall_open.mp h _ (unop U).1.2 (unop U).2 : _)⟩) | /-- If `x` specializes to `y`, then there is a natural map `F.stalk y ⟶ F.stalk x`. -/
noncomputable def stalkSpecializes (F : X.Presheaf C) {x y : X} (h : x ⤳ y) :
F.stalk y ⟶ F.stalk x := by
refine' colimit.desc _ ⟨_, fun U => _, _⟩
· | Mathlib.Topology.Sheaves.Stalks.310_0.hsVUPKIHRY0xmFk | /-- If `x` specializes to `y`, then there is a natural map `F.stalk y ⟶ F.stalk x`. -/
noncomputable def stalkSpecializes (F : X.Presheaf C) {x y : X} (h : x ⤳ y) :
F.stalk y ⟶ F.stalk x | Mathlib_Topology_Sheaves_Stalks |
case refine'_2
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X Y Z : TopCat
F : Presheaf C X
x y : ↑X
h : x ⤳ y
⊢ ∀ ⦃X_1 Y : (OpenNhds y)ᵒᵖ⦄ (f : X_1 ⟶ Y),
(((whiskeringLeft (OpenNhds y)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion y).op).obj F).map f ≫
(fun U =>
colimit.ι ((OpenNhds.... | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Topology.Category.TopCat.OpenNhds
import Mathlib.Topology.Sheaves.Presheaf
import Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing... | intro U V i | /-- If `x` specializes to `y`, then there is a natural map `F.stalk y ⟶ F.stalk x`. -/
noncomputable def stalkSpecializes (F : X.Presheaf C) {x y : X} (h : x ⤳ y) :
F.stalk y ⟶ F.stalk x := by
refine' colimit.desc _ ⟨_, fun U => _, _⟩
· exact
colimit.ι ((OpenNhds.inclusion x).op ⋙ F)
(op ⟨(unop U)... | Mathlib.Topology.Sheaves.Stalks.310_0.hsVUPKIHRY0xmFk | /-- If `x` specializes to `y`, then there is a natural map `F.stalk y ⟶ F.stalk x`. -/
noncomputable def stalkSpecializes (F : X.Presheaf C) {x y : X} (h : x ⤳ y) :
F.stalk y ⟶ F.stalk x | Mathlib_Topology_Sheaves_Stalks |
case refine'_2
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X Y Z : TopCat
F : Presheaf C X
x y : ↑X
h : x ⤳ y
U V : (OpenNhds y)ᵒᵖ
i : U ⟶ V
⊢ (((whiskeringLeft (OpenNhds y)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion y).op).obj F).map i ≫
(fun U =>
colimit.ι ((OpenNhds.inclusion x).op ⋙ F... | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Topology.Category.TopCat.OpenNhds
import Mathlib.Topology.Sheaves.Presheaf
import Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing... | dsimp | /-- If `x` specializes to `y`, then there is a natural map `F.stalk y ⟶ F.stalk x`. -/
noncomputable def stalkSpecializes (F : X.Presheaf C) {x y : X} (h : x ⤳ y) :
F.stalk y ⟶ F.stalk x := by
refine' colimit.desc _ ⟨_, fun U => _, _⟩
· exact
colimit.ι ((OpenNhds.inclusion x).op ⋙ F)
(op ⟨(unop U)... | Mathlib.Topology.Sheaves.Stalks.310_0.hsVUPKIHRY0xmFk | /-- If `x` specializes to `y`, then there is a natural map `F.stalk y ⟶ F.stalk x`. -/
noncomputable def stalkSpecializes (F : X.Presheaf C) {x y : X} (h : x ⤳ y) :
F.stalk y ⟶ F.stalk x | Mathlib_Topology_Sheaves_Stalks |
case refine'_2
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X Y Z : TopCat
F : Presheaf C X
x y : ↑X
h : x ⤳ y
U V : (OpenNhds y)ᵒᵖ
i : U ⟶ V
⊢ F.map ((OpenNhds.inclusion y).map i.unop).op ≫
colimit.ι ((OpenNhds.inclusion x).op ⋙ F) (op { obj := V.unop.obj, property := (_ : x ∈ ↑V.unop.obj) }) =
... | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Topology.Category.TopCat.OpenNhds
import Mathlib.Topology.Sheaves.Presheaf
import Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing... | rw [Category.comp_id] | /-- If `x` specializes to `y`, then there is a natural map `F.stalk y ⟶ F.stalk x`. -/
noncomputable def stalkSpecializes (F : X.Presheaf C) {x y : X} (h : x ⤳ y) :
F.stalk y ⟶ F.stalk x := by
refine' colimit.desc _ ⟨_, fun U => _, _⟩
· exact
colimit.ι ((OpenNhds.inclusion x).op ⋙ F)
(op ⟨(unop U)... | Mathlib.Topology.Sheaves.Stalks.310_0.hsVUPKIHRY0xmFk | /-- If `x` specializes to `y`, then there is a natural map `F.stalk y ⟶ F.stalk x`. -/
noncomputable def stalkSpecializes (F : X.Presheaf C) {x y : X} (h : x ⤳ y) :
F.stalk y ⟶ F.stalk x | Mathlib_Topology_Sheaves_Stalks |
case refine'_2
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X Y Z : TopCat
F : Presheaf C X
x y : ↑X
h : x ⤳ y
U V : (OpenNhds y)ᵒᵖ
i : U ⟶ V
⊢ F.map ((OpenNhds.inclusion y).map i.unop).op ≫
colimit.ι ((OpenNhds.inclusion x).op ⋙ F) (op { obj := V.unop.obj, property := (_ : x ∈ ↑V.unop.obj) }) =
... | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Topology.Category.TopCat.OpenNhds
import Mathlib.Topology.Sheaves.Presheaf
import Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing... | let U' : OpenNhds x := ⟨_, (specializes_iff_forall_open.mp h _ (unop U).1.2 (unop U).2 : _)⟩ | /-- If `x` specializes to `y`, then there is a natural map `F.stalk y ⟶ F.stalk x`. -/
noncomputable def stalkSpecializes (F : X.Presheaf C) {x y : X} (h : x ⤳ y) :
F.stalk y ⟶ F.stalk x := by
refine' colimit.desc _ ⟨_, fun U => _, _⟩
· exact
colimit.ι ((OpenNhds.inclusion x).op ⋙ F)
(op ⟨(unop U)... | Mathlib.Topology.Sheaves.Stalks.310_0.hsVUPKIHRY0xmFk | /-- If `x` specializes to `y`, then there is a natural map `F.stalk y ⟶ F.stalk x`. -/
noncomputable def stalkSpecializes (F : X.Presheaf C) {x y : X} (h : x ⤳ y) :
F.stalk y ⟶ F.stalk x | Mathlib_Topology_Sheaves_Stalks |
case refine'_2
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X Y Z : TopCat
F : Presheaf C X
x y : ↑X
h : x ⤳ y
U V : (OpenNhds y)ᵒᵖ
i : U ⟶ V
U' : OpenNhds x := { obj := U.unop.obj, property := (_ : x ∈ U.unop.obj.carrier) }
⊢ F.map ((OpenNhds.inclusion y).map i.unop).op ≫
colimit.ι ((OpenNhds.incl... | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Topology.Category.TopCat.OpenNhds
import Mathlib.Topology.Sheaves.Presheaf
import Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing... | let V' : OpenNhds x := ⟨_, (specializes_iff_forall_open.mp h _ (unop V).1.2 (unop V).2 : _)⟩ | /-- If `x` specializes to `y`, then there is a natural map `F.stalk y ⟶ F.stalk x`. -/
noncomputable def stalkSpecializes (F : X.Presheaf C) {x y : X} (h : x ⤳ y) :
F.stalk y ⟶ F.stalk x := by
refine' colimit.desc _ ⟨_, fun U => _, _⟩
· exact
colimit.ι ((OpenNhds.inclusion x).op ⋙ F)
(op ⟨(unop U)... | Mathlib.Topology.Sheaves.Stalks.310_0.hsVUPKIHRY0xmFk | /-- If `x` specializes to `y`, then there is a natural map `F.stalk y ⟶ F.stalk x`. -/
noncomputable def stalkSpecializes (F : X.Presheaf C) {x y : X} (h : x ⤳ y) :
F.stalk y ⟶ F.stalk x | Mathlib_Topology_Sheaves_Stalks |
case refine'_2
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X Y Z : TopCat
F : Presheaf C X
x y : ↑X
h : x ⤳ y
U V : (OpenNhds y)ᵒᵖ
i : U ⟶ V
U' : OpenNhds x := { obj := U.unop.obj, property := (_ : x ∈ U.unop.obj.carrier) }
V' : OpenNhds x := { obj := V.unop.obj, property := (_ : x ∈ V.unop.obj.carrier)... | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Topology.Category.TopCat.OpenNhds
import Mathlib.Topology.Sheaves.Presheaf
import Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing... | exact colimit.w ((OpenNhds.inclusion x).op ⋙ F) (show V' ⟶ U' from i.unop).op | /-- If `x` specializes to `y`, then there is a natural map `F.stalk y ⟶ F.stalk x`. -/
noncomputable def stalkSpecializes (F : X.Presheaf C) {x y : X} (h : x ⤳ y) :
F.stalk y ⟶ F.stalk x := by
refine' colimit.desc _ ⟨_, fun U => _, _⟩
· exact
colimit.ι ((OpenNhds.inclusion x).op ⋙ F)
(op ⟨(unop U)... | Mathlib.Topology.Sheaves.Stalks.310_0.hsVUPKIHRY0xmFk | /-- If `x` specializes to `y`, then there is a natural map `F.stalk y ⟶ F.stalk x`. -/
noncomputable def stalkSpecializes (F : X.Presheaf C) {x y : X} (h : x ⤳ y) :
F.stalk y ⟶ F.stalk x | Mathlib_Topology_Sheaves_Stalks |
C✝ : Type u
inst✝³ : Category.{v, u} C✝
inst✝² : HasColimits C✝
X✝ Y Z : TopCat
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : HasColimits C
X : TopCat
F : Presheaf C X
x : ↑X
⊢ stalkSpecializes F (_ : x ⤳ x) = 𝟙 (stalk F x) | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Topology.Category.TopCat.OpenNhds
import Mathlib.Topology.Sheaves.Presheaf
import Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing... | ext | @[simp]
theorem stalkSpecializes_refl {C : Type*} [Category C] [Limits.HasColimits C] {X : TopCat}
(F : X.Presheaf C) (x : X) : F.stalkSpecializes (specializes_refl x) = 𝟙 _ := by
| Mathlib.Topology.Sheaves.Stalks.340_0.hsVUPKIHRY0xmFk | @[simp]
theorem stalkSpecializes_refl {C : Type*} [Category C] [Limits.HasColimits C] {X : TopCat}
(F : X.Presheaf C) (x : X) : F.stalkSpecializes (specializes_refl x) = 𝟙 _ | Mathlib_Topology_Sheaves_Stalks |
case ih
C✝ : Type u
inst✝³ : Category.{v, u} C✝
inst✝² : HasColimits C✝
X✝ Y Z : TopCat
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : HasColimits C
X : TopCat
F : Presheaf C X
x : ↑X
U✝ : Opens ↑X
hxU✝ : x ∈ U✝
⊢ germ F { val := x, property := hxU✝ } ≫ stalkSpecializes F (_ : x ⤳ x) =
germ F { val := x, prope... | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Topology.Category.TopCat.OpenNhds
import Mathlib.Topology.Sheaves.Presheaf
import Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing... | simp | @[simp]
theorem stalkSpecializes_refl {C : Type*} [Category C] [Limits.HasColimits C] {X : TopCat}
(F : X.Presheaf C) (x : X) : F.stalkSpecializes (specializes_refl x) = 𝟙 _ := by
ext
| Mathlib.Topology.Sheaves.Stalks.340_0.hsVUPKIHRY0xmFk | @[simp]
theorem stalkSpecializes_refl {C : Type*} [Category C] [Limits.HasColimits C] {X : TopCat}
(F : X.Presheaf C) (x : X) : F.stalkSpecializes (specializes_refl x) = 𝟙 _ | Mathlib_Topology_Sheaves_Stalks |
C✝ : Type u
inst✝³ : Category.{v, u} C✝
inst✝² : HasColimits C✝
X✝ Y Z : TopCat
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : HasColimits C
X : TopCat
F : Presheaf C X
x y z : ↑X
h : x ⤳ y
h' : y ⤳ z
⊢ stalkSpecializes F h' ≫ stalkSpecializes F h = stalkSpecializes F (_ : x ⤳ z) | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Topology.Category.TopCat.OpenNhds
import Mathlib.Topology.Sheaves.Presheaf
import Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing... | ext | @[reassoc (attr := simp), elementwise (attr := simp)]
theorem stalkSpecializes_comp {C : Type*} [Category C] [Limits.HasColimits C] {X : TopCat}
(F : X.Presheaf C) {x y z : X} (h : x ⤳ y) (h' : y ⤳ z) :
F.stalkSpecializes h' ≫ F.stalkSpecializes h = F.stalkSpecializes (h.trans h') := by
| Mathlib.Topology.Sheaves.Stalks.348_0.hsVUPKIHRY0xmFk | @[reassoc (attr | Mathlib_Topology_Sheaves_Stalks |
case ih
C✝ : Type u
inst✝³ : Category.{v, u} C✝
inst✝² : HasColimits C✝
X✝ Y Z : TopCat
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : HasColimits C
X : TopCat
F : Presheaf C X
x y z : ↑X
h : x ⤳ y
h' : y ⤳ z
U✝ : Opens ↑X
hxU✝ : z ∈ U✝
⊢ germ F { val := z, property := hxU✝ } ≫ stalkSpecializes F h' ≫ stalkSpecial... | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Topology.Category.TopCat.OpenNhds
import Mathlib.Topology.Sheaves.Presheaf
import Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing... | simp | @[reassoc (attr := simp), elementwise (attr := simp)]
theorem stalkSpecializes_comp {C : Type*} [Category C] [Limits.HasColimits C] {X : TopCat}
(F : X.Presheaf C) {x y z : X} (h : x ⤳ y) (h' : y ⤳ z) :
F.stalkSpecializes h' ≫ F.stalkSpecializes h = F.stalkSpecializes (h.trans h') := by
ext
| Mathlib.Topology.Sheaves.Stalks.348_0.hsVUPKIHRY0xmFk | @[reassoc (attr | Mathlib_Topology_Sheaves_Stalks |
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X Y Z : TopCat
F G : Presheaf C X
f : F ⟶ G
x y : ↑X
h : x ⤳ y
⊢ stalkSpecializes F h ≫ (stalkFunctor C x).map f = (stalkFunctor C y).map f ≫ stalkSpecializes G h | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Topology.Category.TopCat.OpenNhds
import Mathlib.Topology.Sheaves.Presheaf
import Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing... | change (_ : colimit _ ⟶ _) = (_ : colimit _ ⟶ _) | @[reassoc (attr := simp), elementwise (attr := simp)]
theorem stalkSpecializes_stalkFunctor_map {F G : X.Presheaf C} (f : F ⟶ G) {x y : X} (h : x ⤳ y) :
F.stalkSpecializes h ≫ (stalkFunctor C x).map f =
(stalkFunctor C y).map f ≫ G.stalkSpecializes h := by
| Mathlib.Topology.Sheaves.Stalks.357_0.hsVUPKIHRY0xmFk | @[reassoc (attr | Mathlib_Topology_Sheaves_Stalks |
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X Y Z : TopCat
F G : Presheaf C X
f : F ⟶ G
x y : ↑X
h : x ⤳ y
⊢ stalkSpecializes F h ≫ (stalkFunctor C x).map f = (stalkFunctor C y).map f ≫ stalkSpecializes G h | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Topology.Category.TopCat.OpenNhds
import Mathlib.Topology.Sheaves.Presheaf
import Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing... | ext | @[reassoc (attr := simp), elementwise (attr := simp)]
theorem stalkSpecializes_stalkFunctor_map {F G : X.Presheaf C} (f : F ⟶ G) {x y : X} (h : x ⤳ y) :
F.stalkSpecializes h ≫ (stalkFunctor C x).map f =
(stalkFunctor C y).map f ≫ G.stalkSpecializes h := by
change (_ : colimit _ ⟶ _) = (_ : colimit _ ⟶ _)
| Mathlib.Topology.Sheaves.Stalks.357_0.hsVUPKIHRY0xmFk | @[reassoc (attr | Mathlib_Topology_Sheaves_Stalks |
case w
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X Y Z : TopCat
F G : Presheaf C X
f : F ⟶ G
x y : ↑X
h : x ⤳ y
j✝ : (OpenNhds y)ᵒᵖ
⊢ colimit.ι (((whiskeringLeft (OpenNhds y)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion y).op).obj F) j✝ ≫
stalkSpecializes F h ≫ (stalkFunctor C x).map f =
colimi... | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Topology.Category.TopCat.OpenNhds
import Mathlib.Topology.Sheaves.Presheaf
import Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing... | delta stalkFunctor | @[reassoc (attr := simp), elementwise (attr := simp)]
theorem stalkSpecializes_stalkFunctor_map {F G : X.Presheaf C} (f : F ⟶ G) {x y : X} (h : x ⤳ y) :
F.stalkSpecializes h ≫ (stalkFunctor C x).map f =
(stalkFunctor C y).map f ≫ G.stalkSpecializes h := by
change (_ : colimit _ ⟶ _) = (_ : colimit _ ⟶ _)
... | Mathlib.Topology.Sheaves.Stalks.357_0.hsVUPKIHRY0xmFk | @[reassoc (attr | Mathlib_Topology_Sheaves_Stalks |
case w
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X Y Z : TopCat
F G : Presheaf C X
f : F ⟶ G
x y : ↑X
h : x ⤳ y
j✝ : (OpenNhds y)ᵒᵖ
⊢ colimit.ι (((whiskeringLeft (OpenNhds y)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion y).op).obj F) j✝ ≫
stalkSpecializes F h ≫
((whiskeringLeft (OpenNhds x)... | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Topology.Category.TopCat.OpenNhds
import Mathlib.Topology.Sheaves.Presheaf
import Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing... | simpa [stalkSpecializes] using by rfl | @[reassoc (attr := simp), elementwise (attr := simp)]
theorem stalkSpecializes_stalkFunctor_map {F G : X.Presheaf C} (f : F ⟶ G) {x y : X} (h : x ⤳ y) :
F.stalkSpecializes h ≫ (stalkFunctor C x).map f =
(stalkFunctor C y).map f ≫ G.stalkSpecializes h := by
change (_ : colimit _ ⟶ _) = (_ : colimit _ ⟶ _)
... | Mathlib.Topology.Sheaves.Stalks.357_0.hsVUPKIHRY0xmFk | @[reassoc (attr | Mathlib_Topology_Sheaves_Stalks |
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X Y Z : TopCat
F G : Presheaf C X
f : F ⟶ G
x y : ↑X
h : x ⤳ y
j✝ : (OpenNhds y)ᵒᵖ
⊢ f.app (op j✝.unop.obj) ≫
colimit.ι ((OpenNhds.inclusion x).op ⋙ G) (op { obj := j✝.unop.obj, property := (_ : x ∈ j✝.unop.obj.carrier) }) =
f.app (op ((OpenNhds.incl... | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Topology.Category.TopCat.OpenNhds
import Mathlib.Topology.Sheaves.Presheaf
import Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing... | rfl | @[reassoc (attr := simp), elementwise (attr := simp)]
theorem stalkSpecializes_stalkFunctor_map {F G : X.Presheaf C} (f : F ⟶ G) {x y : X} (h : x ⤳ y) :
F.stalkSpecializes h ≫ (stalkFunctor C x).map f =
(stalkFunctor C y).map f ≫ G.stalkSpecializes h := by
change (_ : colimit _ ⟶ _) = (_ : colimit _ ⟶ _)
... | Mathlib.Topology.Sheaves.Stalks.357_0.hsVUPKIHRY0xmFk | @[reassoc (attr | Mathlib_Topology_Sheaves_Stalks |
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X Y Z : TopCat
f : X ⟶ Y
F : Presheaf C X
x y : ↑X
h : x ⤳ y
⊢ stalkSpecializes (f _* F) (_ : f x ⤳ f y) ≫ stalkPushforward C f F x = stalkPushforward C f F y ≫ stalkSpecializes F h | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Topology.Category.TopCat.OpenNhds
import Mathlib.Topology.Sheaves.Presheaf
import Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing... | change (_ : colimit _ ⟶ _) = (_ : colimit _ ⟶ _) | @[reassoc, elementwise, simp, nolint simpNF] -- see std4#365 for the simpNF issue
theorem stalkSpecializes_stalkPushforward (f : X ⟶ Y) (F : X.Presheaf C) {x y : X} (h : x ⤳ y) :
(f _* F).stalkSpecializes (f.map_specializes h) ≫ F.stalkPushforward _ f x =
F.stalkPushforward _ f y ≫ F.stalkSpecializes h := by
... | Mathlib.Topology.Sheaves.Stalks.366_0.hsVUPKIHRY0xmFk | @[reassoc, elementwise, simp, nolint simpNF] -- see std4#365 for the simpNF issue
theorem stalkSpecializes_stalkPushforward (f : X ⟶ Y) (F : X.Presheaf C) {x y : X} (h : x ⤳ y) :
(f _* F).stalkSpecializes (f.map_specializes h) ≫ F.stalkPushforward _ f x =
F.stalkPushforward _ f y ≫ F.stalkSpecializes h | Mathlib_Topology_Sheaves_Stalks |
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X Y Z : TopCat
f : X ⟶ Y
F : Presheaf C X
x y : ↑X
h : x ⤳ y
⊢ stalkSpecializes (f _* F) (_ : f x ⤳ f y) ≫ stalkPushforward C f F x = stalkPushforward C f F y ≫ stalkSpecializes F h | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Topology.Category.TopCat.OpenNhds
import Mathlib.Topology.Sheaves.Presheaf
import Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing... | ext | @[reassoc, elementwise, simp, nolint simpNF] -- see std4#365 for the simpNF issue
theorem stalkSpecializes_stalkPushforward (f : X ⟶ Y) (F : X.Presheaf C) {x y : X} (h : x ⤳ y) :
(f _* F).stalkSpecializes (f.map_specializes h) ≫ F.stalkPushforward _ f x =
F.stalkPushforward _ f y ≫ F.stalkSpecializes h := by
... | Mathlib.Topology.Sheaves.Stalks.366_0.hsVUPKIHRY0xmFk | @[reassoc, elementwise, simp, nolint simpNF] -- see std4#365 for the simpNF issue
theorem stalkSpecializes_stalkPushforward (f : X ⟶ Y) (F : X.Presheaf C) {x y : X} (h : x ⤳ y) :
(f _* F).stalkSpecializes (f.map_specializes h) ≫ F.stalkPushforward _ f x =
F.stalkPushforward _ f y ≫ F.stalkSpecializes h | Mathlib_Topology_Sheaves_Stalks |
case w
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X Y Z : TopCat
f : X ⟶ Y
F : Presheaf C X
x y : ↑X
h : x ⤳ y
j✝ : (OpenNhds (f y))ᵒᵖ
⊢ colimit.ι (((whiskeringLeft (OpenNhds (f y))ᵒᵖ (Opens ↑Y)ᵒᵖ C).obj (OpenNhds.inclusion (f y)).op).obj (f _* F)) j✝ ≫
stalkSpecializes (f _* F) (_ : f x ⤳ f y) ≫... | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Topology.Category.TopCat.OpenNhds
import Mathlib.Topology.Sheaves.Presheaf
import Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing... | delta stalkPushforward | @[reassoc, elementwise, simp, nolint simpNF] -- see std4#365 for the simpNF issue
theorem stalkSpecializes_stalkPushforward (f : X ⟶ Y) (F : X.Presheaf C) {x y : X} (h : x ⤳ y) :
(f _* F).stalkSpecializes (f.map_specializes h) ≫ F.stalkPushforward _ f x =
F.stalkPushforward _ f y ≫ F.stalkSpecializes h := by
... | Mathlib.Topology.Sheaves.Stalks.366_0.hsVUPKIHRY0xmFk | @[reassoc, elementwise, simp, nolint simpNF] -- see std4#365 for the simpNF issue
theorem stalkSpecializes_stalkPushforward (f : X ⟶ Y) (F : X.Presheaf C) {x y : X} (h : x ⤳ y) :
(f _* F).stalkSpecializes (f.map_specializes h) ≫ F.stalkPushforward _ f x =
F.stalkPushforward _ f y ≫ F.stalkSpecializes h | Mathlib_Topology_Sheaves_Stalks |
case w
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X Y Z : TopCat
f : X ⟶ Y
F : Presheaf C X
x y : ↑X
h : x ⤳ y
j✝ : (OpenNhds (f y))ᵒᵖ
⊢ colimit.ι (((whiskeringLeft (OpenNhds (f y))ᵒᵖ (Opens ↑Y)ᵒᵖ C).obj (OpenNhds.inclusion (f y)).op).obj (f _* F)) j✝ ≫
stalkSpecializes (f _* F) (_ : f x ⤳ f y) ≫... | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Topology.Category.TopCat.OpenNhds
import Mathlib.Topology.Sheaves.Presheaf
import Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing... | simp only [stalkSpecializes, colimit.ι_desc_assoc, colimit.ι_map_assoc, colimit.ι_pre,
Category.assoc, colimit.pre_desc, colimit.ι_desc] | @[reassoc, elementwise, simp, nolint simpNF] -- see std4#365 for the simpNF issue
theorem stalkSpecializes_stalkPushforward (f : X ⟶ Y) (F : X.Presheaf C) {x y : X} (h : x ⤳ y) :
(f _* F).stalkSpecializes (f.map_specializes h) ≫ F.stalkPushforward _ f x =
F.stalkPushforward _ f y ≫ F.stalkSpecializes h := by
... | Mathlib.Topology.Sheaves.Stalks.366_0.hsVUPKIHRY0xmFk | @[reassoc, elementwise, simp, nolint simpNF] -- see std4#365 for the simpNF issue
theorem stalkSpecializes_stalkPushforward (f : X ⟶ Y) (F : X.Presheaf C) {x y : X} (h : x ⤳ y) :
(f _* F).stalkSpecializes (f.map_specializes h) ≫ F.stalkPushforward _ f x =
F.stalkPushforward _ f y ≫ F.stalkSpecializes h | Mathlib_Topology_Sheaves_Stalks |
case w
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X Y Z : TopCat
f : X ⟶ Y
F : Presheaf C X
x y : ↑X
h : x ⤳ y
j✝ : (OpenNhds (f y))ᵒᵖ
⊢ (whiskerRight (NatTrans.op (OpenNhds.inclusionMapIso f x).inv) F).app
(op { obj := j✝.unop.obj, property := (_ : f x ∈ j✝.unop.obj.carrier) }) ≫
colimit... | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Topology.Category.TopCat.OpenNhds
import Mathlib.Topology.Sheaves.Presheaf
import Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing... | rfl | @[reassoc, elementwise, simp, nolint simpNF] -- see std4#365 for the simpNF issue
theorem stalkSpecializes_stalkPushforward (f : X ⟶ Y) (F : X.Presheaf C) {x y : X} (h : x ⤳ y) :
(f _* F).stalkSpecializes (f.map_specializes h) ≫ F.stalkPushforward _ f x =
F.stalkPushforward _ f y ≫ F.stalkSpecializes h := by
... | Mathlib.Topology.Sheaves.Stalks.366_0.hsVUPKIHRY0xmFk | @[reassoc, elementwise, simp, nolint simpNF] -- see std4#365 for the simpNF issue
theorem stalkSpecializes_stalkPushforward (f : X ⟶ Y) (F : X.Presheaf C) {x y : X} (h : x ⤳ y) :
(f _* F).stalkSpecializes (f.map_specializes h) ≫ F.stalkPushforward _ f x =
F.stalkPushforward _ f y ≫ F.stalkSpecializes h | Mathlib_Topology_Sheaves_Stalks |
C✝ : Type u
inst✝³ : Category.{v, u} C✝
inst✝² : HasColimits C✝
X✝ Y Z : TopCat
X : TopCat
C : Type u_1
inst✝¹ : Category.{?u.683138, u_1} C
inst✝ : HasColimits C
F : Presheaf C X
x y : ↑X
e : Inseparable x y
⊢ stalkSpecializes F (_ : nhds y ≤ nhds x) ≫ stalkSpecializes F (_ : nhds x ≤ nhds y) = 𝟙 (stalk F x) | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Topology.Category.TopCat.OpenNhds
import Mathlib.Topology.Sheaves.Presheaf
import Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing... | simp | /-- The stalks are isomorphic on inseparable points -/
@[simps]
def stalkCongr {X : TopCat} {C : Type*} [Category C] [HasColimits C] (F : X.Presheaf C) {x y : X}
(e : Inseparable x y) : F.stalk x ≅ F.stalk y :=
⟨F.stalkSpecializes e.ge, F.stalkSpecializes e.le, by | Mathlib.Topology.Sheaves.Stalks.378_0.hsVUPKIHRY0xmFk | /-- The stalks are isomorphic on inseparable points -/
@[simps]
def stalkCongr {X : TopCat} {C : Type*} [Category C] [HasColimits C] (F : X.Presheaf C) {x y : X}
(e : Inseparable x y) : F.stalk x ≅ F.stalk y | Mathlib_Topology_Sheaves_Stalks |
C✝ : Type u
inst✝³ : Category.{v, u} C✝
inst✝² : HasColimits C✝
X✝ Y Z : TopCat
X : TopCat
C : Type u_1
inst✝¹ : Category.{?u.683138, u_1} C
inst✝ : HasColimits C
F : Presheaf C X
x y : ↑X
e : Inseparable x y
⊢ stalkSpecializes F (_ : nhds x ≤ nhds y) ≫ stalkSpecializes F (_ : nhds y ≤ nhds x) = 𝟙 (stalk F y) | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Topology.Category.TopCat.OpenNhds
import Mathlib.Topology.Sheaves.Presheaf
import Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing... | simp | /-- The stalks are isomorphic on inseparable points -/
@[simps]
def stalkCongr {X : TopCat} {C : Type*} [Category C] [HasColimits C] (F : X.Presheaf C) {x y : X}
(e : Inseparable x y) : F.stalk x ≅ F.stalk y :=
⟨F.stalkSpecializes e.ge, F.stalkSpecializes e.le, by simp, by | Mathlib.Topology.Sheaves.Stalks.378_0.hsVUPKIHRY0xmFk | /-- The stalks are isomorphic on inseparable points -/
@[simps]
def stalkCongr {X : TopCat} {C : Type*} [Category C] [HasColimits C] (F : X.Presheaf C) {x y : X}
(e : Inseparable x y) : F.stalk x ≅ F.stalk y | Mathlib_Topology_Sheaves_Stalks |
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasColimits C
X Y Z : TopCat
inst✝ : ConcreteCategory C
F : Presheaf C X
U V : Opens ↑X
x : ↑X
hxU : x ∈ U
hxV : x ∈ V
W : Opens ↑X
hxW : x ∈ W
iWU : W ⟶ U
iWV : W ⟶ V
sU : (forget C).obj (F.obj (op U))
sV : (forget C).obj (F.obj (op V))
ih : (F.map iWU.op) sU = (F.map iWV... | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Topology.Category.TopCat.OpenNhds
import Mathlib.Topology.Sheaves.Presheaf
import Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing... | erw [← F.germ_res iWU ⟨x, hxW⟩, ← F.germ_res iWV ⟨x, hxW⟩, comp_apply, comp_apply, ih] | theorem germ_ext (F : X.Presheaf C) {U V : Opens X} {x : X} {hxU : x ∈ U} {hxV : x ∈ V}
(W : Opens X) (hxW : x ∈ W) (iWU : W ⟶ U) (iWV : W ⟶ V) {sU : F.obj (op U)} {sV : F.obj (op V)}
(ih : F.map iWU.op sU = F.map iWV.op sV) :
F.germ ⟨x, hxU⟩ sU = F.germ ⟨x, hxV⟩ sV := by
| Mathlib.Topology.Sheaves.Stalks.400_0.hsVUPKIHRY0xmFk | theorem germ_ext (F : X.Presheaf C) {U V : Opens X} {x : X} {hxU : x ∈ U} {hxV : x ∈ V}
(W : Opens X) (hxW : x ∈ W) (iWU : W ⟶ U) (iWV : W ⟶ V) {sU : F.obj (op U)} {sV : F.obj (op V)}
(ih : F.map iWU.op sU = F.map iWV.op sV) :
F.germ ⟨x, hxU⟩ sU = F.germ ⟨x, hxV⟩ sV | Mathlib_Topology_Sheaves_Stalks |
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasColimits C
X Y Z : TopCat
inst✝¹ : ConcreteCategory C
inst✝ : PreservesFilteredColimits (forget C)
F : Presheaf C X
x : ↑X
t : (forget C).obj (stalk F x)
⊢ ∃ U, ∃ (m : x ∈ U), ∃ s, (germ F { val := x, property := m }) s = t | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Topology.Category.TopCat.OpenNhds
import Mathlib.Topology.Sheaves.Presheaf
import Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing... | obtain ⟨U, s, e⟩ :=
Types.jointly_surjective.{v, v} _ (isColimitOfPreserves (forget C) (colimit.isColimit _)) t | /--
For presheaves valued in a concrete category whose forgetful functor preserves filtered colimits,
every element of the stalk is the germ of a section.
-/
theorem germ_exist (F : X.Presheaf C) (x : X) (t : (stalk.{v, u} F x : Type v)) :
∃ (U : Opens X) (m : x ∈ U) (s : F.obj (op U)), F.germ ⟨x, m⟩ s = t := by
| Mathlib.Topology.Sheaves.Stalks.410_0.hsVUPKIHRY0xmFk | /--
For presheaves valued in a concrete category whose forgetful functor preserves filtered colimits,
every element of the stalk is the germ of a section.
-/
theorem germ_exist (F : X.Presheaf C) (x : X) (t : (stalk.{v, u} F x : Type v)) :
∃ (U : Opens X) (m : x ∈ U) (s : F.obj (op U)), F.germ ⟨x, m⟩ s = t | Mathlib_Topology_Sheaves_Stalks |
case intro.intro
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasColimits C
X Y Z : TopCat
inst✝¹ : ConcreteCategory C
inst✝ : PreservesFilteredColimits (forget C)
F : Presheaf C X
x : ↑X
t : (forget C).obj (stalk F x)
U : (OpenNhds x)ᵒᵖ
s : (((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion x).... | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Topology.Category.TopCat.OpenNhds
import Mathlib.Topology.Sheaves.Presheaf
import Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing... | revert s e | /--
For presheaves valued in a concrete category whose forgetful functor preserves filtered colimits,
every element of the stalk is the germ of a section.
-/
theorem germ_exist (F : X.Presheaf C) (x : X) (t : (stalk.{v, u} F x : Type v)) :
∃ (U : Opens X) (m : x ∈ U) (s : F.obj (op U)), F.germ ⟨x, m⟩ s = t := by
... | Mathlib.Topology.Sheaves.Stalks.410_0.hsVUPKIHRY0xmFk | /--
For presheaves valued in a concrete category whose forgetful functor preserves filtered colimits,
every element of the stalk is the germ of a section.
-/
theorem germ_exist (F : X.Presheaf C) (x : X) (t : (stalk.{v, u} F x : Type v)) :
∃ (U : Opens X) (m : x ∈ U) (s : F.obj (op U)), F.germ ⟨x, m⟩ s = t | Mathlib_Topology_Sheaves_Stalks |
case intro.intro
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasColimits C
X Y Z : TopCat
inst✝¹ : ConcreteCategory C
inst✝ : PreservesFilteredColimits (forget C)
F : Presheaf C X
x : ↑X
t : (forget C).obj (stalk F x)
U : (OpenNhds x)ᵒᵖ
⊢ ∀ (s : (((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusio... | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Topology.Category.TopCat.OpenNhds
import Mathlib.Topology.Sheaves.Presheaf
import Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing... | induction U with | h U => ?_ | /--
For presheaves valued in a concrete category whose forgetful functor preserves filtered colimits,
every element of the stalk is the germ of a section.
-/
theorem germ_exist (F : X.Presheaf C) (x : X) (t : (stalk.{v, u} F x : Type v)) :
∃ (U : Opens X) (m : x ∈ U) (s : F.obj (op U)), F.germ ⟨x, m⟩ s = t := by
... | Mathlib.Topology.Sheaves.Stalks.410_0.hsVUPKIHRY0xmFk | /--
For presheaves valued in a concrete category whose forgetful functor preserves filtered colimits,
every element of the stalk is the germ of a section.
-/
theorem germ_exist (F : X.Presheaf C) (x : X) (t : (stalk.{v, u} F x : Type v)) :
∃ (U : Opens X) (m : x ∈ U) (s : F.obj (op U)), F.germ ⟨x, m⟩ s = t | Mathlib_Topology_Sheaves_Stalks |
case intro.intro
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasColimits C
X Y Z : TopCat
inst✝¹ : ConcreteCategory C
inst✝ : PreservesFilteredColimits (forget C)
F : Presheaf C X
x : ↑X
t : (forget C).obj (stalk F x)
U : (OpenNhds x)ᵒᵖ
⊢ ∀ (s : (((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusio... | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Topology.Category.TopCat.OpenNhds
import Mathlib.Topology.Sheaves.Presheaf
import Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing... | induction U with | h U => ?_ | /--
For presheaves valued in a concrete category whose forgetful functor preserves filtered colimits,
every element of the stalk is the germ of a section.
-/
theorem germ_exist (F : X.Presheaf C) (x : X) (t : (stalk.{v, u} F x : Type v)) :
∃ (U : Opens X) (m : x ∈ U) (s : F.obj (op U)), F.germ ⟨x, m⟩ s = t := by
... | Mathlib.Topology.Sheaves.Stalks.410_0.hsVUPKIHRY0xmFk | /--
For presheaves valued in a concrete category whose forgetful functor preserves filtered colimits,
every element of the stalk is the germ of a section.
-/
theorem germ_exist (F : X.Presheaf C) (x : X) (t : (stalk.{v, u} F x : Type v)) :
∃ (U : Opens X) (m : x ∈ U) (s : F.obj (op U)), F.germ ⟨x, m⟩ s = t | Mathlib_Topology_Sheaves_Stalks |
case intro.intro.h
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasColimits C
X Y Z : TopCat
inst✝¹ : ConcreteCategory C
inst✝ : PreservesFilteredColimits (forget C)
F : Presheaf C X
x : ↑X
t : (forget C).obj (stalk F x)
U : OpenNhds x
⊢ ∀ (s : (((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion ... | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Topology.Category.TopCat.OpenNhds
import Mathlib.Topology.Sheaves.Presheaf
import Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing... | cases' U with V m | /--
For presheaves valued in a concrete category whose forgetful functor preserves filtered colimits,
every element of the stalk is the germ of a section.
-/
theorem germ_exist (F : X.Presheaf C) (x : X) (t : (stalk.{v, u} F x : Type v)) :
∃ (U : Opens X) (m : x ∈ U) (s : F.obj (op U)), F.germ ⟨x, m⟩ s = t := by
... | Mathlib.Topology.Sheaves.Stalks.410_0.hsVUPKIHRY0xmFk | /--
For presheaves valued in a concrete category whose forgetful functor preserves filtered colimits,
every element of the stalk is the germ of a section.
-/
theorem germ_exist (F : X.Presheaf C) (x : X) (t : (stalk.{v, u} F x : Type v)) :
∃ (U : Opens X) (m : x ∈ U) (s : F.obj (op U)), F.germ ⟨x, m⟩ s = t | Mathlib_Topology_Sheaves_Stalks |
case intro.intro.h.mk
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasColimits C
X Y Z : TopCat
inst✝¹ : ConcreteCategory C
inst✝ : PreservesFilteredColimits (forget C)
F : Presheaf C X
x : ↑X
t : (forget C).obj (stalk F x)
V : Opens ↑X
m : x ∈ V
⊢ ∀
(s :
(((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj... | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Topology.Category.TopCat.OpenNhds
import Mathlib.Topology.Sheaves.Presheaf
import Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing... | intro s e | /--
For presheaves valued in a concrete category whose forgetful functor preserves filtered colimits,
every element of the stalk is the germ of a section.
-/
theorem germ_exist (F : X.Presheaf C) (x : X) (t : (stalk.{v, u} F x : Type v)) :
∃ (U : Opens X) (m : x ∈ U) (s : F.obj (op U)), F.germ ⟨x, m⟩ s = t := by
... | Mathlib.Topology.Sheaves.Stalks.410_0.hsVUPKIHRY0xmFk | /--
For presheaves valued in a concrete category whose forgetful functor preserves filtered colimits,
every element of the stalk is the germ of a section.
-/
theorem germ_exist (F : X.Presheaf C) (x : X) (t : (stalk.{v, u} F x : Type v)) :
∃ (U : Opens X) (m : x ∈ U) (s : F.obj (op U)), F.germ ⟨x, m⟩ s = t | Mathlib_Topology_Sheaves_Stalks |
case intro.intro.h.mk
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasColimits C
X Y Z : TopCat
inst✝¹ : ConcreteCategory C
inst✝ : PreservesFilteredColimits (forget C)
F : Presheaf C X
x : ↑X
t : (forget C).obj (stalk F x)
V : Opens ↑X
m : x ∈ V
s :
(((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.in... | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Topology.Category.TopCat.OpenNhds
import Mathlib.Topology.Sheaves.Presheaf
import Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing... | exact ⟨V, m, s, e⟩ | /--
For presheaves valued in a concrete category whose forgetful functor preserves filtered colimits,
every element of the stalk is the germ of a section.
-/
theorem germ_exist (F : X.Presheaf C) (x : X) (t : (stalk.{v, u} F x : Type v)) :
∃ (U : Opens X) (m : x ∈ U) (s : F.obj (op U)), F.germ ⟨x, m⟩ s = t := by
... | Mathlib.Topology.Sheaves.Stalks.410_0.hsVUPKIHRY0xmFk | /--
For presheaves valued in a concrete category whose forgetful functor preserves filtered colimits,
every element of the stalk is the germ of a section.
-/
theorem germ_exist (F : X.Presheaf C) (x : X) (t : (stalk.{v, u} F x : Type v)) :
∃ (U : Opens X) (m : x ∈ U) (s : F.obj (op U)), F.germ ⟨x, m⟩ s = t | Mathlib_Topology_Sheaves_Stalks |
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasColimits C
X Y Z : TopCat
inst✝¹ : ConcreteCategory C
inst✝ : PreservesFilteredColimits (forget C)
F : Presheaf C X
U V : Opens ↑X
x : ↑X
mU : x ∈ U
mV : x ∈ V
s : (forget C).obj (F.obj (op U))
t : (forget C).obj (F.obj (op V))
h : (germ F { val := x, property := mU }) ... | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Topology.Category.TopCat.OpenNhds
import Mathlib.Topology.Sheaves.Presheaf
import Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing... | obtain ⟨W, iU, iV, e⟩ :=
(Types.FilteredColimit.isColimit_eq_iff.{v, v} _
(isColimitOfPreserves _ (colimit.isColimit ((OpenNhds.inclusion x).op ⋙ F)))).mp h | theorem germ_eq (F : X.Presheaf C) {U V : Opens X} (x : X) (mU : x ∈ U) (mV : x ∈ V)
(s : F.obj (op U)) (t : F.obj (op V)) (h : germ F ⟨x, mU⟩ s = germ F ⟨x, mV⟩ t) :
∃ (W : Opens X) (_m : x ∈ W) (iU : W ⟶ U) (iV : W ⟶ V), F.map iU.op s = F.map iV.op t := by
| Mathlib.Topology.Sheaves.Stalks.426_0.hsVUPKIHRY0xmFk | theorem germ_eq (F : X.Presheaf C) {U V : Opens X} (x : X) (mU : x ∈ U) (mV : x ∈ V)
(s : F.obj (op U)) (t : F.obj (op V)) (h : germ F ⟨x, mU⟩ s = germ F ⟨x, mV⟩ t) :
∃ (W : Opens X) (_m : x ∈ W) (iU : W ⟶ U) (iV : W ⟶ V), F.map iU.op s = F.map iV.op t | Mathlib_Topology_Sheaves_Stalks |
case intro.intro.intro
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasColimits C
X Y Z : TopCat
inst✝¹ : ConcreteCategory C
inst✝ : PreservesFilteredColimits (forget C)
F : Presheaf C X
U V : Opens ↑X
x : ↑X
mU : x ∈ U
mV : x ∈ V
s : (forget C).obj (F.obj (op U))
t : (forget C).obj (F.obj (op V))
h : (germ F { val :... | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Topology.Category.TopCat.OpenNhds
import Mathlib.Topology.Sheaves.Presheaf
import Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing... | exact ⟨(unop W).1, (unop W).2, iU.unop, iV.unop, e⟩ | theorem germ_eq (F : X.Presheaf C) {U V : Opens X} (x : X) (mU : x ∈ U) (mV : x ∈ V)
(s : F.obj (op U)) (t : F.obj (op V)) (h : germ F ⟨x, mU⟩ s = germ F ⟨x, mV⟩ t) :
∃ (W : Opens X) (_m : x ∈ W) (iU : W ⟶ U) (iV : W ⟶ V), F.map iU.op s = F.map iV.op t := by
obtain ⟨W, iU, iV, e⟩ :=
(Types.FilteredColimit... | Mathlib.Topology.Sheaves.Stalks.426_0.hsVUPKIHRY0xmFk | theorem germ_eq (F : X.Presheaf C) {U V : Opens X} (x : X) (mU : x ∈ U) (mV : x ∈ V)
(s : F.obj (op U)) (t : F.obj (op V)) (h : germ F ⟨x, mU⟩ s = germ F ⟨x, mV⟩ t) :
∃ (W : Opens X) (_m : x ∈ W) (iU : W ⟶ U) (iV : W ⟶ V), F.map iU.op s = F.map iV.op t | Mathlib_Topology_Sheaves_Stalks |
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasColimits C
X Y Z : TopCat
inst✝¹ : ConcreteCategory C
inst✝ : PreservesFilteredColimits (forget C)
F G : Presheaf C X
f : F ⟶ G
h : ∀ (U : Opens ↑X), Function.Injective ⇑(f.app (op U))
x : ↑X
s t : (forget C).obj ((stalkFunctor C x).obj F)
hst : ((stalkFunctor C x).map ... | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Topology.Category.TopCat.OpenNhds
import Mathlib.Topology.Sheaves.Presheaf
import Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing... | rcases germ_exist F x s with ⟨U₁, hxU₁, s, rfl⟩ | theorem stalkFunctor_map_injective_of_app_injective {F G : Presheaf C X} (f : F ⟶ G)
(h : ∀ U : Opens X, Function.Injective (f.app (op U))) (x : X) :
Function.Injective ((stalkFunctor C x).map f) := fun s t hst => by
| Mathlib.Topology.Sheaves.Stalks.436_0.hsVUPKIHRY0xmFk | theorem stalkFunctor_map_injective_of_app_injective {F G : Presheaf C X} (f : F ⟶ G)
(h : ∀ U : Opens X, Function.Injective (f.app (op U))) (x : X) :
Function.Injective ((stalkFunctor C x).map f) | Mathlib_Topology_Sheaves_Stalks |
case intro.intro.intro
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasColimits C
X Y Z : TopCat
inst✝¹ : ConcreteCategory C
inst✝ : PreservesFilteredColimits (forget C)
F G : Presheaf C X
f : F ⟶ G
h : ∀ (U : Opens ↑X), Function.Injective ⇑(f.app (op U))
x : ↑X
t : (forget C).obj ((stalkFunctor C x).obj F)
U₁ : Open... | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Topology.Category.TopCat.OpenNhds
import Mathlib.Topology.Sheaves.Presheaf
import Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing... | rcases germ_exist F x t with ⟨U₂, hxU₂, t, rfl⟩ | theorem stalkFunctor_map_injective_of_app_injective {F G : Presheaf C X} (f : F ⟶ G)
(h : ∀ U : Opens X, Function.Injective (f.app (op U))) (x : X) :
Function.Injective ((stalkFunctor C x).map f) := fun s t hst => by
rcases germ_exist F x s with ⟨U₁, hxU₁, s, rfl⟩
| Mathlib.Topology.Sheaves.Stalks.436_0.hsVUPKIHRY0xmFk | theorem stalkFunctor_map_injective_of_app_injective {F G : Presheaf C X} (f : F ⟶ G)
(h : ∀ U : Opens X, Function.Injective (f.app (op U))) (x : X) :
Function.Injective ((stalkFunctor C x).map f) | Mathlib_Topology_Sheaves_Stalks |
case intro.intro.intro.intro.intro.intro
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasColimits C
X Y Z : TopCat
inst✝¹ : ConcreteCategory C
inst✝ : PreservesFilteredColimits (forget C)
F G : Presheaf C X
f : F ⟶ G
h : ∀ (U : Opens ↑X), Function.Injective ⇑(f.app (op U))
x : ↑X
U₁ : Opens ↑X
hxU₁ : x ∈ U₁
s : (forg... | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Topology.Category.TopCat.OpenNhds
import Mathlib.Topology.Sheaves.Presheaf
import Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing... | erw [stalkFunctor_map_germ_apply _ ⟨x, _⟩] at hst | theorem stalkFunctor_map_injective_of_app_injective {F G : Presheaf C X} (f : F ⟶ G)
(h : ∀ U : Opens X, Function.Injective (f.app (op U))) (x : X) :
Function.Injective ((stalkFunctor C x).map f) := fun s t hst => by
rcases germ_exist F x s with ⟨U₁, hxU₁, s, rfl⟩
rcases germ_exist F x t with ⟨U₂, hxU₂, t, ... | Mathlib.Topology.Sheaves.Stalks.436_0.hsVUPKIHRY0xmFk | theorem stalkFunctor_map_injective_of_app_injective {F G : Presheaf C X} (f : F ⟶ G)
(h : ∀ U : Opens X, Function.Injective (f.app (op U))) (x : X) :
Function.Injective ((stalkFunctor C x).map f) | Mathlib_Topology_Sheaves_Stalks |
case intro.intro.intro.intro.intro.intro
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasColimits C
X Y Z : TopCat
inst✝¹ : ConcreteCategory C
inst✝ : PreservesFilteredColimits (forget C)
F G : Presheaf C X
f : F ⟶ G
h : ∀ (U : Opens ↑X), Function.Injective ⇑(f.app (op U))
x : ↑X
U₁ : Opens ↑X
hxU₁ : x ∈ U₁
s : (forg... | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Topology.Category.TopCat.OpenNhds
import Mathlib.Topology.Sheaves.Presheaf
import Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing... | erw [stalkFunctor_map_germ_apply _ ⟨x, _⟩] at hst | theorem stalkFunctor_map_injective_of_app_injective {F G : Presheaf C X} (f : F ⟶ G)
(h : ∀ U : Opens X, Function.Injective (f.app (op U))) (x : X) :
Function.Injective ((stalkFunctor C x).map f) := fun s t hst => by
rcases germ_exist F x s with ⟨U₁, hxU₁, s, rfl⟩
rcases germ_exist F x t with ⟨U₂, hxU₂, t, ... | Mathlib.Topology.Sheaves.Stalks.436_0.hsVUPKIHRY0xmFk | theorem stalkFunctor_map_injective_of_app_injective {F G : Presheaf C X} (f : F ⟶ G)
(h : ∀ U : Opens X, Function.Injective (f.app (op U))) (x : X) :
Function.Injective ((stalkFunctor C x).map f) | Mathlib_Topology_Sheaves_Stalks |
case intro.intro.intro.intro.intro.intro
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasColimits C
X Y Z : TopCat
inst✝¹ : ConcreteCategory C
inst✝ : PreservesFilteredColimits (forget C)
F G : Presheaf C X
f : F ⟶ G
h : ∀ (U : Opens ↑X), Function.Injective ⇑(f.app (op U))
x : ↑X
U₁ : Opens ↑X
hxU₁ : x ∈ U₁
s : (forg... | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Topology.Category.TopCat.OpenNhds
import Mathlib.Topology.Sheaves.Presheaf
import Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing... | obtain ⟨W, hxW, iWU₁, iWU₂, heq⟩ := G.germ_eq x hxU₁ hxU₂ _ _ hst | theorem stalkFunctor_map_injective_of_app_injective {F G : Presheaf C X} (f : F ⟶ G)
(h : ∀ U : Opens X, Function.Injective (f.app (op U))) (x : X) :
Function.Injective ((stalkFunctor C x).map f) := fun s t hst => by
rcases germ_exist F x s with ⟨U₁, hxU₁, s, rfl⟩
rcases germ_exist F x t with ⟨U₂, hxU₂, t, ... | Mathlib.Topology.Sheaves.Stalks.436_0.hsVUPKIHRY0xmFk | theorem stalkFunctor_map_injective_of_app_injective {F G : Presheaf C X} (f : F ⟶ G)
(h : ∀ U : Opens X, Function.Injective (f.app (op U))) (x : X) :
Function.Injective ((stalkFunctor C x).map f) | Mathlib_Topology_Sheaves_Stalks |
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasColimits C
X Y Z : TopCat
inst✝¹ : ConcreteCategory C
inst✝ : PreservesFilteredColimits (forget C)
F G : Presheaf C X
f : F ⟶ G
h : ∀ (U : Opens ↑X), Function.Injective ⇑(f.app (op U))
x : ↑X
U₁ : Opens ↑X... | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Topology.Category.TopCat.OpenNhds
import Mathlib.Topology.Sheaves.Presheaf
import Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing... | rw [← comp_apply, ← comp_apply, ← f.naturality, ← f.naturality, comp_apply, comp_apply] at heq | theorem stalkFunctor_map_injective_of_app_injective {F G : Presheaf C X} (f : F ⟶ G)
(h : ∀ U : Opens X, Function.Injective (f.app (op U))) (x : X) :
Function.Injective ((stalkFunctor C x).map f) := fun s t hst => by
rcases germ_exist F x s with ⟨U₁, hxU₁, s, rfl⟩
rcases germ_exist F x t with ⟨U₂, hxU₂, t, ... | Mathlib.Topology.Sheaves.Stalks.436_0.hsVUPKIHRY0xmFk | theorem stalkFunctor_map_injective_of_app_injective {F G : Presheaf C X} (f : F ⟶ G)
(h : ∀ U : Opens X, Function.Injective (f.app (op U))) (x : X) :
Function.Injective ((stalkFunctor C x).map f) | Mathlib_Topology_Sheaves_Stalks |
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasColimits C
X Y Z : TopCat
inst✝¹ : ConcreteCategory C
inst✝ : PreservesFilteredColimits (forget C)
F G : Presheaf C X
f : F ⟶ G
h : ∀ (U : Opens ↑X), Function.Injective ⇑(f.app (op U))
x : ↑X
U₁ : Opens ↑X... | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Topology.Category.TopCat.OpenNhds
import Mathlib.Topology.Sheaves.Presheaf
import Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing... | replace heq := h W heq | theorem stalkFunctor_map_injective_of_app_injective {F G : Presheaf C X} (f : F ⟶ G)
(h : ∀ U : Opens X, Function.Injective (f.app (op U))) (x : X) :
Function.Injective ((stalkFunctor C x).map f) := fun s t hst => by
rcases germ_exist F x s with ⟨U₁, hxU₁, s, rfl⟩
rcases germ_exist F x t with ⟨U₂, hxU₂, t, ... | Mathlib.Topology.Sheaves.Stalks.436_0.hsVUPKIHRY0xmFk | theorem stalkFunctor_map_injective_of_app_injective {F G : Presheaf C X} (f : F ⟶ G)
(h : ∀ U : Opens X, Function.Injective (f.app (op U))) (x : X) :
Function.Injective ((stalkFunctor C x).map f) | Mathlib_Topology_Sheaves_Stalks |
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasColimits C
X Y Z : TopCat
inst✝¹ : ConcreteCategory C
inst✝ : PreservesFilteredColimits (forget C)
F G : Presheaf C X
f : F ⟶ G
h : ∀ (U : Opens ↑X), Function.Injective ⇑(f.app (op U))
x : ↑X
U₁ : Opens ↑X... | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Topology.Category.TopCat.OpenNhds
import Mathlib.Topology.Sheaves.Presheaf
import Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing... | convert congr_arg (F.germ ⟨x, hxW⟩) heq using 1 | theorem stalkFunctor_map_injective_of_app_injective {F G : Presheaf C X} (f : F ⟶ G)
(h : ∀ U : Opens X, Function.Injective (f.app (op U))) (x : X) :
Function.Injective ((stalkFunctor C x).map f) := fun s t hst => by
rcases germ_exist F x s with ⟨U₁, hxU₁, s, rfl⟩
rcases germ_exist F x t with ⟨U₂, hxU₂, t, ... | Mathlib.Topology.Sheaves.Stalks.436_0.hsVUPKIHRY0xmFk | theorem stalkFunctor_map_injective_of_app_injective {F G : Presheaf C X} (f : F ⟶ G)
(h : ∀ U : Opens X, Function.Injective (f.app (op U))) (x : X) :
Function.Injective ((stalkFunctor C x).map f) | Mathlib_Topology_Sheaves_Stalks |
case h.e'_2.h
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasColimits C
X Y Z : TopCat
inst✝¹ : ConcreteCategory C
inst✝ : PreservesFilteredColimits (forget C)
F G : Presheaf C X
f : F ⟶ G
h : ∀ (U : Opens ↑X), Function.Injective ⇑(f.app (op U))
x : ↑X
U₁ : Opens ↑X
hxU₁ : x ∈ U₁
s : (forget C).obj (F.obj (op U₁))
U... | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Topology.Category.TopCat.OpenNhds
import Mathlib.Topology.Sheaves.Presheaf
import Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing... | exacts [(F.germ_res_apply iWU₁ ⟨x, hxW⟩ s).symm, (F.germ_res_apply iWU₂ ⟨x, hxW⟩ t).symm] | theorem stalkFunctor_map_injective_of_app_injective {F G : Presheaf C X} (f : F ⟶ G)
(h : ∀ U : Opens X, Function.Injective (f.app (op U))) (x : X) :
Function.Injective ((stalkFunctor C x).map f) := fun s t hst => by
rcases germ_exist F x s with ⟨U₁, hxU₁, s, rfl⟩
rcases germ_exist F x t with ⟨U₂, hxU₂, t, ... | Mathlib.Topology.Sheaves.Stalks.436_0.hsVUPKIHRY0xmFk | theorem stalkFunctor_map_injective_of_app_injective {F G : Presheaf C X} (f : F ⟶ G)
(h : ∀ U : Opens X, Function.Injective (f.app (op U))) (x : X) :
Function.Injective ((stalkFunctor C x).map f) | Mathlib_Topology_Sheaves_Stalks |
C : Type u
inst✝⁶ : Category.{v, u} C
inst✝⁵ : HasColimits C
X Y Z : TopCat
inst✝⁴ : ConcreteCategory C
inst✝³ : PreservesFilteredColimits (forget C)
inst✝² : HasLimits C
inst✝¹ : PreservesLimits (forget C)
inst✝ : ReflectsIsomorphisms (forget C)
F : Sheaf C X
U : Opens ↑X
s t : (forget C).obj (F.val.obj (op U))
h : ∀ ... | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Topology.Category.TopCat.OpenNhds
import Mathlib.Topology.Sheaves.Presheaf
import Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing... | choose V m i₁ i₂ heq using fun x : U => F.presheaf.germ_eq x.1 x.2 x.2 s t (h x) | /-- Let `F` be a sheaf valued in a concrete category, whose forgetful functor reflects isomorphisms,
preserves limits and filtered colimits. Then two sections who agree on every stalk must be equal.
-/
theorem section_ext (F : Sheaf C X) (U : Opens X) (s t : F.1.obj (op U))
(h : ∀ x : U, F.presheaf.germ x s = F.pre... | Mathlib.Topology.Sheaves.Stalks.453_0.hsVUPKIHRY0xmFk | /-- Let `F` be a sheaf valued in a concrete category, whose forgetful functor reflects isomorphisms,
preserves limits and filtered colimits. Then two sections who agree on every stalk must be equal.
-/
theorem section_ext (F : Sheaf C X) (U : Opens X) (s t : F.1.obj (op U))
(h : ∀ x : U, F.presheaf.germ x s = F.pre... | Mathlib_Topology_Sheaves_Stalks |
C : Type u
inst✝⁶ : Category.{v, u} C
inst✝⁵ : HasColimits C
X Y Z : TopCat
inst✝⁴ : ConcreteCategory C
inst✝³ : PreservesFilteredColimits (forget C)
inst✝² : HasLimits C
inst✝¹ : PreservesLimits (forget C)
inst✝ : ReflectsIsomorphisms (forget C)
F : Sheaf C X
U : Opens ↑X
s t : (forget C).obj (F.val.obj (op U))
h : ∀ ... | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Topology.Category.TopCat.OpenNhds
import Mathlib.Topology.Sheaves.Presheaf
import Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing... | apply F.eq_of_locally_eq' V U i₁ | /-- Let `F` be a sheaf valued in a concrete category, whose forgetful functor reflects isomorphisms,
preserves limits and filtered colimits. Then two sections who agree on every stalk must be equal.
-/
theorem section_ext (F : Sheaf C X) (U : Opens X) (s t : F.1.obj (op U))
(h : ∀ x : U, F.presheaf.germ x s = F.pre... | Mathlib.Topology.Sheaves.Stalks.453_0.hsVUPKIHRY0xmFk | /-- Let `F` be a sheaf valued in a concrete category, whose forgetful functor reflects isomorphisms,
preserves limits and filtered colimits. Then two sections who agree on every stalk must be equal.
-/
theorem section_ext (F : Sheaf C X) (U : Opens X) (s t : F.1.obj (op U))
(h : ∀ x : U, F.presheaf.germ x s = F.pre... | Mathlib_Topology_Sheaves_Stalks |
case hcover
C : Type u
inst✝⁶ : Category.{v, u} C
inst✝⁵ : HasColimits C
X Y Z : TopCat
inst✝⁴ : ConcreteCategory C
inst✝³ : PreservesFilteredColimits (forget C)
inst✝² : HasLimits C
inst✝¹ : PreservesLimits (forget C)
inst✝ : ReflectsIsomorphisms (forget C)
F : Sheaf C X
U : Opens ↑X
s t : (forget C).obj (F.val.obj (o... | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Topology.Category.TopCat.OpenNhds
import Mathlib.Topology.Sheaves.Presheaf
import Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing... | intro x hxU | /-- Let `F` be a sheaf valued in a concrete category, whose forgetful functor reflects isomorphisms,
preserves limits and filtered colimits. Then two sections who agree on every stalk must be equal.
-/
theorem section_ext (F : Sheaf C X) (U : Opens X) (s t : F.1.obj (op U))
(h : ∀ x : U, F.presheaf.germ x s = F.pre... | Mathlib.Topology.Sheaves.Stalks.453_0.hsVUPKIHRY0xmFk | /-- Let `F` be a sheaf valued in a concrete category, whose forgetful functor reflects isomorphisms,
preserves limits and filtered colimits. Then two sections who agree on every stalk must be equal.
-/
theorem section_ext (F : Sheaf C X) (U : Opens X) (s t : F.1.obj (op U))
(h : ∀ x : U, F.presheaf.germ x s = F.pre... | Mathlib_Topology_Sheaves_Stalks |
case hcover
C : Type u
inst✝⁶ : Category.{v, u} C
inst✝⁵ : HasColimits C
X Y Z : TopCat
inst✝⁴ : ConcreteCategory C
inst✝³ : PreservesFilteredColimits (forget C)
inst✝² : HasLimits C
inst✝¹ : PreservesLimits (forget C)
inst✝ : ReflectsIsomorphisms (forget C)
F : Sheaf C X
U : Opens ↑X
s t : (forget C).obj (F.val.obj (o... | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Topology.Category.TopCat.OpenNhds
import Mathlib.Topology.Sheaves.Presheaf
import Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing... | simp only [Opens.coe_iSup, Set.mem_iUnion, SetLike.mem_coe] | /-- Let `F` be a sheaf valued in a concrete category, whose forgetful functor reflects isomorphisms,
preserves limits and filtered colimits. Then two sections who agree on every stalk must be equal.
-/
theorem section_ext (F : Sheaf C X) (U : Opens X) (s t : F.1.obj (op U))
(h : ∀ x : U, F.presheaf.germ x s = F.pre... | Mathlib.Topology.Sheaves.Stalks.453_0.hsVUPKIHRY0xmFk | /-- Let `F` be a sheaf valued in a concrete category, whose forgetful functor reflects isomorphisms,
preserves limits and filtered colimits. Then two sections who agree on every stalk must be equal.
-/
theorem section_ext (F : Sheaf C X) (U : Opens X) (s t : F.1.obj (op U))
(h : ∀ x : U, F.presheaf.germ x s = F.pre... | Mathlib_Topology_Sheaves_Stalks |
case hcover
C : Type u
inst✝⁶ : Category.{v, u} C
inst✝⁵ : HasColimits C
X Y Z : TopCat
inst✝⁴ : ConcreteCategory C
inst✝³ : PreservesFilteredColimits (forget C)
inst✝² : HasLimits C
inst✝¹ : PreservesLimits (forget C)
inst✝ : ReflectsIsomorphisms (forget C)
F : Sheaf C X
U : Opens ↑X
s t : (forget C).obj (F.val.obj (o... | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Topology.Category.TopCat.OpenNhds
import Mathlib.Topology.Sheaves.Presheaf
import Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing... | exact ⟨⟨x, hxU⟩, m ⟨x, hxU⟩⟩ | /-- Let `F` be a sheaf valued in a concrete category, whose forgetful functor reflects isomorphisms,
preserves limits and filtered colimits. Then two sections who agree on every stalk must be equal.
-/
theorem section_ext (F : Sheaf C X) (U : Opens X) (s t : F.1.obj (op U))
(h : ∀ x : U, F.presheaf.germ x s = F.pre... | Mathlib.Topology.Sheaves.Stalks.453_0.hsVUPKIHRY0xmFk | /-- Let `F` be a sheaf valued in a concrete category, whose forgetful functor reflects isomorphisms,
preserves limits and filtered colimits. Then two sections who agree on every stalk must be equal.
-/
theorem section_ext (F : Sheaf C X) (U : Opens X) (s t : F.1.obj (op U))
(h : ∀ x : U, F.presheaf.germ x s = F.pre... | Mathlib_Topology_Sheaves_Stalks |
case h
C : Type u
inst✝⁶ : Category.{v, u} C
inst✝⁵ : HasColimits C
X Y Z : TopCat
inst✝⁴ : ConcreteCategory C
inst✝³ : PreservesFilteredColimits (forget C)
inst✝² : HasLimits C
inst✝¹ : PreservesLimits (forget C)
inst✝ : ReflectsIsomorphisms (forget C)
F : Sheaf C X
U : Opens ↑X
s t : (forget C).obj (F.val.obj (op U))... | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Topology.Category.TopCat.OpenNhds
import Mathlib.Topology.Sheaves.Presheaf
import Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing... | intro x | /-- Let `F` be a sheaf valued in a concrete category, whose forgetful functor reflects isomorphisms,
preserves limits and filtered colimits. Then two sections who agree on every stalk must be equal.
-/
theorem section_ext (F : Sheaf C X) (U : Opens X) (s t : F.1.obj (op U))
(h : ∀ x : U, F.presheaf.germ x s = F.pre... | Mathlib.Topology.Sheaves.Stalks.453_0.hsVUPKIHRY0xmFk | /-- Let `F` be a sheaf valued in a concrete category, whose forgetful functor reflects isomorphisms,
preserves limits and filtered colimits. Then two sections who agree on every stalk must be equal.
-/
theorem section_ext (F : Sheaf C X) (U : Opens X) (s t : F.1.obj (op U))
(h : ∀ x : U, F.presheaf.germ x s = F.pre... | Mathlib_Topology_Sheaves_Stalks |
case h
C : Type u
inst✝⁶ : Category.{v, u} C
inst✝⁵ : HasColimits C
X Y Z : TopCat
inst✝⁴ : ConcreteCategory C
inst✝³ : PreservesFilteredColimits (forget C)
inst✝² : HasLimits C
inst✝¹ : PreservesLimits (forget C)
inst✝ : ReflectsIsomorphisms (forget C)
F : Sheaf C X
U : Opens ↑X
s t : (forget C).obj (F.val.obj (op U))... | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Topology.Category.TopCat.OpenNhds
import Mathlib.Topology.Sheaves.Presheaf
import Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing... | rw [heq, Subsingleton.elim (i₁ x) (i₂ x)] | /-- Let `F` be a sheaf valued in a concrete category, whose forgetful functor reflects isomorphisms,
preserves limits and filtered colimits. Then two sections who agree on every stalk must be equal.
-/
theorem section_ext (F : Sheaf C X) (U : Opens X) (s t : F.1.obj (op U))
(h : ∀ x : U, F.presheaf.germ x s = F.pre... | Mathlib.Topology.Sheaves.Stalks.453_0.hsVUPKIHRY0xmFk | /-- Let `F` be a sheaf valued in a concrete category, whose forgetful functor reflects isomorphisms,
preserves limits and filtered colimits. Then two sections who agree on every stalk must be equal.
-/
theorem section_ext (F : Sheaf C X) (U : Opens X) (s t : F.1.obj (op U))
(h : ∀ x : U, F.presheaf.germ x s = F.pre... | Mathlib_Topology_Sheaves_Stalks |
C : Type u
inst✝⁶ : Category.{v, u} C
inst✝⁵ : HasColimits C
X Y Z : TopCat
inst✝⁴ : ConcreteCategory C
inst✝³ : PreservesFilteredColimits (forget C)
inst✝² : HasLimits C
inst✝¹ : PreservesLimits (forget C)
inst✝ : ReflectsIsomorphisms (forget C)
F : Sheaf C X
G : Presheaf C X
f : F.val ⟶ G
U : Opens ↑X
h : ∀ (x : ↥U),... | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Topology.Category.TopCat.OpenNhds
import Mathlib.Topology.Sheaves.Presheaf
import Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing... | erw [stalkFunctor_map_germ_apply, stalkFunctor_map_germ_apply, hst] | theorem app_injective_of_stalkFunctor_map_injective {F : Sheaf C X} {G : Presheaf C X} (f : F.1 ⟶ G)
(U : Opens X) (h : ∀ x : U, Function.Injective ((stalkFunctor C x.val).map f)) :
Function.Injective (f.app (op U)) := fun s t hst =>
section_ext F _ _ _ fun x =>
h x <| by | Mathlib.Topology.Sheaves.Stalks.477_0.hsVUPKIHRY0xmFk | theorem app_injective_of_stalkFunctor_map_injective {F : Sheaf C X} {G : Presheaf C X} (f : F.1 ⟶ G)
(U : Opens X) (h : ∀ x : U, Function.Injective ((stalkFunctor C x.val).map f)) :
Function.Injective (f.app (op U)) | Mathlib_Topology_Sheaves_Stalks |
C : Type u
inst✝⁶ : Category.{v, u} C
inst✝⁵ : HasColimits C
X Y Z : TopCat
inst✝⁴ : ConcreteCategory C
inst✝³ : PreservesFilteredColimits (forget C)
inst✝² : HasLimits C
inst✝¹ : PreservesLimits (forget C)
inst✝ : ReflectsIsomorphisms (forget C)
F G : Sheaf C X
f : F ⟶ G
U : Opens ↑X
hinj : ∀ (x : ↥U), Function.Inject... | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Topology.Category.TopCat.OpenNhds
import Mathlib.Topology.Sheaves.Presheaf
import Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing... | intro t | /-- For surjectivity, we are given an arbitrary section `t` and need to find a preimage for it.
We claim that it suffices to find preimages *locally*. That is, for each `x : U` we construct
a neighborhood `V ≤ U` and a section `s : F.obj (op V))` such that `f.app (op V) s` and `t`
agree on `V`. -/
theorem app_surjectiv... | Mathlib.Topology.Sheaves.Stalks.530_0.hsVUPKIHRY0xmFk | /-- For surjectivity, we are given an arbitrary section `t` and need to find a preimage for it.
We claim that it suffices to find preimages *locally*. That is, for each `x : U` we construct
a neighborhood `V ≤ U` and a section `s : F.obj (op V))` such that `f.app (op V) s` and `t`
agree on `V`. -/
theorem app_surjectiv... | Mathlib_Topology_Sheaves_Stalks |
C : Type u
inst✝⁶ : Category.{v, u} C
inst✝⁵ : HasColimits C
X Y Z : TopCat
inst✝⁴ : ConcreteCategory C
inst✝³ : PreservesFilteredColimits (forget C)
inst✝² : HasLimits C
inst✝¹ : PreservesLimits (forget C)
inst✝ : ReflectsIsomorphisms (forget C)
F G : Sheaf C X
f : F ⟶ G
U : Opens ↑X
hinj : ∀ (x : ↥U), Function.Inject... | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Topology.Category.TopCat.OpenNhds
import Mathlib.Topology.Sheaves.Presheaf
import Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing... | choose V mV iVU sf heq using hsurj t | /-- For surjectivity, we are given an arbitrary section `t` and need to find a preimage for it.
We claim that it suffices to find preimages *locally*. That is, for each `x : U` we construct
a neighborhood `V ≤ U` and a section `s : F.obj (op V))` such that `f.app (op V) s` and `t`
agree on `V`. -/
theorem app_surjectiv... | Mathlib.Topology.Sheaves.Stalks.530_0.hsVUPKIHRY0xmFk | /-- For surjectivity, we are given an arbitrary section `t` and need to find a preimage for it.
We claim that it suffices to find preimages *locally*. That is, for each `x : U` we construct
a neighborhood `V ≤ U` and a section `s : F.obj (op V))` such that `f.app (op V) s` and `t`
agree on `V`. -/
theorem app_surjectiv... | Mathlib_Topology_Sheaves_Stalks |
C : Type u
inst✝⁶ : Category.{v, u} C
inst✝⁵ : HasColimits C
X Y Z : TopCat
inst✝⁴ : ConcreteCategory C
inst✝³ : PreservesFilteredColimits (forget C)
inst✝² : HasLimits C
inst✝¹ : PreservesLimits (forget C)
inst✝ : ReflectsIsomorphisms (forget C)
F G : Sheaf C X
f : F ⟶ G
U : Opens ↑X
hinj : ∀ (x : ↥U), Function.Inject... | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Topology.Category.TopCat.OpenNhds
import Mathlib.Topology.Sheaves.Presheaf
import Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing... | have V_cover : U ≤ iSup V := by
intro x hxU
simp only [Opens.coe_iSup, Set.mem_iUnion, SetLike.mem_coe]
exact ⟨⟨x, hxU⟩, mV ⟨x, hxU⟩⟩ | /-- For surjectivity, we are given an arbitrary section `t` and need to find a preimage for it.
We claim that it suffices to find preimages *locally*. That is, for each `x : U` we construct
a neighborhood `V ≤ U` and a section `s : F.obj (op V))` such that `f.app (op V) s` and `t`
agree on `V`. -/
theorem app_surjectiv... | Mathlib.Topology.Sheaves.Stalks.530_0.hsVUPKIHRY0xmFk | /-- For surjectivity, we are given an arbitrary section `t` and need to find a preimage for it.
We claim that it suffices to find preimages *locally*. That is, for each `x : U` we construct
a neighborhood `V ≤ U` and a section `s : F.obj (op V))` such that `f.app (op V) s` and `t`
agree on `V`. -/
theorem app_surjectiv... | Mathlib_Topology_Sheaves_Stalks |
C : Type u
inst✝⁶ : Category.{v, u} C
inst✝⁵ : HasColimits C
X Y Z : TopCat
inst✝⁴ : ConcreteCategory C
inst✝³ : PreservesFilteredColimits (forget C)
inst✝² : HasLimits C
inst✝¹ : PreservesLimits (forget C)
inst✝ : ReflectsIsomorphisms (forget C)
F G : Sheaf C X
f : F ⟶ G
U : Opens ↑X
hinj : ∀ (x : ↥U), Function.Inject... | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Topology.Category.TopCat.OpenNhds
import Mathlib.Topology.Sheaves.Presheaf
import Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing... | intro x hxU | /-- For surjectivity, we are given an arbitrary section `t` and need to find a preimage for it.
We claim that it suffices to find preimages *locally*. That is, for each `x : U` we construct
a neighborhood `V ≤ U` and a section `s : F.obj (op V))` such that `f.app (op V) s` and `t`
agree on `V`. -/
theorem app_surjectiv... | Mathlib.Topology.Sheaves.Stalks.530_0.hsVUPKIHRY0xmFk | /-- For surjectivity, we are given an arbitrary section `t` and need to find a preimage for it.
We claim that it suffices to find preimages *locally*. That is, for each `x : U` we construct
a neighborhood `V ≤ U` and a section `s : F.obj (op V))` such that `f.app (op V) s` and `t`
agree on `V`. -/
theorem app_surjectiv... | Mathlib_Topology_Sheaves_Stalks |
C : Type u
inst✝⁶ : Category.{v, u} C
inst✝⁵ : HasColimits C
X Y Z : TopCat
inst✝⁴ : ConcreteCategory C
inst✝³ : PreservesFilteredColimits (forget C)
inst✝² : HasLimits C
inst✝¹ : PreservesLimits (forget C)
inst✝ : ReflectsIsomorphisms (forget C)
F G : Sheaf C X
f : F ⟶ G
U : Opens ↑X
hinj : ∀ (x : ↥U), Function.Inject... | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Topology.Category.TopCat.OpenNhds
import Mathlib.Topology.Sheaves.Presheaf
import Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing... | simp only [Opens.coe_iSup, Set.mem_iUnion, SetLike.mem_coe] | /-- For surjectivity, we are given an arbitrary section `t` and need to find a preimage for it.
We claim that it suffices to find preimages *locally*. That is, for each `x : U` we construct
a neighborhood `V ≤ U` and a section `s : F.obj (op V))` such that `f.app (op V) s` and `t`
agree on `V`. -/
theorem app_surjectiv... | Mathlib.Topology.Sheaves.Stalks.530_0.hsVUPKIHRY0xmFk | /-- For surjectivity, we are given an arbitrary section `t` and need to find a preimage for it.
We claim that it suffices to find preimages *locally*. That is, for each `x : U` we construct
a neighborhood `V ≤ U` and a section `s : F.obj (op V))` such that `f.app (op V) s` and `t`
agree on `V`. -/
theorem app_surjectiv... | Mathlib_Topology_Sheaves_Stalks |
C : Type u
inst✝⁶ : Category.{v, u} C
inst✝⁵ : HasColimits C
X Y Z : TopCat
inst✝⁴ : ConcreteCategory C
inst✝³ : PreservesFilteredColimits (forget C)
inst✝² : HasLimits C
inst✝¹ : PreservesLimits (forget C)
inst✝ : ReflectsIsomorphisms (forget C)
F G : Sheaf C X
f : F ⟶ G
U : Opens ↑X
hinj : ∀ (x : ↥U), Function.Inject... | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Topology.Category.TopCat.OpenNhds
import Mathlib.Topology.Sheaves.Presheaf
import Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing... | exact ⟨⟨x, hxU⟩, mV ⟨x, hxU⟩⟩ | /-- For surjectivity, we are given an arbitrary section `t` and need to find a preimage for it.
We claim that it suffices to find preimages *locally*. That is, for each `x : U` we construct
a neighborhood `V ≤ U` and a section `s : F.obj (op V))` such that `f.app (op V) s` and `t`
agree on `V`. -/
theorem app_surjectiv... | Mathlib.Topology.Sheaves.Stalks.530_0.hsVUPKIHRY0xmFk | /-- For surjectivity, we are given an arbitrary section `t` and need to find a preimage for it.
We claim that it suffices to find preimages *locally*. That is, for each `x : U` we construct
a neighborhood `V ≤ U` and a section `s : F.obj (op V))` such that `f.app (op V) s` and `t`
agree on `V`. -/
theorem app_surjectiv... | Mathlib_Topology_Sheaves_Stalks |
C : Type u
inst✝⁶ : Category.{v, u} C
inst✝⁵ : HasColimits C
X Y Z : TopCat
inst✝⁴ : ConcreteCategory C
inst✝³ : PreservesFilteredColimits (forget C)
inst✝² : HasLimits C
inst✝¹ : PreservesLimits (forget C)
inst✝ : ReflectsIsomorphisms (forget C)
F G : Sheaf C X
f : F ⟶ G
U : Opens ↑X
hinj : ∀ (x : ↥U), Function.Inject... | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Topology.Category.TopCat.OpenNhds
import Mathlib.Topology.Sheaves.Presheaf
import Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing... | suffices IsCompatible F.val V sf by
-- Since `F` is a sheaf, we can glue all the local preimages together to get a global preimage.
obtain ⟨s, s_spec, -⟩ := F.existsUnique_gluing' V U iVU V_cover sf this
· use s
apply G.eq_of_locally_eq' V U iVU V_cover
intro x
rw [← comp_apply, ← f.1.natu... | /-- For surjectivity, we are given an arbitrary section `t` and need to find a preimage for it.
We claim that it suffices to find preimages *locally*. That is, for each `x : U` we construct
a neighborhood `V ≤ U` and a section `s : F.obj (op V))` such that `f.app (op V) s` and `t`
agree on `V`. -/
theorem app_surjectiv... | Mathlib.Topology.Sheaves.Stalks.530_0.hsVUPKIHRY0xmFk | /-- For surjectivity, we are given an arbitrary section `t` and need to find a preimage for it.
We claim that it suffices to find preimages *locally*. That is, for each `x : U` we construct
a neighborhood `V ≤ U` and a section `s : F.obj (op V))` such that `f.app (op V) s` and `t`
agree on `V`. -/
theorem app_surjectiv... | Mathlib_Topology_Sheaves_Stalks |
C : Type u
inst✝⁶ : Category.{v, u} C
inst✝⁵ : HasColimits C
X Y Z : TopCat
inst✝⁴ : ConcreteCategory C
inst✝³ : PreservesFilteredColimits (forget C)
inst✝² : HasLimits C
inst✝¹ : PreservesLimits (forget C)
inst✝ : ReflectsIsomorphisms (forget C)
F G : Sheaf C X
f : F ⟶ G
U : Opens ↑X
hinj : ∀ (x : ↥U), Function.Inject... | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Topology.Category.TopCat.OpenNhds
import Mathlib.Topology.Sheaves.Presheaf
import Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing... | obtain ⟨s, s_spec, -⟩ := F.existsUnique_gluing' V U iVU V_cover sf this | /-- For surjectivity, we are given an arbitrary section `t` and need to find a preimage for it.
We claim that it suffices to find preimages *locally*. That is, for each `x : U` we construct
a neighborhood `V ≤ U` and a section `s : F.obj (op V))` such that `f.app (op V) s` and `t`
agree on `V`. -/
theorem app_surjectiv... | Mathlib.Topology.Sheaves.Stalks.530_0.hsVUPKIHRY0xmFk | /-- For surjectivity, we are given an arbitrary section `t` and need to find a preimage for it.
We claim that it suffices to find preimages *locally*. That is, for each `x : U` we construct
a neighborhood `V ≤ U` and a section `s : F.obj (op V))` such that `f.app (op V) s` and `t`
agree on `V`. -/
theorem app_surjectiv... | Mathlib_Topology_Sheaves_Stalks |
case intro.intro
C : Type u
inst✝⁶ : Category.{v, u} C
inst✝⁵ : HasColimits C
X Y Z : TopCat
inst✝⁴ : ConcreteCategory C
inst✝³ : PreservesFilteredColimits (forget C)
inst✝² : HasLimits C
inst✝¹ : PreservesLimits (forget C)
inst✝ : ReflectsIsomorphisms (forget C)
F G : Sheaf C X
f : F ⟶ G
U : Opens ↑X
hinj : ∀ (x : ↥U)... | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Topology.Category.TopCat.OpenNhds
import Mathlib.Topology.Sheaves.Presheaf
import Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing... | use s | /-- For surjectivity, we are given an arbitrary section `t` and need to find a preimage for it.
We claim that it suffices to find preimages *locally*. That is, for each `x : U` we construct
a neighborhood `V ≤ U` and a section `s : F.obj (op V))` such that `f.app (op V) s` and `t`
agree on `V`. -/
theorem app_surjectiv... | Mathlib.Topology.Sheaves.Stalks.530_0.hsVUPKIHRY0xmFk | /-- For surjectivity, we are given an arbitrary section `t` and need to find a preimage for it.
We claim that it suffices to find preimages *locally*. That is, for each `x : U` we construct
a neighborhood `V ≤ U` and a section `s : F.obj (op V))` such that `f.app (op V) s` and `t`
agree on `V`. -/
theorem app_surjectiv... | Mathlib_Topology_Sheaves_Stalks |
case h
C : Type u
inst✝⁶ : Category.{v, u} C
inst✝⁵ : HasColimits C
X Y Z : TopCat
inst✝⁴ : ConcreteCategory C
inst✝³ : PreservesFilteredColimits (forget C)
inst✝² : HasLimits C
inst✝¹ : PreservesLimits (forget C)
inst✝ : ReflectsIsomorphisms (forget C)
F G : Sheaf C X
f : F ⟶ G
U : Opens ↑X
hinj : ∀ (x : ↥U), Function... | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Topology.Category.TopCat.OpenNhds
import Mathlib.Topology.Sheaves.Presheaf
import Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing... | apply G.eq_of_locally_eq' V U iVU V_cover | /-- For surjectivity, we are given an arbitrary section `t` and need to find a preimage for it.
We claim that it suffices to find preimages *locally*. That is, for each `x : U` we construct
a neighborhood `V ≤ U` and a section `s : F.obj (op V))` such that `f.app (op V) s` and `t`
agree on `V`. -/
theorem app_surjectiv... | Mathlib.Topology.Sheaves.Stalks.530_0.hsVUPKIHRY0xmFk | /-- For surjectivity, we are given an arbitrary section `t` and need to find a preimage for it.
We claim that it suffices to find preimages *locally*. That is, for each `x : U` we construct
a neighborhood `V ≤ U` and a section `s : F.obj (op V))` such that `f.app (op V) s` and `t`
agree on `V`. -/
theorem app_surjectiv... | Mathlib_Topology_Sheaves_Stalks |
case h.h
C : Type u
inst✝⁶ : Category.{v, u} C
inst✝⁵ : HasColimits C
X Y Z : TopCat
inst✝⁴ : ConcreteCategory C
inst✝³ : PreservesFilteredColimits (forget C)
inst✝² : HasLimits C
inst✝¹ : PreservesLimits (forget C)
inst✝ : ReflectsIsomorphisms (forget C)
F G : Sheaf C X
f : F ⟶ G
U : Opens ↑X
hinj : ∀ (x : ↥U), Functi... | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Topology.Category.TopCat.OpenNhds
import Mathlib.Topology.Sheaves.Presheaf
import Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing... | intro x | /-- For surjectivity, we are given an arbitrary section `t` and need to find a preimage for it.
We claim that it suffices to find preimages *locally*. That is, for each `x : U` we construct
a neighborhood `V ≤ U` and a section `s : F.obj (op V))` such that `f.app (op V) s` and `t`
agree on `V`. -/
theorem app_surjectiv... | Mathlib.Topology.Sheaves.Stalks.530_0.hsVUPKIHRY0xmFk | /-- For surjectivity, we are given an arbitrary section `t` and need to find a preimage for it.
We claim that it suffices to find preimages *locally*. That is, for each `x : U` we construct
a neighborhood `V ≤ U` and a section `s : F.obj (op V))` such that `f.app (op V) s` and `t`
agree on `V`. -/
theorem app_surjectiv... | Mathlib_Topology_Sheaves_Stalks |
case h.h
C : Type u
inst✝⁶ : Category.{v, u} C
inst✝⁵ : HasColimits C
X Y Z : TopCat
inst✝⁴ : ConcreteCategory C
inst✝³ : PreservesFilteredColimits (forget C)
inst✝² : HasLimits C
inst✝¹ : PreservesLimits (forget C)
inst✝ : ReflectsIsomorphisms (forget C)
F G : Sheaf C X
f : F ⟶ G
U : Opens ↑X
hinj : ∀ (x : ↥U), Functi... | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Topology.Category.TopCat.OpenNhds
import Mathlib.Topology.Sheaves.Presheaf
import Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing... | rw [← comp_apply, ← f.1.naturality, comp_apply, s_spec, heq] | /-- For surjectivity, we are given an arbitrary section `t` and need to find a preimage for it.
We claim that it suffices to find preimages *locally*. That is, for each `x : U` we construct
a neighborhood `V ≤ U` and a section `s : F.obj (op V))` such that `f.app (op V) s` and `t`
agree on `V`. -/
theorem app_surjectiv... | Mathlib.Topology.Sheaves.Stalks.530_0.hsVUPKIHRY0xmFk | /-- For surjectivity, we are given an arbitrary section `t` and need to find a preimage for it.
We claim that it suffices to find preimages *locally*. That is, for each `x : U` we construct
a neighborhood `V ≤ U` and a section `s : F.obj (op V))` such that `f.app (op V) s` and `t`
agree on `V`. -/
theorem app_surjectiv... | Mathlib_Topology_Sheaves_Stalks |
C : Type u
inst✝⁶ : Category.{v, u} C
inst✝⁵ : HasColimits C
X Y Z : TopCat
inst✝⁴ : ConcreteCategory C
inst✝³ : PreservesFilteredColimits (forget C)
inst✝² : HasLimits C
inst✝¹ : PreservesLimits (forget C)
inst✝ : ReflectsIsomorphisms (forget C)
F G : Sheaf C X
f : F ⟶ G
U : Opens ↑X
hinj : ∀ (x : ↥U), Function.Inject... | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Topology.Category.TopCat.OpenNhds
import Mathlib.Topology.Sheaves.Presheaf
import Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing... | intro x y | /-- For surjectivity, we are given an arbitrary section `t` and need to find a preimage for it.
We claim that it suffices to find preimages *locally*. That is, for each `x : U` we construct
a neighborhood `V ≤ U` and a section `s : F.obj (op V))` such that `f.app (op V) s` and `t`
agree on `V`. -/
theorem app_surjectiv... | Mathlib.Topology.Sheaves.Stalks.530_0.hsVUPKIHRY0xmFk | /-- For surjectivity, we are given an arbitrary section `t` and need to find a preimage for it.
We claim that it suffices to find preimages *locally*. That is, for each `x : U` we construct
a neighborhood `V ≤ U` and a section `s : F.obj (op V))` such that `f.app (op V) s` and `t`
agree on `V`. -/
theorem app_surjectiv... | Mathlib_Topology_Sheaves_Stalks |
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