topology.sheaves.sheafify
⟷
Mathlib.Topology.Sheaves.Sheafify
The following section lists changes to this file in mathlib3 and mathlib4 that occured after the initial port. Most recent changes are shown first. Hovering over a commit will show all commits associated with the same mathlib3 commit.
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(last sync)
mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83
@@ -123,11 +123,11 @@ theorem stalkToFiber_injective (x : X) : Function.Injective (F.stalkToFiber x) :
rcases hU ⟨x, U.2⟩ with ⟨U', mU, iU, gU, wU⟩
rcases hV ⟨x, V.2⟩ with ⟨V', mV, iV, gV, wV⟩
have wUx := wU ⟨x, mU⟩
- dsimp at wUx ; erw [wUx] at e ; clear wUx
+ dsimp at wUx; erw [wUx] at e; clear wUx
have wVx := wV ⟨x, mV⟩
- dsimp at wVx ; erw [wVx] at e ; clear wVx
+ dsimp at wVx; erw [wVx] at e; clear wVx
rcases F.germ_eq x mU mV gU gV e with ⟨W, mW, iU', iV', e'⟩
- dsimp at e'
+ dsimp at e'
use⟨W ⊓ (U' ⊓ V'), ⟨mW, mU, mV⟩⟩
refine' ⟨_, _, _⟩
· change W ⊓ (U' ⊓ V') ⟶ U.obj
@@ -137,9 +137,9 @@ theorem stalkToFiber_injective (x : X) : Function.Injective (F.stalkToFiber x) :
· intro w
dsimp
specialize wU ⟨w.1, w.2.2.1⟩
- dsimp at wU
+ dsimp at wU
specialize wV ⟨w.1, w.2.2.2⟩
- dsimp at wV
+ dsimp at wV
erw [wU, ← F.germ_res iU' ⟨w, w.2.1⟩, wV, ← F.germ_res iV' ⟨w, w.2.1⟩,
CategoryTheory.types_comp_apply, CategoryTheory.types_comp_apply, e']
#align Top.presheaf.stalk_to_fiber_injective TopCat.Presheaf.stalkToFiber_injective
mathlib commit https://github.com/leanprover-community/mathlib/commit/ce64cd319bb6b3e82f31c2d38e79080d377be451
@@ -3,8 +3,8 @@ Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
-import Mathbin.Topology.Sheaves.LocalPredicate
-import Mathbin.Topology.Sheaves.Stalks
+import Topology.Sheaves.LocalPredicate
+import Topology.Sheaves.Stalks
#align_import topology.sheaves.sheafify from "leanprover-community/mathlib"@"5c1efce12ba86d4901463f61019832f6a4b1a0d0"
mathlib commit https://github.com/leanprover-community/mathlib/commit/63721b2c3eba6c325ecf8ae8cca27155a4f6306f
@@ -108,7 +108,7 @@ theorem stalkToFiber_surjective (x : X) : Function.Surjective (F.stalkToFiber x)
apply stalk_to_fiber_surjective
intro t
obtain ⟨U, m, s, rfl⟩ := F.germ_exist _ t
- · use ⟨U, m⟩
+ · use⟨U, m⟩
fconstructor
· exact fun y => F.germ y s
· exact ⟨prelocal_predicate.sheafify_of ⟨s, fun _ => rfl⟩, rfl⟩
@@ -128,7 +128,7 @@ theorem stalkToFiber_injective (x : X) : Function.Injective (F.stalkToFiber x) :
dsimp at wVx ; erw [wVx] at e ; clear wVx
rcases F.germ_eq x mU mV gU gV e with ⟨W, mW, iU', iV', e'⟩
dsimp at e'
- use ⟨W ⊓ (U' ⊓ V'), ⟨mW, mU, mV⟩⟩
+ use⟨W ⊓ (U' ⊓ V'), ⟨mW, mU, mV⟩⟩
refine' ⟨_, _, _⟩
· change W ⊓ (U' ⊓ V') ⟶ U.obj
exact opens.inf_le_right _ _ ≫ opens.inf_le_left _ _ ≫ iU
mathlib commit https://github.com/leanprover-community/mathlib/commit/8ea5598db6caeddde6cb734aa179cc2408dbd345
@@ -2,15 +2,12 @@
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-
-! This file was ported from Lean 3 source module topology.sheaves.sheafify
-! leanprover-community/mathlib commit 5c1efce12ba86d4901463f61019832f6a4b1a0d0
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
-/
import Mathbin.Topology.Sheaves.LocalPredicate
import Mathbin.Topology.Sheaves.Stalks
+#align_import topology.sheaves.sheafify from "leanprover-community/mathlib"@"5c1efce12ba86d4901463f61019832f6a4b1a0d0"
+
/-!
# Sheafification of `Type` valued presheaves
mathlib commit https://github.com/leanprover-community/mathlib/commit/2a0ce625dbb0ffbc7d1316597de0b25c1ec75303
@@ -92,7 +92,7 @@ sending each section to its germs.
def toSheafify : F ⟶ F.sheafify.1
where
app U f := ⟨fun x => F.germ x f, PrelocalPredicate.sheafifyOf ⟨f, fun x => rfl⟩⟩
- naturality' U U' f := by ext (x⟨u, m⟩); exact germ_res_apply F f.unop ⟨u, m⟩ x
+ naturality' U U' f := by ext x ⟨u, m⟩; exact germ_res_apply F f.unop ⟨u, m⟩ x
#align Top.presheaf.to_sheafify TopCat.Presheaf.toSheafify
-/
mathlib commit https://github.com/leanprover-community/mathlib/commit/9fb8964792b4237dac6200193a0d533f1b3f7423
@@ -84,6 +84,7 @@ def sheafify : Sheaf (Type v) X :=
#align Top.presheaf.sheafify TopCat.Presheaf.sheafify
-/
+#print TopCat.Presheaf.toSheafify /-
/-- The morphism from a presheaf to its sheafification,
sending each section to its germs.
(This forms the unit of the adjunction.)
@@ -93,6 +94,7 @@ def toSheafify : F ⟶ F.sheafify.1
app U f := ⟨fun x => F.germ x f, PrelocalPredicate.sheafifyOf ⟨f, fun x => rfl⟩⟩
naturality' U U' f := by ext (x⟨u, m⟩); exact germ_res_apply F f.unop ⟨u, m⟩ x
#align Top.presheaf.to_sheafify TopCat.Presheaf.toSheafify
+-/
#print TopCat.Presheaf.stalkToFiber /-
/-- The natural morphism from the stalk of the sheafification to the original stalk.
mathlib commit https://github.com/leanprover-community/mathlib/commit/5f25c089cb34db4db112556f23c50d12da81b297
@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
! This file was ported from Lean 3 source module topology.sheaves.sheafify
-! leanprover-community/mathlib commit bb103f356534a9a7d3596a672097e375290a4c3a
+! leanprover-community/mathlib commit 5c1efce12ba86d4901463f61019832f6a4b1a0d0
! Please do not edit these lines, except to modify the commit id
! if you have ported upstream changes.
-/
@@ -14,6 +14,9 @@ import Mathbin.Topology.Sheaves.Stalks
/-!
# Sheafification of `Type` valued presheaves
+> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
+> Any changes to this file require a corresponding PR to mathlib4.
+
We construct the sheafification of a `Type` valued presheaf,
as the subsheaf of dependent functions into the stalks
consisting of functions which are locally germs.
@@ -49,6 +52,7 @@ namespace TopCat.Presheaf
namespace Sheafify
+#print TopCat.Presheaf.Sheafify.isGerm /-
/--
The prelocal predicate on functions into the stalks, asserting that the function is equal to a germ.
-/
@@ -58,22 +62,27 @@ def isGerm : PrelocalPredicate fun x => F.stalk x
res := fun V U i f ⟨g, p⟩ =>
⟨F.map i.op g, fun x => (p (i x)).trans (F.germ_res_apply _ _ _).symm⟩
#align Top.presheaf.sheafify.is_germ TopCat.Presheaf.Sheafify.isGerm
+-/
+#print TopCat.Presheaf.Sheafify.isLocallyGerm /-
/-- The local predicate on functions into the stalks,
asserting that the function is locally equal to a germ.
-/
def isLocallyGerm : LocalPredicate fun x => F.stalk x :=
(isGerm F).sheafify
#align Top.presheaf.sheafify.is_locally_germ TopCat.Presheaf.Sheafify.isLocallyGerm
+-/
end Sheafify
+#print TopCat.Presheaf.sheafify /-
/-- The sheafification of a `Type` valued presheaf, defined as the functions into the stalks which
are locally equal to germs.
-/
def sheafify : Sheaf (Type v) X :=
subsheafToTypes (Sheafify.isLocallyGerm F)
#align Top.presheaf.sheafify TopCat.Presheaf.sheafify
+-/
/-- The morphism from a presheaf to its sheafification,
sending each section to its germs.
@@ -85,13 +94,16 @@ def toSheafify : F ⟶ F.sheafify.1
naturality' U U' f := by ext (x⟨u, m⟩); exact germ_res_apply F f.unop ⟨u, m⟩ x
#align Top.presheaf.to_sheafify TopCat.Presheaf.toSheafify
+#print TopCat.Presheaf.stalkToFiber /-
/-- The natural morphism from the stalk of the sheafification to the original stalk.
In `sheafify_stalk_iso` we show this is an isomorphism.
-/
def stalkToFiber (x : X) : F.sheafify.Presheaf.stalk x ⟶ F.stalk x :=
stalkToFiber (Sheafify.isLocallyGerm F) x
#align Top.presheaf.stalk_to_fiber TopCat.Presheaf.stalkToFiber
+-/
+#print TopCat.Presheaf.stalkToFiber_surjective /-
theorem stalkToFiber_surjective (x : X) : Function.Surjective (F.stalkToFiber x) :=
by
apply stalk_to_fiber_surjective
@@ -102,7 +114,9 @@ theorem stalkToFiber_surjective (x : X) : Function.Surjective (F.stalkToFiber x)
· exact fun y => F.germ y s
· exact ⟨prelocal_predicate.sheafify_of ⟨s, fun _ => rfl⟩, rfl⟩
#align Top.presheaf.stalk_to_fiber_surjective TopCat.Presheaf.stalkToFiber_surjective
+-/
+#print TopCat.Presheaf.stalkToFiber_injective /-
theorem stalkToFiber_injective (x : X) : Function.Injective (F.stalkToFiber x) :=
by
apply stalk_to_fiber_injective
@@ -130,12 +144,15 @@ theorem stalkToFiber_injective (x : X) : Function.Injective (F.stalkToFiber x) :
erw [wU, ← F.germ_res iU' ⟨w, w.2.1⟩, wV, ← F.germ_res iV' ⟨w, w.2.1⟩,
CategoryTheory.types_comp_apply, CategoryTheory.types_comp_apply, e']
#align Top.presheaf.stalk_to_fiber_injective TopCat.Presheaf.stalkToFiber_injective
+-/
+#print TopCat.Presheaf.sheafifyStalkIso /-
/-- The isomorphism betweeen a stalk of the sheafification and the original stalk.
-/
def sheafifyStalkIso (x : X) : F.sheafify.Presheaf.stalk x ≅ F.stalk x :=
(Equiv.ofBijective _ ⟨stalkToFiber_injective _ _, stalkToFiber_surjective _ _⟩).toIso
#align Top.presheaf.sheafify_stalk_iso TopCat.Presheaf.sheafifyStalkIso
+-/
-- PROJECT functoriality, and that sheafification is the left adjoint of the forgetful functor.
end TopCat.Presheaf
mathlib commit https://github.com/leanprover-community/mathlib/commit/cca40788df1b8755d5baf17ab2f27dacc2e17acb
@@ -110,11 +110,11 @@ theorem stalkToFiber_injective (x : X) : Function.Injective (F.stalkToFiber x) :
rcases hU ⟨x, U.2⟩ with ⟨U', mU, iU, gU, wU⟩
rcases hV ⟨x, V.2⟩ with ⟨V', mV, iV, gV, wV⟩
have wUx := wU ⟨x, mU⟩
- dsimp at wUx; erw [wUx] at e; clear wUx
+ dsimp at wUx ; erw [wUx] at e ; clear wUx
have wVx := wV ⟨x, mV⟩
- dsimp at wVx; erw [wVx] at e; clear wVx
+ dsimp at wVx ; erw [wVx] at e ; clear wVx
rcases F.germ_eq x mU mV gU gV e with ⟨W, mW, iU', iV', e'⟩
- dsimp at e'
+ dsimp at e'
use ⟨W ⊓ (U' ⊓ V'), ⟨mW, mU, mV⟩⟩
refine' ⟨_, _, _⟩
· change W ⊓ (U' ⊓ V') ⟶ U.obj
@@ -124,9 +124,9 @@ theorem stalkToFiber_injective (x : X) : Function.Injective (F.stalkToFiber x) :
· intro w
dsimp
specialize wU ⟨w.1, w.2.2.1⟩
- dsimp at wU
+ dsimp at wU
specialize wV ⟨w.1, w.2.2.2⟩
- dsimp at wV
+ dsimp at wV
erw [wU, ← F.germ_res iU' ⟨w, w.2.1⟩, wV, ← F.germ_res iV' ⟨w, w.2.1⟩,
CategoryTheory.types_comp_apply, CategoryTheory.types_comp_apply, e']
#align Top.presheaf.stalk_to_fiber_injective TopCat.Presheaf.stalkToFiber_injective
mathlib commit https://github.com/leanprover-community/mathlib/commit/917c3c072e487b3cccdbfeff17e75b40e45f66cb
@@ -82,9 +82,7 @@ sending each section to its germs.
def toSheafify : F ⟶ F.sheafify.1
where
app U f := ⟨fun x => F.germ x f, PrelocalPredicate.sheafifyOf ⟨f, fun x => rfl⟩⟩
- naturality' U U' f := by
- ext (x⟨u, m⟩)
- exact germ_res_apply F f.unop ⟨u, m⟩ x
+ naturality' U U' f := by ext (x⟨u, m⟩); exact germ_res_apply F f.unop ⟨u, m⟩ x
#align Top.presheaf.to_sheafify TopCat.Presheaf.toSheafify
/-- The natural morphism from the stalk of the sheafification to the original stalk.
mathlib commit https://github.com/leanprover-community/mathlib/commit/9b2b58d6b14b895b2f375108e765cb47de71aebd
@@ -81,7 +81,7 @@ sending each section to its germs.
-/
def toSheafify : F ⟶ F.sheafify.1
where
- app U f := ⟨fun x => F.germ x f, PrelocalPredicate.sheafify_of ⟨f, fun x => rfl⟩⟩
+ app U f := ⟨fun x => F.germ x f, PrelocalPredicate.sheafifyOf ⟨f, fun x => rfl⟩⟩
naturality' U U' f := by
ext (x⟨u, m⟩)
exact germ_res_apply F f.unop ⟨u, m⟩ x
mathlib commit https://github.com/leanprover-community/mathlib/commit/bd9851ca476957ea4549eb19b40e7b5ade9428cc
@@ -2,15 +2,12 @@
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-
-! This file was ported from Lean 3 source module topology.sheaves.sheafify
-! leanprover-community/mathlib commit bb103f356534a9a7d3596a672097e375290a4c3a
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
-/
import Mathlib.Topology.Sheaves.LocalPredicate
import Mathlib.Topology.Sheaves.Stalks
+#align_import topology.sheaves.sheafify from "leanprover-community/mathlib"@"bb103f356534a9a7d3596a672097e375290a4c3a"
+
/-!
# Sheafification of `Type` valued presheaves
This is the second half of the changes originally in #5699, removing all occurrences of ;
after a space and implementing a linter rule to enforce it.
In most cases this 2-character substring has a space after it, so the following command was run first:
find . -type f -name "*.lean" -exec sed -i -E 's/ ; /; /g' {} \;
The remaining cases were few enough in number that they were done manually.
@@ -106,9 +106,9 @@ theorem stalkToFiber_injective (x : X) : Function.Injective (F.stalkToFiber x) :
rcases hU ⟨x, U.2⟩ with ⟨U', mU, iU, gU, wU⟩
rcases hV ⟨x, V.2⟩ with ⟨V', mV, iV, gV, wV⟩
have wUx := wU ⟨x, mU⟩
- dsimp at wUx ; erw [wUx] at e ; clear wUx
+ dsimp at wUx; erw [wUx] at e; clear wUx
have wVx := wV ⟨x, mV⟩
- dsimp at wVx ; erw [wVx] at e ; clear wVx
+ dsimp at wVx; erw [wVx] at e; clear wVx
rcases F.germ_eq x mU mV gU gV e with ⟨W, mW, iU', iV', (e' : F.map iU'.op gU = F.map iV'.op gV)⟩
use ⟨W ⊓ (U' ⊓ V'), ⟨mW, mU, mV⟩⟩
refine' ⟨_, _, _⟩
Make sure in particular one can import Mathlib.Tactic for teaching without getting category theory notations in scope. Note that only two files needed an extra open.
@@ -37,11 +37,7 @@ universe v
noncomputable section
-open TopCat
-
-open Opposite
-
-open TopologicalSpace
+open TopCat Opposite TopologicalSpace CategoryTheory
variable {X : TopCat.{v}} (F : Presheaf (Type v) X)
@@ -113,8 +113,7 @@ theorem stalkToFiber_injective (x : X) : Function.Injective (F.stalkToFiber x) :
dsimp at wUx ; erw [wUx] at e ; clear wUx
have wVx := wV ⟨x, mV⟩
dsimp at wVx ; erw [wVx] at e ; clear wVx
- rcases F.germ_eq x mU mV gU gV e with ⟨W, mW, iU', iV', e'⟩
- dsimp at e'
+ rcases F.germ_eq x mU mV gU gV e with ⟨W, mW, iU', iV', (e' : F.map iU'.op gU = F.map iV'.op gV)⟩
use ⟨W ⊓ (U' ⊓ V'), ⟨mW, mU, mV⟩⟩
refine' ⟨_, _, _⟩
· change W ⊓ (U' ⊓ V') ⟶ U.obj
@@ -126,7 +125,7 @@ theorem stalkToFiber_injective (x : X) : Function.Injective (F.stalkToFiber x) :
specialize wV ⟨w.1, w.2.2.2⟩
dsimp at wU wV ⊢
erw [wU, ← F.germ_res iU' ⟨w, w.2.1⟩, wV, ← F.germ_res iV' ⟨w, w.2.1⟩,
- CategoryTheory.types_comp_apply, CategoryTheory.types_comp_apply, e']; rfl
+ CategoryTheory.types_comp_apply, CategoryTheory.types_comp_apply, e']
#align Top.presheaf.stalk_to_fiber_injective TopCat.Presheaf.stalkToFiber_injective
/-- The isomorphism between a stalk of the sheafification and the original stalk.
@@ -129,7 +129,7 @@ theorem stalkToFiber_injective (x : X) : Function.Injective (F.stalkToFiber x) :
CategoryTheory.types_comp_apply, CategoryTheory.types_comp_apply, e']; rfl
#align Top.presheaf.stalk_to_fiber_injective TopCat.Presheaf.stalkToFiber_injective
-/-- The isomorphism betweeen a stalk of the sheafification and the original stalk.
+/-- The isomorphism between a stalk of the sheafification and the original stalk.
-/
def sheafifyStalkIso (x : X) : F.sheafify.presheaf.stalk x ≅ F.stalk x :=
(Equiv.ofBijective _ ⟨stalkToFiber_injective _ _, stalkToFiber_surjective _ _⟩).toIso
The unported dependencies are