category_theory.sites.canonical
⟷
Mathlib.CategoryTheory.Sites.Canonical
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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mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83
@@ -3,7 +3,7 @@ Copyright (c) 2020 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta
-/
-import CategoryTheory.Sites.SheafOfTypes
+import CategoryTheory.Sites.IsSheafFor
#align_import category_theory.sites.canonical from "leanprover-community/mathlib"@"36938f775671ff28bea1c0310f1608e4afbb22e0"
mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83
@@ -93,12 +93,12 @@ theorem isSheafFor_bind (P : Cᵒᵖ ⥤ Type v) (U : Sieve X) (B : ∀ ⦃Y⦄
simpa using this
trans s (m ≫ l ≫ h ≫ f) this
· have := ht (U.downward_closed hf h) _ ((B _).downward_closed hl m)
- rw [op_comp, functor_to_types.map_comp_apply] at this
+ rw [op_comp, functor_to_types.map_comp_apply] at this
rw [this]
change s _ _ = s _ _
simp
· have : s _ _ = _ := (ht hf _ hm).symm
- simp only [assoc] at this
+ simp only [assoc] at this
rw [this]
simp
refine' ⟨hU.amalgamate t hT, _, _⟩
@@ -181,7 +181,7 @@ def finestTopologySingle (P : Cᵒᵖ ⥤ Type v) : GrothendieckTopology C
apply (hS _ _).IsSeparatedFor
· intro Y f hf
have := hR hf _ (𝟙 _)
- rw [pullback_id, pullback_comp] at this
+ rw [pullback_id, pullback_comp] at this
apply this
#align category_theory.sheaf.finest_topology_single CategoryTheory.Sheaf.finestTopologySingle
-/
mathlib commit https://github.com/leanprover-community/mathlib/commit/ce64cd319bb6b3e82f31c2d38e79080d377be451
@@ -3,7 +3,7 @@ Copyright (c) 2020 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta
-/
-import Mathbin.CategoryTheory.Sites.SheafOfTypes
+import CategoryTheory.Sites.SheafOfTypes
#align_import category_theory.sites.canonical from "leanprover-community/mathlib"@"36938f775671ff28bea1c0310f1608e4afbb22e0"
mathlib commit https://github.com/leanprover-community/mathlib/commit/8ea5598db6caeddde6cb734aa179cc2408dbd345
@@ -2,14 +2,11 @@
Copyright (c) 2020 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta
-
-! This file was ported from Lean 3 source module category_theory.sites.canonical
-! leanprover-community/mathlib commit 36938f775671ff28bea1c0310f1608e4afbb22e0
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
-/
import Mathbin.CategoryTheory.Sites.SheafOfTypes
+#align_import category_theory.sites.canonical from "leanprover-community/mathlib"@"36938f775671ff28bea1c0310f1608e4afbb22e0"
+
/-!
# The canonical topology on a category
mathlib commit https://github.com/leanprover-community/mathlib/commit/2a0ce625dbb0ffbc7d1316597de0b25c1ec75303
@@ -146,7 +146,7 @@ theorem isSheafFor_trans (P : Cᵒᵖ ⥤ Type v) (R S : Sieve X) (hR : Presieve
· intro Y f hf
have : sieve.pullback f (bind R fun T (k : T ⟶ X) (hf : R k) => pullback k S) = R.pullback f :=
by
- ext (Z g)
+ ext Z g
constructor
· rintro ⟨W, k, l, hl, _, comm⟩
rw [pullback_apply, ← comm]
mathlib commit https://github.com/leanprover-community/mathlib/commit/9fb8964792b4237dac6200193a0d533f1b3f7423
@@ -233,10 +233,12 @@ def canonicalTopology (C : Type u) [Category.{v} C] : GrothendieckTopology C :=
#align category_theory.sheaf.canonical_topology CategoryTheory.Sheaf.canonicalTopology
-/
+#print CategoryTheory.Sheaf.isSheaf_yoneda_obj /-
/-- `yoneda.obj X` is a sheaf for the canonical topology. -/
theorem isSheaf_yoneda_obj (X : C) : Presieve.IsSheaf (canonicalTopology C) (yoneda.obj X) :=
fun Y S hS => sheaf_for_finestTopology _ (Set.mem_range_self _) _ hS
#align category_theory.sheaf.is_sheaf_yoneda_obj CategoryTheory.Sheaf.isSheaf_yoneda_obj
+-/
#print CategoryTheory.Sheaf.isSheaf_of_representable /-
/-- A representable functor is a sheaf for the canonical topology. -/
@@ -257,11 +259,13 @@ def Subcanonical (J : GrothendieckTopology C) : Prop :=
namespace Subcanonical
+#print CategoryTheory.Sheaf.Subcanonical.of_yoneda_isSheaf /-
/-- If every functor `yoneda.obj X` is a `J`-sheaf, then `J` is subcanonical. -/
theorem of_yoneda_isSheaf (J : GrothendieckTopology C)
(h : ∀ X, Presieve.IsSheaf J (yoneda.obj X)) : Subcanonical J :=
le_finestTopology _ _ (by rintro P ⟨X, rfl⟩; apply h)
#align category_theory.sheaf.subcanonical.of_yoneda_is_sheaf CategoryTheory.Sheaf.Subcanonical.of_yoneda_isSheaf
+-/
#print CategoryTheory.Sheaf.Subcanonical.isSheaf_of_representable /-
/-- If `J` is subcanonical, then any representable is a `J`-sheaf. -/
mathlib commit https://github.com/leanprover-community/mathlib/commit/58a272265b5e05f258161260dd2c5d247213cbd3
@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta
! This file was ported from Lean 3 source module category_theory.sites.canonical
-! leanprover-community/mathlib commit 9e7c80f638149bfb3504ba8ff48dfdbfc949fb1a
+! leanprover-community/mathlib commit 36938f775671ff28bea1c0310f1608e4afbb22e0
! Please do not edit these lines, except to modify the commit id
! if you have ported upstream changes.
-/
@@ -13,6 +13,9 @@ import Mathbin.CategoryTheory.Sites.SheafOfTypes
/-!
# The canonical topology on a category
+> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
+> Any changes to this file require a corresponding PR to mathlib4.
+
We define the finest (largest) Grothendieck topology for which a given presheaf `P` is a sheaf.
This is well defined since if `P` is a sheaf for a topology `J`, then it is a sheaf for any
coarser (smaller) topology. Nonetheless we define the topology explicitly by specifying its sieves:
mathlib commit https://github.com/leanprover-community/mathlib/commit/13361559d66b84f80b6d5a1c4a26aa5054766725
@@ -53,6 +53,7 @@ variable {X Y : C} {S : Sieve X} {R : Presieve X}
variable (J J₂ : GrothendieckTopology C)
+#print CategoryTheory.Sheaf.isSheafFor_bind /-
/--
To show `P` is a sheaf for the binding of `U` with `B`, it suffices to show that `P` is a sheaf for
`U`, that `P` is a sheaf for each sieve in `B`, and that it is separated for any pullback of any
@@ -112,7 +113,9 @@ theorem isSheafFor_bind (P : Cᵒᵖ ⥤ Type v) (U : Sieve X) (B : ∀ ⦃Y⦄
rw [← functor_to_types.map_comp_apply, ← op_comp, hy _ (presieve.bind_comp _ _ hg),
hU.valid_glue _ _ hf, ht hf _ hg]
#align category_theory.sheaf.is_sheaf_for_bind CategoryTheory.Sheaf.isSheafFor_bind
+-/
+#print CategoryTheory.Sheaf.isSheafFor_trans /-
/-- Given two sieves `R` and `S`, to show that `P` is a sheaf for `S`, we can show:
* `P` is a sheaf for `R`
* `P` is a sheaf for the pullback of `S` along any arrow in `R`
@@ -151,7 +154,9 @@ theorem isSheafFor_trans (P : Cᵒᵖ ⥤ Type v) (R S : Sieve X) (hR : Presieve
rw [this]
apply hR' hf
#align category_theory.sheaf.is_sheaf_for_trans CategoryTheory.Sheaf.isSheafFor_trans
+-/
+#print CategoryTheory.Sheaf.finestTopologySingle /-
/-- Construct the finest (largest) Grothendieck topology for which the given presheaf is a sheaf.
This is a special case of https://stacks.math.columbia.edu/tag/00Z9, but following a different
@@ -179,7 +184,9 @@ def finestTopologySingle (P : Cᵒᵖ ⥤ Type v) : GrothendieckTopology C
rw [pullback_id, pullback_comp] at this
apply this
#align category_theory.sheaf.finest_topology_single CategoryTheory.Sheaf.finestTopologySingle
+-/
+#print CategoryTheory.Sheaf.finestTopology /-
/--
Construct the finest (largest) Grothendieck topology for which all the given presheaves are sheaves.
@@ -188,13 +195,17 @@ This is equal to the construction of <https://stacks.math.columbia.edu/tag/00Z9>
def finestTopology (Ps : Set (Cᵒᵖ ⥤ Type v)) : GrothendieckTopology C :=
sInf (finestTopologySingle '' Ps)
#align category_theory.sheaf.finest_topology CategoryTheory.Sheaf.finestTopology
+-/
+#print CategoryTheory.Sheaf.sheaf_for_finestTopology /-
/-- Check that if `P ∈ Ps`, then `P` is indeed a sheaf for the finest topology on `Ps`. -/
theorem sheaf_for_finestTopology (Ps : Set (Cᵒᵖ ⥤ Type v)) (h : P ∈ Ps) :
Presieve.IsSheaf (finestTopology Ps) P := fun X S hS => by
simpa using hS _ ⟨⟨_, _, ⟨_, h, rfl⟩, rfl⟩, rfl⟩ _ (𝟙 _)
#align category_theory.sheaf.sheaf_for_finest_topology CategoryTheory.Sheaf.sheaf_for_finestTopology
+-/
+#print CategoryTheory.Sheaf.le_finestTopology /-
/--
Check that if each `P ∈ Ps` is a sheaf for `J`, then `J` is a subtopology of `finest_topology Ps`.
-/
@@ -206,7 +217,9 @@ theorem le_finestTopology (Ps : Set (Cᵒᵖ ⥤ Type v)) (J : GrothendieckTopol
-- this can't be combined with the previous because the `subst` is applied at the end
exact hJ P hP (S.pullback f) (J.pullback_stable f hS)
#align category_theory.sheaf.le_finest_topology CategoryTheory.Sheaf.le_finestTopology
+-/
+#print CategoryTheory.Sheaf.canonicalTopology /-
/-- The `canonical_topology` on a category is the finest (largest) topology for which every
representable presheaf is a sheaf.
@@ -215,24 +228,29 @@ See <https://stacks.math.columbia.edu/tag/00ZA>
def canonicalTopology (C : Type u) [Category.{v} C] : GrothendieckTopology C :=
finestTopology (Set.range yoneda.obj)
#align category_theory.sheaf.canonical_topology CategoryTheory.Sheaf.canonicalTopology
+-/
/-- `yoneda.obj X` is a sheaf for the canonical topology. -/
theorem isSheaf_yoneda_obj (X : C) : Presieve.IsSheaf (canonicalTopology C) (yoneda.obj X) :=
fun Y S hS => sheaf_for_finestTopology _ (Set.mem_range_self _) _ hS
#align category_theory.sheaf.is_sheaf_yoneda_obj CategoryTheory.Sheaf.isSheaf_yoneda_obj
+#print CategoryTheory.Sheaf.isSheaf_of_representable /-
/-- A representable functor is a sheaf for the canonical topology. -/
theorem isSheaf_of_representable (P : Cᵒᵖ ⥤ Type v) [P.Representable] :
Presieve.IsSheaf (canonicalTopology C) P :=
Presieve.isSheaf_iso (canonicalTopology C) P.reprW (isSheaf_yoneda_obj _)
#align category_theory.sheaf.is_sheaf_of_representable CategoryTheory.Sheaf.isSheaf_of_representable
+-/
+#print CategoryTheory.Sheaf.Subcanonical /-
/-- A subcanonical topology is a topology which is smaller than the canonical topology.
Equivalently, a topology is subcanonical iff every representable is a sheaf.
-/
def Subcanonical (J : GrothendieckTopology C) : Prop :=
J ≤ canonicalTopology C
#align category_theory.sheaf.subcanonical CategoryTheory.Sheaf.Subcanonical
+-/
namespace Subcanonical
@@ -242,11 +260,13 @@ theorem of_yoneda_isSheaf (J : GrothendieckTopology C)
le_finestTopology _ _ (by rintro P ⟨X, rfl⟩; apply h)
#align category_theory.sheaf.subcanonical.of_yoneda_is_sheaf CategoryTheory.Sheaf.Subcanonical.of_yoneda_isSheaf
+#print CategoryTheory.Sheaf.Subcanonical.isSheaf_of_representable /-
/-- If `J` is subcanonical, then any representable is a `J`-sheaf. -/
theorem isSheaf_of_representable {J : GrothendieckTopology C} (hJ : Subcanonical J)
(P : Cᵒᵖ ⥤ Type v) [P.Representable] : Presieve.IsSheaf J P :=
Presieve.isSheaf_of_le _ hJ (isSheaf_of_representable P)
#align category_theory.sheaf.subcanonical.is_sheaf_of_representable CategoryTheory.Sheaf.Subcanonical.isSheaf_of_representable
+-/
end Subcanonical
mathlib commit https://github.com/leanprover-community/mathlib/commit/cca40788df1b8755d5baf17ab2f27dacc2e17acb
@@ -92,12 +92,12 @@ theorem isSheafFor_bind (P : Cᵒᵖ ⥤ Type v) (U : Sieve X) (B : ∀ ⦃Y⦄
simpa using this
trans s (m ≫ l ≫ h ≫ f) this
· have := ht (U.downward_closed hf h) _ ((B _).downward_closed hl m)
- rw [op_comp, functor_to_types.map_comp_apply] at this
+ rw [op_comp, functor_to_types.map_comp_apply] at this
rw [this]
change s _ _ = s _ _
simp
· have : s _ _ = _ := (ht hf _ hm).symm
- simp only [assoc] at this
+ simp only [assoc] at this
rw [this]
simp
refine' ⟨hU.amalgamate t hT, _, _⟩
@@ -176,7 +176,7 @@ def finestTopologySingle (P : Cᵒᵖ ⥤ Type v) : GrothendieckTopology C
apply (hS _ _).IsSeparatedFor
· intro Y f hf
have := hR hf _ (𝟙 _)
- rw [pullback_id, pullback_comp] at this
+ rw [pullback_id, pullback_comp] at this
apply this
#align category_theory.sheaf.finest_topology_single CategoryTheory.Sheaf.finestTopologySingle
mathlib commit https://github.com/leanprover-community/mathlib/commit/917c3c072e487b3cccdbfeff17e75b40e45f66cb
@@ -239,10 +239,7 @@ namespace Subcanonical
/-- If every functor `yoneda.obj X` is a `J`-sheaf, then `J` is subcanonical. -/
theorem of_yoneda_isSheaf (J : GrothendieckTopology C)
(h : ∀ X, Presieve.IsSheaf J (yoneda.obj X)) : Subcanonical J :=
- le_finestTopology _ _
- (by
- rintro P ⟨X, rfl⟩
- apply h)
+ le_finestTopology _ _ (by rintro P ⟨X, rfl⟩; apply h)
#align category_theory.sheaf.subcanonical.of_yoneda_is_sheaf CategoryTheory.Sheaf.Subcanonical.of_yoneda_isSheaf
/-- If `J` is subcanonical, then any representable is a `J`-sheaf. -/
mathlib commit https://github.com/leanprover-community/mathlib/commit/e3fb84046afd187b710170887195d50bada934ee
@@ -186,7 +186,7 @@ Construct the finest (largest) Grothendieck topology for which all the given pre
This is equal to the construction of <https://stacks.math.columbia.edu/tag/00Z9>.
-/
def finestTopology (Ps : Set (Cᵒᵖ ⥤ Type v)) : GrothendieckTopology C :=
- infₛ (finestTopologySingle '' Ps)
+ sInf (finestTopologySingle '' Ps)
#align category_theory.sheaf.finest_topology CategoryTheory.Sheaf.finestTopology
/-- Check that if `P ∈ Ps`, then `P` is indeed a sheaf for the finest topology on `Ps`. -/
mathlib commit https://github.com/leanprover-community/mathlib/commit/bd9851ca476957ea4549eb19b40e7b5ade9428cc
@@ -131,8 +131,8 @@ theorem isSheafFor_trans (P : Cᵒᵖ ⥤ Type v) (R S : Sieve X)
rintro Z f ⟨W, f, g, hg, hf : S _, rfl⟩
apply hf
apply Presieve.isSheafFor_subsieve_aux P this
- apply isSheafFor_bind _ _ _ hR hS
- · intro Y f hf Z g
+ · apply isSheafFor_bind _ _ _ hR hS
+ intro Y f hf Z g
rw [← pullback_comp]
apply (hS (R.downward_closed hf _)).isSeparatedFor
· intro Y f hf
Empty lines were removed by executing the following Python script twice
import os
import re
# Loop through each file in the repository
for dir_path, dirs, files in os.walk('.'):
for filename in files:
if filename.endswith('.lean'):
file_path = os.path.join(dir_path, filename)
# Open the file and read its contents
with open(file_path, 'r') as file:
content = file.read()
# Use a regular expression to replace sequences of "variable" lines separated by empty lines
# with sequences without empty lines
modified_content = re.sub(r'(variable.*\n)\n(variable(?! .* in))', r'\1\2', content)
# Write the modified content back to the file
with open(file_path, 'w') as file:
file.write(modified_content)
@@ -46,9 +46,7 @@ variable {C : Type u} [Category.{v} C]
namespace Sheaf
variable {P : Cᵒᵖ ⥤ Type v}
-
variable {X Y : C} {S : Sieve X} {R : Presieve X}
-
variable (J J₂ : GrothendieckTopology C)
/--
open Classical
(#11199)
We remove all but one open Classical
s, instead preferring to use open scoped Classical
. The only real side-effect this led to is moving a couple declarations to use Exists.choose
instead of Classical.choose
.
The first few commits are explicitly labelled regex replaces for ease of review.
@@ -38,7 +38,8 @@ universe v u
namespace CategoryTheory
-open CategoryTheory Category Limits Sieve Classical
+open scoped Classical
+open CategoryTheory Category Limits Sieve
variable {C : Type u} [Category.{v} C]
Homogenises porting notes via capitalisation and addition of whitespace.
It makes the following changes:
@@ -84,7 +84,7 @@ theorem isSheafFor_bind (P : Cᵒᵖ ⥤ Type v) (U : Sieve X) (B : ∀ ⦃Y⦄
apply (hB' hf (l ≫ h)).ext
intro M m hm
have : bind U B (m ≫ l ≫ h ≫ f) := by
- -- porting note: had to make explicit the parameter `((m ≫ l ≫ h) ≫ f)` and
+ -- Porting note: had to make explicit the parameter `((m ≫ l ≫ h) ≫ f)` and
-- using `by exact`
have : bind U B ((m ≫ l ≫ h) ≫ f) := by exact Presieve.bind_comp f hf hm
simpa using this
@@ -93,11 +93,11 @@ theorem isSheafFor_bind (P : Cᵒᵖ ⥤ Type v) (U : Sieve X) (B : ∀ ⦃Y⦄
rw [op_comp, FunctorToTypes.map_comp_apply] at this
rw [this]
change s _ _ = s _ _
- -- porting note: the proof was `by simp`
+ -- Porting note: the proof was `by simp`
congr 1
simp only [assoc]
· have h : s _ _ = _ := (ht hf _ hm).symm
- -- porting note: this was done by `simp only [assoc] at`
+ -- Porting note: this was done by `simp only [assoc] at`
conv_lhs at h => congr; rw [assoc, assoc]
rw [h]
simp only [op_comp, assoc, FunctorToTypes.map_comp_apply]
@@ -134,7 +134,6 @@ theorem isSheafFor_trans (P : Cᵒᵖ ⥤ Type v) (R S : Sieve X)
apply Presieve.isSheafFor_subsieve_aux P this
apply isSheafFor_bind _ _ _ hR hS
· intro Y f hf Z g
- dsimp
rw [← pullback_comp]
apply (hS (R.downward_closed hf _)).isSeparatedFor
· intro Y f hf
@@ -2,14 +2,11 @@
Copyright (c) 2020 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta
-
-! This file was ported from Lean 3 source module category_theory.sites.canonical
-! leanprover-community/mathlib commit 9e7c80f638149bfb3504ba8ff48dfdbfc949fb1a
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
+#align_import category_theory.sites.canonical from "leanprover-community/mathlib"@"9e7c80f638149bfb3504ba8ff48dfdbfc949fb1a"
+
/-!
# The canonical topology on a category
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.
@@ -101,7 +101,7 @@ theorem isSheafFor_bind (P : Cᵒᵖ ⥤ Type v) (U : Sieve X) (B : ∀ ⦃Y⦄
simp only [assoc]
· have h : s _ _ = _ := (ht hf _ hm).symm
-- porting note: this was done by `simp only [assoc] at`
- conv_lhs at h => congr ; rw [assoc, assoc]
+ conv_lhs at h => congr; rw [assoc, assoc]
rw [h]
simp only [op_comp, assoc, FunctorToTypes.map_comp_apply]
refine' ⟨hU.amalgamate t hT, _, _⟩
@@ -143,8 +143,7 @@ theorem isSheafFor_trans (P : Cᵒᵖ ⥤ Type v) (R S : Sieve X)
· intro Y f hf
have : Sieve.pullback f (bind R fun T (k : T ⟶ X) (_ : R k) => pullback k S) =
R.pullback f := by
- ext Z
- intro g
+ ext Z g
constructor
· rintro ⟨W, k, l, hl, _, comm⟩
rw [pullback_apply, ← comm]
The unported dependencies are