category_theory.sites.surjective
⟷
Mathlib.CategoryTheory.Sites.Surjective
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/ce64cd319bb6b3e82f31c2d38e79080d377be451
@@ -3,8 +3,8 @@ Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
-import Mathbin.CategoryTheory.Sites.Subsheaf
-import Mathbin.CategoryTheory.Sites.CompatibleSheafification
+import CategoryTheory.Sites.Subsheaf
+import CategoryTheory.Sites.CompatibleSheafification
#align_import category_theory.sites.surjective from "leanprover-community/mathlib"@"087c325ae0ab42dbdd5dee55bc37d3d5a0bf2197"
mathlib commit https://github.com/leanprover-community/mathlib/commit/8ea5598db6caeddde6cb734aa179cc2408dbd345
@@ -2,15 +2,12 @@
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-
-! This file was ported from Lean 3 source module category_theory.sites.surjective
-! leanprover-community/mathlib commit 087c325ae0ab42dbdd5dee55bc37d3d5a0bf2197
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
-/
import Mathbin.CategoryTheory.Sites.Subsheaf
import Mathbin.CategoryTheory.Sites.CompatibleSheafification
+#align_import category_theory.sites.surjective from "leanprover-community/mathlib"@"087c325ae0ab42dbdd5dee55bc37d3d5a0bf2197"
+
/-!
# Locally surjective morphisms
mathlib commit https://github.com/leanprover-community/mathlib/commit/2a0ce625dbb0ffbc7d1316597de0b25c1ec75303
@@ -74,7 +74,7 @@ theorem imageSieve_whisker_forget {F G : Cᵒᵖ ⥤ A} (f : F ⟶ G) {U : C} (s
#print CategoryTheory.imageSieve_app /-
theorem imageSieve_app {F G : Cᵒᵖ ⥤ A} (f : F ⟶ G) {U : C} (s : F.obj (op U)) :
imageSieve f (f.app _ s) = ⊤ := by
- ext (V i)
+ ext V i
simp only [sieve.top_apply, iff_true_iff, image_sieve_apply]
have := elementwise_of (f.naturality i.op)
exact ⟨F.map i.op s, this s⟩
mathlib commit https://github.com/leanprover-community/mathlib/commit/9fb8964792b4237dac6200193a0d533f1b3f7423
@@ -43,6 +43,7 @@ attribute [local instance] concrete_category.has_coe_to_sort concrete_category.h
variable {A : Type u'} [Category.{v'} A] [ConcreteCategory.{w'} A]
+#print CategoryTheory.imageSieve /-
/-- Given `f : F ⟶ G`, a morphism between presieves, and `s : G.obj (op U)`, this is the sieve
of `U` consisting of the `i : V ⟶ U` such that `s` restricted along `i` is in the image of `f`. -/
@[simps (config := lemmasOnly)]
@@ -54,17 +55,23 @@ def imageSieve {F G : Cᵒᵖ ⥤ A} (f : F ⟶ G) {U : C} (s : G.obj (op U)) :
refine' ⟨F.map j.op t, _⟩
rw [op_comp, G.map_comp, comp_apply, ← ht, elementwise_of f.naturality]
#align category_theory.image_sieve CategoryTheory.imageSieve
+-/
+#print CategoryTheory.imageSieve_eq_sieveOfSection /-
theorem imageSieve_eq_sieveOfSection {F G : Cᵒᵖ ⥤ A} (f : F ⟶ G) {U : C} (s : G.obj (op U)) :
imageSieve f s = (imagePresheaf (whiskerRight f (forget A))).sieveOfSection s :=
rfl
#align category_theory.image_sieve_eq_sieve_of_section CategoryTheory.imageSieve_eq_sieveOfSection
+-/
+#print CategoryTheory.imageSieve_whisker_forget /-
theorem imageSieve_whisker_forget {F G : Cᵒᵖ ⥤ A} (f : F ⟶ G) {U : C} (s : G.obj (op U)) :
imageSieve (whiskerRight f (forget A)) s = imageSieve f s :=
rfl
#align category_theory.image_sieve_whisker_forget CategoryTheory.imageSieve_whisker_forget
+-/
+#print CategoryTheory.imageSieve_app /-
theorem imageSieve_app {F G : Cᵒᵖ ⥤ A} (f : F ⟶ G) {U : C} (s : F.obj (op U)) :
imageSieve f (f.app _ s) = ⊤ := by
ext (V i)
@@ -72,6 +79,7 @@ theorem imageSieve_app {F G : Cᵒᵖ ⥤ A} (f : F ⟶ G) {U : C} (s : F.obj (o
have := elementwise_of (f.naturality i.op)
exact ⟨F.map i.op s, this s⟩
#align category_theory.image_sieve_app CategoryTheory.imageSieve_app
+-/
#print CategoryTheory.IsLocallySurjective /-
/-- A morphism of presheaves `f : F ⟶ G` is locally surjective with respect to a grothendieck
@@ -81,6 +89,7 @@ def IsLocallySurjective {F G : Cᵒᵖ ⥤ A} (f : F ⟶ G) : Prop :=
#align category_theory.is_locally_surjective CategoryTheory.IsLocallySurjective
-/
+#print CategoryTheory.isLocallySurjective_iff_imagePresheaf_sheafify_eq_top /-
theorem isLocallySurjective_iff_imagePresheaf_sheafify_eq_top {F G : Cᵒᵖ ⥤ A} (f : F ⟶ G) :
IsLocallySurjective J f ↔ (imagePresheaf (whiskerRight f (forget A))).sheafify J = ⊤ :=
by
@@ -88,7 +97,9 @@ theorem isLocallySurjective_iff_imagePresheaf_sheafify_eq_top {F G : Cᵒᵖ ⥤
Set.top_eq_univ, Set.mem_univ, iff_true_iff]
exact ⟨fun H U => H (unop U), fun H U => H (op U)⟩
#align category_theory.is_locally_surjective_iff_image_presheaf_sheafify_eq_top CategoryTheory.isLocallySurjective_iff_imagePresheaf_sheafify_eq_top
+-/
+#print CategoryTheory.isLocallySurjective_iff_imagePresheaf_sheafify_eq_top' /-
theorem isLocallySurjective_iff_imagePresheaf_sheafify_eq_top' {F G : Cᵒᵖ ⥤ Type w} (f : F ⟶ G) :
IsLocallySurjective J f ↔ (imagePresheaf f).sheafify J = ⊤ :=
by
@@ -96,6 +107,7 @@ theorem isLocallySurjective_iff_imagePresheaf_sheafify_eq_top' {F G : Cᵒᵖ
Set.top_eq_univ, Set.mem_univ, iff_true_iff]
exact ⟨fun H U => H (unop U), fun H U => H (op U)⟩
#align category_theory.is_locally_surjective_iff_image_presheaf_sheafify_eq_top' CategoryTheory.isLocallySurjective_iff_imagePresheaf_sheafify_eq_top'
+-/
#print CategoryTheory.isLocallySurjective_iff_isIso /-
theorem isLocallySurjective_iff_isIso {F G : Sheaf J (Type w)} (f : F ⟶ G) :
@@ -116,6 +128,7 @@ theorem isLocallySurjective_iff_whisker_forget {F G : Cᵒᵖ ⥤ A} (f : F ⟶
#align category_theory.is_locally_surjective_iff_whisker_forget CategoryTheory.isLocallySurjective_iff_whisker_forget
-/
+#print CategoryTheory.isLocallySurjective_of_surjective /-
theorem isLocallySurjective_of_surjective {F G : Cᵒᵖ ⥤ A} (f : F ⟶ G)
(H : ∀ U, Function.Surjective (f.app U)) : IsLocallySurjective J f :=
by
@@ -124,6 +137,7 @@ theorem isLocallySurjective_of_surjective {F G : Cᵒᵖ ⥤ A} (f : F ⟶ G)
rw [image_sieve_app]
exact J.top_mem _
#align category_theory.is_locally_surjective_of_surjective CategoryTheory.isLocallySurjective_of_surjective
+-/
#print CategoryTheory.isLocallySurjective_of_iso /-
theorem isLocallySurjective_of_iso {F G : Cᵒᵖ ⥤ A} (f : F ⟶ G) [IsIso f] :
@@ -193,6 +207,7 @@ variable {B : Type w} [Category.{max u v} B] [ConcreteCategory.{max u v} B]
[∀ (α β : Type max u v) (fst snd : β → α),
Limits.HasLimitsOfShape (Limits.WalkingMulticospan fst snd) B]
+#print CategoryTheory.toSheafify_isLocallySurjective /-
theorem toSheafify_isLocallySurjective (F : Cᵒᵖ ⥤ B) : IsLocallySurjective J (J.toSheafify F) :=
by
rw [is_locally_surjective_iff_whisker_forget, ← to_sheafify_comp_sheafify_comp_iso_inv]
@@ -201,6 +216,7 @@ theorem toSheafify_isLocallySurjective (F : Cᵒᵖ ⥤ B) : IsLocallySurjective
exact is_iso.of_iso_inv (sheafification_iso_image_presheaf J (F ⋙ forget B))
· exact is_locally_surjective_of_iso _ _
#align category_theory.to_sheafify_is_locally_surjective CategoryTheory.toSheafify_isLocallySurjective
+-/
end
mathlib commit https://github.com/leanprover-community/mathlib/commit/9fb8964792b4237dac6200193a0d533f1b3f7423
@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
! This file was ported from Lean 3 source module category_theory.sites.surjective
-! leanprover-community/mathlib commit 70fd9563a21e7b963887c9360bd29b2393e6225a
+! leanprover-community/mathlib commit 087c325ae0ab42dbdd5dee55bc37d3d5a0bf2197
! Please do not edit these lines, except to modify the commit id
! if you have ported upstream changes.
-/
@@ -15,6 +15,9 @@ import Mathbin.CategoryTheory.Sites.CompatibleSheafification
# Locally surjective morphisms
+> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
+> Any changes to this file require a corresponding PR to mathlib4.
+
## Main definitions
- `is_locally_surjective` : A morphism of presheaves valued in a concrete category is locally
mathlib commit https://github.com/leanprover-community/mathlib/commit/c471da714c044131b90c133701e51b877c246677
@@ -70,11 +70,13 @@ theorem imageSieve_app {F G : Cᵒᵖ ⥤ A} (f : F ⟶ G) {U : C} (s : F.obj (o
exact ⟨F.map i.op s, this s⟩
#align category_theory.image_sieve_app CategoryTheory.imageSieve_app
+#print CategoryTheory.IsLocallySurjective /-
/-- A morphism of presheaves `f : F ⟶ G` is locally surjective with respect to a grothendieck
topology if every section of `G` is locally in the image of `f`. -/
def IsLocallySurjective {F G : Cᵒᵖ ⥤ A} (f : F ⟶ G) : Prop :=
∀ (U : C) (s : G.obj (op U)), imageSieve f s ∈ J U
#align category_theory.is_locally_surjective CategoryTheory.IsLocallySurjective
+-/
theorem isLocallySurjective_iff_imagePresheaf_sheafify_eq_top {F G : Cᵒᵖ ⥤ A} (f : F ⟶ G) :
IsLocallySurjective J f ↔ (imagePresheaf (whiskerRight f (forget A))).sheafify J = ⊤ :=
@@ -92,6 +94,7 @@ theorem isLocallySurjective_iff_imagePresheaf_sheafify_eq_top' {F G : Cᵒᵖ
exact ⟨fun H U => H (unop U), fun H U => H (op U)⟩
#align category_theory.is_locally_surjective_iff_image_presheaf_sheafify_eq_top' CategoryTheory.isLocallySurjective_iff_imagePresheaf_sheafify_eq_top'
+#print CategoryTheory.isLocallySurjective_iff_isIso /-
theorem isLocallySurjective_iff_isIso {F G : Sheaf J (Type w)} (f : F ⟶ G) :
IsLocallySurjective J f.1 ↔ IsIso (imageSheafι f) :=
by
@@ -101,11 +104,14 @@ theorem isLocallySurjective_iff_isIso {F G : Sheaf J (Type w)} (f : F ⟶ G) :
⟨fun h => @is_iso_of_reflects_iso _ _ (image_sheaf_ι f) (Sheaf_to_presheaf J _) h _, fun h =>
@functor.map_is_iso _ _ (Sheaf_to_presheaf J _) _ h⟩
#align category_theory.is_locally_surjective_iff_is_iso CategoryTheory.isLocallySurjective_iff_isIso
+-/
+#print CategoryTheory.isLocallySurjective_iff_whisker_forget /-
theorem isLocallySurjective_iff_whisker_forget {F G : Cᵒᵖ ⥤ A} (f : F ⟶ G) :
IsLocallySurjective J f ↔ IsLocallySurjective J (whiskerRight f (forget A)) := by
simpa only [is_locally_surjective_iff_image_presheaf_sheafify_eq_top]
#align category_theory.is_locally_surjective_iff_whisker_forget CategoryTheory.isLocallySurjective_iff_whisker_forget
+-/
theorem isLocallySurjective_of_surjective {F G : Cᵒᵖ ⥤ A} (f : F ⟶ G)
(H : ∀ U, Function.Surjective (f.app U)) : IsLocallySurjective J f :=
@@ -116,6 +122,7 @@ theorem isLocallySurjective_of_surjective {F G : Cᵒᵖ ⥤ A} (f : F ⟶ G)
exact J.top_mem _
#align category_theory.is_locally_surjective_of_surjective CategoryTheory.isLocallySurjective_of_surjective
+#print CategoryTheory.isLocallySurjective_of_iso /-
theorem isLocallySurjective_of_iso {F G : Cᵒᵖ ⥤ A} (f : F ⟶ G) [IsIso f] :
IsLocallySurjective J f :=
by
@@ -125,7 +132,9 @@ theorem isLocallySurjective_of_iso {F G : Cᵒᵖ ⥤ A} (f : F ⟶ G) [IsIso f]
rw [← is_iso_iff_bijective]
infer_instance
#align category_theory.is_locally_surjective_of_iso CategoryTheory.isLocallySurjective_of_iso
+-/
+#print CategoryTheory.IsLocallySurjective.comp /-
theorem IsLocallySurjective.comp {F₁ F₂ F₃ : Cᵒᵖ ⥤ A} {f₁ : F₁ ⟶ F₂} {f₂ : F₂ ⟶ F₃}
(h₁ : IsLocallySurjective J f₁) (h₂ : IsLocallySurjective J f₂) :
IsLocallySurjective J (f₁ ≫ f₂) := by
@@ -142,11 +151,13 @@ theorem IsLocallySurjective.comp {F₁ F₂ F₃ : Cᵒᵖ ⥤ A} {f₁ : F₁
· apply h₂
· intros; apply h₁
#align category_theory.is_locally_surjective.comp CategoryTheory.IsLocallySurjective.comp
+-/
section
variable (F : Cᵒᵖ ⥤ Type max u v)
+#print CategoryTheory.sheafificationIsoImagePresheaf /-
/-- The image of `F` in `J.sheafify F` is isomorphic to the sheafification. -/
noncomputable def sheafificationIsoImagePresheaf :
J.sheafify F ≅ ((imagePresheaf (J.toSheafify F)).sheafify J).toPresheaf
@@ -167,6 +178,7 @@ noncomputable def sheafificationIsoImagePresheaf :
exact J.sheafify_hom_ext _ _ (J.sheafify_is_sheaf _) (by simp [to_image_presheaf_sheafify])
infer_instance
#align category_theory.sheafification_iso_image_presheaf CategoryTheory.sheafificationIsoImagePresheaf
+-/
-- We need to sheafify
variable {B : Type w} [Category.{max u v} B] [ConcreteCategory.{max u v} B]
mathlib commit https://github.com/leanprover-community/mathlib/commit/cca40788df1b8755d5baf17ab2f27dacc2e17acb
@@ -140,7 +140,7 @@ theorem IsLocallySurjective.comp {F₁ F₂ F₃ : Cᵒᵖ ⥤ A} {f₁ : F₁
apply J.superset_covering this
apply J.bind_covering
· apply h₂
- · intros ; apply h₁
+ · intros; apply h₁
#align category_theory.is_locally_surjective.comp CategoryTheory.IsLocallySurjective.comp
section
mathlib commit https://github.com/leanprover-community/mathlib/commit/917c3c072e487b3cccdbfeff17e75b40e45f66cb
@@ -140,8 +140,7 @@ theorem IsLocallySurjective.comp {F₁ F₂ F₃ : Cᵒᵖ ⥤ A} {f₁ : F₁
apply J.superset_covering this
apply J.bind_covering
· apply h₂
- · intros
- apply h₁
+ · intros ; apply h₁
#align category_theory.is_locally_surjective.comp CategoryTheory.IsLocallySurjective.comp
section
mathlib commit https://github.com/leanprover-community/mathlib/commit/bd9851ca476957ea4549eb19b40e7b5ade9428cc
FunLike
(#9833)
This follows up from #9785, which renamed FunLike
to DFunLike
, by introducing a new abbreviation FunLike F α β := DFunLike F α (fun _ => β)
, to make the non-dependent use of FunLike
easier.
I searched for the pattern DFunLike.*fun
and DFunLike.*λ
in all files to replace expressions of the form DFunLike F α (fun _ => β)
with FunLike F α β
. I did this everywhere except for extends
clauses for two reasons: it would conflict with #8386, and more importantly extends
must directly refer to a structure with no unfolding of def
s or abbrev
s.
@@ -32,7 +32,7 @@ namespace CategoryTheory
variable {C : Type u} [Category.{v} C] (J : GrothendieckTopology C)
-attribute [local instance] ConcreteCategory.hasCoeToSort ConcreteCategory.instDFunLike
+attribute [local instance] ConcreteCategory.hasCoeToSort ConcreteCategory.instFunLike
variable {A : Type u'} [Category.{v'} A] [ConcreteCategory.{w'} A]
FunLike
to DFunLike
(#9785)
This prepares for the introduction of a non-dependent synonym of FunLike, which helps a lot with keeping #8386 readable.
This is entirely search-and-replace in 680197f combined with manual fixes in 4145626, e900597 and b8428f8. The commands that generated this change:
sed -i 's/\bFunLike\b/DFunLike/g' {Archive,Counterexamples,Mathlib,test}/**/*.lean
sed -i 's/\btoFunLike\b/toDFunLike/g' {Archive,Counterexamples,Mathlib,test}/**/*.lean
sed -i 's/import Mathlib.Data.DFunLike/import Mathlib.Data.FunLike/g' {Archive,Counterexamples,Mathlib,test}/**/*.lean
sed -i 's/\bHom_FunLike\b/Hom_DFunLike/g' {Archive,Counterexamples,Mathlib,test}/**/*.lean
sed -i 's/\binstFunLike\b/instDFunLike/g' {Archive,Counterexamples,Mathlib,test}/**/*.lean
sed -i 's/\bfunLike\b/instDFunLike/g' {Archive,Counterexamples,Mathlib,test}/**/*.lean
sed -i 's/\btoo many metavariables to apply `fun_like.has_coe_to_fun`/too many metavariables to apply `DFunLike.hasCoeToFun`/g' {Archive,Counterexamples,Mathlib,test}/**/*.lean
Co-authored-by: Anne Baanen <Vierkantor@users.noreply.github.com>
@@ -32,7 +32,7 @@ namespace CategoryTheory
variable {C : Type u} [Category.{v} C] (J : GrothendieckTopology C)
-attribute [local instance] ConcreteCategory.hasCoeToSort ConcreteCategory.funLike
+attribute [local instance] ConcreteCategory.hasCoeToSort ConcreteCategory.instDFunLike
variable {A : Type u'} [Category.{v'} A] [ConcreteCategory.{w'} A]
We define a typeclass HasSheafify
which says that presheaves on a site with values in some category can be sheafified, i.e. that the inclusion functor from sheaves to presheaves has a left exact left adjoint. We redefine presheafToSheaf
as an arbitrary choice of such a left adjoint.
@@ -144,7 +144,7 @@ noncomputable def sheafificationIsoImagePresheaf :
J.sheafifyLift (toImagePresheafSheafify J _)
((isSheaf_iff_isSheaf_of_type J _).mpr <|
Subpresheaf.sheafify_isSheaf _ <|
- (isSheaf_iff_isSheaf_of_type J _).mp <| sheafify_isSheaf J _)
+ (isSheaf_iff_isSheaf_of_type J _).mp <| GrothendieckTopology.sheafify_isSheaf J _)
inv := Subpresheaf.ι _
hom_inv_id :=
J.sheafify_hom_ext _ _ (J.sheafify_isSheaf _) (by simp [toImagePresheafSheafify])
@@ -113,7 +113,7 @@ theorem isLocallySurjective_of_iso {F G : Cᵒᵖ ⥤ A} (f : F ⟶ G) [IsIso f]
apply isLocallySurjective_of_surjective
intro U
apply Function.Bijective.surjective
- rw [← isIso_iff_bijective, ←forget_map_eq_coe]
+ rw [← isIso_iff_bijective, ← forget_map_eq_coe]
infer_instance
#align category_theory.is_locally_surjective_of_iso CategoryTheory.isLocallySurjective_of_iso
@@ -2,14 +2,11 @@
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-
-! This file was ported from Lean 3 source module category_theory.sites.surjective
-! leanprover-community/mathlib commit 70fd9563a21e7b963887c9360bd29b2393e6225a
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
-/
import Mathlib.CategoryTheory.Sites.Subsheaf
import Mathlib.CategoryTheory.Sites.CompatibleSheafification
+
+#align_import category_theory.sites.surjective from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
/-!
# Locally surjective morphisms
@@ -62,8 +62,7 @@ theorem imageSieve_whisker_forget {F G : Cᵒᵖ ⥤ A} (f : F ⟶ G) {U : C} (s
theorem imageSieve_app {F G : Cᵒᵖ ⥤ A} (f : F ⟶ G) {U : C} (s : F.obj (op U)) :
imageSieve f (f.app _ s) = ⊤ := by
- ext V
- intro i
+ ext V i
simp only [Sieve.top_apply, iff_true_iff, imageSieve_apply]
have := elementwise_of% (f.naturality i.op)
exact ⟨F.map i.op s, this s⟩
@@ -35,7 +35,7 @@ namespace CategoryTheory
variable {C : Type u} [Category.{v} C] (J : GrothendieckTopology C)
-attribute [local instance] ConcreteCategory.hasCoeToSort ConcreteCategory.hasCoeToFun
+attribute [local instance] ConcreteCategory.hasCoeToSort ConcreteCategory.funLike
variable {A : Type u'} [Category.{v'} A] [ConcreteCategory.{w'} A]
@@ -117,7 +117,7 @@ theorem isLocallySurjective_of_iso {F G : Cᵒᵖ ⥤ A} (f : F ⟶ G) [IsIso f]
apply isLocallySurjective_of_surjective
intro U
apply Function.Bijective.surjective
- rw [← isIso_iff_bijective]
+ rw [← isIso_iff_bijective, ←forget_map_eq_coe]
infer_instance
#align category_theory.is_locally_surjective_of_iso CategoryTheory.isLocallySurjective_of_iso
The unported dependencies are