category_theory.sites.types
⟷
Mathlib.CategoryTheory.Sites.Types
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
@@ -3,7 +3,7 @@ Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
-import Mathbin.CategoryTheory.Sites.Canonical
+import CategoryTheory.Sites.Canonical
#align_import category_theory.sites.types from "leanprover-community/mathlib"@"7e5137f579de09a059a5ce98f364a04e221aabf0"
mathlib commit https://github.com/leanprover-community/mathlib/commit/8ea5598db6caeddde6cb734aa179cc2408dbd345
@@ -2,14 +2,11 @@
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-
-! This file was ported from Lean 3 source module category_theory.sites.types
-! leanprover-community/mathlib commit 7e5137f579de09a059a5ce98f364a04e221aabf0
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
-/
import Mathbin.CategoryTheory.Sites.Canonical
+#align_import category_theory.sites.types from "leanprover-community/mathlib"@"7e5137f579de09a059a5ce98f364a04e221aabf0"
+
/-!
# Grothendieck Topology and Sheaves on the Category of Types
mathlib commit https://github.com/leanprover-community/mathlib/commit/9fb8964792b4237dac6200193a0d533f1b3f7423
@@ -73,6 +73,7 @@ theorem generate_discretePresieve_mem (α : Type u) :
open Presieve
+#print CategoryTheory.isSheaf_yoneda' /-
theorem isSheaf_yoneda' {α : Type u} : IsSheaf typesGrothendieckTopology (yoneda.obj α) :=
fun β S hs x hx =>
⟨fun y => x _ (hs y) PUnit.unit, fun γ f h =>
@@ -83,6 +84,7 @@ theorem isSheaf_yoneda' {α : Type u} : IsSheaf typesGrothendieckTopology (yoned
exact rfl,
fun f hf => funext fun y => by convert congr_fun (hf _ (hs y)) PUnit.unit⟩
#align category_theory.is_sheaf_yoneda' CategoryTheory.isSheaf_yoneda'
+-/
#print CategoryTheory.yoneda' /-
/-- The yoneda functor that sends a type to a sheaf over the category of types -/
@@ -103,12 +105,15 @@ theorem yoneda'_comp : yoneda'.{u} ⋙ sheafOfTypesToPresheaf _ = yoneda :=
open Opposite
+#print CategoryTheory.eval /-
/-- Given a presheaf `P` on the category of types, construct
a map `P(α) → (α → P(*))` for all type `α`. -/
def eval (P : Type uᵒᵖ ⥤ Type u) (α : Type u) (s : P.obj (op α)) (x : α) : P.obj (op PUnit) :=
P.map (↾fun _ => x).op s
#align category_theory.eval CategoryTheory.eval
+-/
+#print CategoryTheory.typesGlue /-
/-- Given a sheaf `S` on the category of types, construct a map
`(α → S(*)) → S(α)` that is inverse to `eval`. -/
noncomputable def typesGlue (S : Type uᵒᵖ ⥤ Type u) (hs : IsSheaf typesGrothendieckTopology S)
@@ -124,20 +129,26 @@ noncomputable def typesGlue (S : Type uᵒᵖ ⥤ Type u) (hs : IsSheaf typesGro
simp_rw [← functor_to_types.map_comp_apply, this, ← op_comp]
rfl
#align category_theory.types_glue CategoryTheory.typesGlue
+-/
+#print CategoryTheory.eval_typesGlue /-
theorem eval_typesGlue {S hs α} (f) : eval.{u} S α (typesGlue S hs α f) = f :=
funext fun x =>
(IsSheafFor.valid_glue _ _ _ <| ⟨PUnit.unit, fun _ => Subsingleton.elim _ _⟩).trans <| by
convert functor_to_types.map_id_apply _ _; rw [← op_id]; congr
#align category_theory.eval_types_glue CategoryTheory.eval_typesGlue
+-/
+#print CategoryTheory.typesGlue_eval /-
theorem typesGlue_eval {S hs α} (s) : typesGlue.{u} S hs α (eval S α s) = s :=
(hs.IsSheafFor _ _ (generate_discretePresieve_mem α)).IsSeparatedFor.ext fun β f hf =>
(IsSheafFor.valid_glue _ _ _ hf).trans <|
(FunctorToTypes.map_comp_apply _ _ _ _).symm.trans <| by rw [← op_comp]; congr 2;
exact funext fun x => congr_arg f (Classical.choose_spec hf x).symm
#align category_theory.types_glue_eval CategoryTheory.typesGlue_eval
+-/
+#print CategoryTheory.evalEquiv /-
/-- Given a sheaf `S`, construct an equivalence `S(α) ≃ (α → S(*))`. -/
@[simps]
noncomputable def evalEquiv (S : Type uᵒᵖ ⥤ Type u) (hs : IsSheaf typesGrothendieckTopology S)
@@ -148,12 +159,16 @@ noncomputable def evalEquiv (S : Type uᵒᵖ ⥤ Type u) (hs : IsSheaf typesGro
left_inv := typesGlue_eval
right_inv := eval_typesGlue
#align category_theory.eval_equiv CategoryTheory.evalEquiv
+-/
+#print CategoryTheory.eval_map /-
theorem eval_map (S : Type uᵒᵖ ⥤ Type u) (α β) (f : β ⟶ α) (s x) :
eval S β (S.map f.op s) x = eval S α s (f x) := by
simp_rw [eval, ← functor_to_types.map_comp_apply, ← op_comp]; rfl
#align category_theory.eval_map CategoryTheory.eval_map
+-/
+#print CategoryTheory.equivYoneda /-
/-- Given a sheaf `S`, construct an isomorphism `S ≅ [-, S(*)]`. -/
@[simps]
noncomputable def equivYoneda (S : Type uᵒᵖ ⥤ Type u) (hs : IsSheaf typesGrothendieckTopology S) :
@@ -161,7 +176,9 @@ noncomputable def equivYoneda (S : Type uᵒᵖ ⥤ Type u) (hs : IsSheaf typesG
NatIso.ofComponents (fun α => Equiv.toIso <| evalEquiv S hs <| unop α) fun α β f =>
funext fun s => funext fun x => eval_map S (unop α) (unop β) f.unop _ _
#align category_theory.equiv_yoneda CategoryTheory.equivYoneda
+-/
+#print CategoryTheory.equivYoneda' /-
/-- Given a sheaf `S`, construct an isomorphism `S ≅ [-, S(*)]`. -/
@[simps]
noncomputable def equivYoneda' (S : SheafOfTypes typesGrothendieckTopology) :
@@ -172,13 +189,17 @@ noncomputable def equivYoneda' (S : SheafOfTypes typesGrothendieckTopology) :
hom_inv_id' := by ext1; apply (equiv_yoneda S.1 S.2).hom_inv_id
inv_hom_id' := by ext1; apply (equiv_yoneda S.1 S.2).inv_hom_id
#align category_theory.equiv_yoneda' CategoryTheory.equivYoneda'
+-/
+#print CategoryTheory.eval_app /-
theorem eval_app (S₁ S₂ : SheafOfTypes.{u} typesGrothendieckTopology) (f : S₁ ⟶ S₂) (α : Type u)
(s : S₁.1.obj (op α)) (x : α) :
eval S₂.1 α (f.val.app (op α) s) x = f.val.app (op PUnit) (eval S₁.1 α s x) :=
(congr_fun (f.val.naturality (↾fun _ : PUnit => x).op) s).symm
#align category_theory.eval_app CategoryTheory.eval_app
+-/
+#print CategoryTheory.typeEquiv /-
/-- `yoneda'` induces an equivalence of category between `Type u` and
`Sheaf types_grothendieck_topology`. -/
@[simps]
@@ -198,6 +219,7 @@ noncomputable def typeEquiv : Type u ≌ SheafOfTypes typesGrothendieckTopology
NatTrans.ext _ _ <|
funext fun α => funext fun s => funext fun x => eval_app S₁ S₂ f (unop α) s x)
#align category_theory.type_equiv CategoryTheory.typeEquiv
+-/
#print CategoryTheory.subcanonical_typesGrothendieckTopology /-
theorem subcanonical_typesGrothendieckTopology : Sheaf.Subcanonical typesGrothendieckTopology.{u} :=
mathlib commit https://github.com/leanprover-community/mathlib/commit/7e5137f579de09a059a5ce98f364a04e221aabf0
@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
! This file was ported from Lean 3 source module category_theory.sites.types
-! leanprover-community/mathlib commit 9f9015c645d85695581237cc761981036be8bd37
+! leanprover-community/mathlib commit 7e5137f579de09a059a5ce98f364a04e221aabf0
! 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.Canonical
/-!
# Grothendieck Topology and Sheaves on the Category of Types
+> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
+> Any changes to this file require a corresponding PR to mathlib4.
+
In this file we define a Grothendieck topology on the category of types,
and construct the canonical functor that sends a type to a sheaf over
the category of types, and make this an equivalence of categories.
mathlib commit https://github.com/leanprover-community/mathlib/commit/58a272265b5e05f258161260dd2c5d247213cbd3
@@ -27,6 +27,7 @@ namespace CategoryTheory
open scoped CategoryTheory.Type
+#print CategoryTheory.typesGrothendieckTopology /-
/-- A Grothendieck topology associated to the category of all types.
A sieve is a covering iff it is jointly surjective. -/
def typesGrothendieckTopology : GrothendieckTopology (Type u)
@@ -36,7 +37,9 @@ def typesGrothendieckTopology : GrothendieckTopology (Type u)
pullback_stable' α β S f hs x := hs (f x)
transitive' α S hs R hr x := hr (hs x) PUnit.unit
#align category_theory.types_grothendieck_topology CategoryTheory.typesGrothendieckTopology
+-/
+#print CategoryTheory.discreteSieve /-
/-- The discrete sieve on a type, which only includes arrows whose image is a subsingleton. -/
@[simps]
def discreteSieve (α : Type u) : Sieve α
@@ -44,19 +47,26 @@ def discreteSieve (α : Type u) : Sieve α
arrows β f := ∃ x, ∀ y, f y = x
downward_closed' := fun β γ f ⟨x, hx⟩ g => ⟨x, fun y => hx <| g y⟩
#align category_theory.discrete_sieve CategoryTheory.discreteSieve
+-/
+#print CategoryTheory.discreteSieve_mem /-
theorem discreteSieve_mem (α : Type u) : discreteSieve α ∈ typesGrothendieckTopology α := fun x =>
⟨x, fun y => rfl⟩
#align category_theory.discrete_sieve_mem CategoryTheory.discreteSieve_mem
+-/
+#print CategoryTheory.discretePresieve /-
/-- The discrete presieve on a type, which only includes arrows whose domain is a singleton. -/
def discretePresieve (α : Type u) : Presieve α := fun β f => ∃ x : β, ∀ y : β, y = x
#align category_theory.discrete_presieve CategoryTheory.discretePresieve
+-/
+#print CategoryTheory.generate_discretePresieve_mem /-
theorem generate_discretePresieve_mem (α : Type u) :
Sieve.generate (discretePresieve α) ∈ typesGrothendieckTopology α := fun x =>
⟨PUnit, id, fun _ => x, ⟨PUnit.unit, fun _ => Subsingleton.elim _ _⟩, rfl⟩
#align category_theory.generate_discrete_presieve_mem CategoryTheory.generate_discretePresieve_mem
+-/
open Presieve
@@ -71,6 +81,7 @@ theorem isSheaf_yoneda' {α : Type u} : IsSheaf typesGrothendieckTopology (yoned
fun f hf => funext fun y => by convert congr_fun (hf _ (hs y)) PUnit.unit⟩
#align category_theory.is_sheaf_yoneda' CategoryTheory.isSheaf_yoneda'
+#print CategoryTheory.yoneda' /-
/-- The yoneda functor that sends a type to a sheaf over the category of types -/
@[simps]
def yoneda' : Type u ⥤ SheafOfTypes typesGrothendieckTopology
@@ -78,11 +89,14 @@ def yoneda' : Type u ⥤ SheafOfTypes typesGrothendieckTopology
obj α := ⟨yoneda.obj α, isSheaf_yoneda'⟩
map α β f := ⟨yoneda.map f⟩
#align category_theory.yoneda' CategoryTheory.yoneda'
+-/
+#print CategoryTheory.yoneda'_comp /-
@[simp]
theorem yoneda'_comp : yoneda'.{u} ⋙ sheafOfTypesToPresheaf _ = yoneda :=
rfl
#align category_theory.yoneda'_comp CategoryTheory.yoneda'_comp
+-/
open Opposite
@@ -182,10 +196,13 @@ noncomputable def typeEquiv : Type u ≌ SheafOfTypes typesGrothendieckTopology
funext fun α => funext fun s => funext fun x => eval_app S₁ S₂ f (unop α) s x)
#align category_theory.type_equiv CategoryTheory.typeEquiv
+#print CategoryTheory.subcanonical_typesGrothendieckTopology /-
theorem subcanonical_typesGrothendieckTopology : Sheaf.Subcanonical typesGrothendieckTopology.{u} :=
Sheaf.Subcanonical.of_yoneda_isSheaf _ fun X => isSheaf_yoneda'
#align category_theory.subcanonical_types_grothendieck_topology CategoryTheory.subcanonical_typesGrothendieckTopology
+-/
+#print CategoryTheory.typesGrothendieckTopology_eq_canonical /-
theorem typesGrothendieckTopology_eq_canonical :
typesGrothendieckTopology.{u} = Sheaf.canonicalTopology (Type u) :=
le_antisymm subcanonical_typesGrothendieckTopology <|
@@ -204,6 +221,7 @@ theorem typesGrothendieckTopology_eq_canonical :
Bool.noConfusion <| ULift.up.inj <| (congr_fun this PUnit.unit : _),
fun hs β f => isSheaf_yoneda' _ fun y => hs _⟩⟩
#align category_theory.types_grothendieck_topology_eq_canonical CategoryTheory.typesGrothendieckTopology_eq_canonical
+-/
end CategoryTheory
mathlib commit https://github.com/leanprover-community/mathlib/commit/917c3c072e487b3cccdbfeff17e75b40e45f66cb
@@ -25,7 +25,7 @@ universe u
namespace CategoryTheory
-open CategoryTheory.Type
+open scoped CategoryTheory.Type
/-- A Grothendieck topology associated to the category of all types.
A sieve is a covering iff it is jointly surjective. -/
mathlib commit https://github.com/leanprover-community/mathlib/commit/917c3c072e487b3cccdbfeff17e75b40e45f66cb
@@ -110,20 +110,14 @@ noncomputable def typesGlue (S : Type uᵒᵖ ⥤ Type u) (hs : IsSheaf typesGro
theorem eval_typesGlue {S hs α} (f) : eval.{u} S α (typesGlue S hs α f) = f :=
funext fun x =>
- (IsSheafFor.valid_glue _ _ _ <| ⟨PUnit.unit, fun _ => Subsingleton.elim _ _⟩).trans <|
- by
- convert functor_to_types.map_id_apply _ _
- rw [← op_id]
- congr
+ (IsSheafFor.valid_glue _ _ _ <| ⟨PUnit.unit, fun _ => Subsingleton.elim _ _⟩).trans <| by
+ convert functor_to_types.map_id_apply _ _; rw [← op_id]; congr
#align category_theory.eval_types_glue CategoryTheory.eval_typesGlue
theorem typesGlue_eval {S hs α} (s) : typesGlue.{u} S hs α (eval S α s) = s :=
(hs.IsSheafFor _ _ (generate_discretePresieve_mem α)).IsSeparatedFor.ext fun β f hf =>
(IsSheafFor.valid_glue _ _ _ hf).trans <|
- (FunctorToTypes.map_comp_apply _ _ _ _).symm.trans <|
- by
- rw [← op_comp]
- congr 2
+ (FunctorToTypes.map_comp_apply _ _ _ _).symm.trans <| by rw [← op_comp]; congr 2;
exact funext fun x => congr_arg f (Classical.choose_spec hf x).symm
#align category_theory.types_glue_eval CategoryTheory.typesGlue_eval
@@ -139,10 +133,8 @@ noncomputable def evalEquiv (S : Type uᵒᵖ ⥤ Type u) (hs : IsSheaf typesGro
#align category_theory.eval_equiv CategoryTheory.evalEquiv
theorem eval_map (S : Type uᵒᵖ ⥤ Type u) (α β) (f : β ⟶ α) (s x) :
- eval S β (S.map f.op s) x = eval S α s (f x) :=
- by
- simp_rw [eval, ← functor_to_types.map_comp_apply, ← op_comp]
- rfl
+ eval S β (S.map f.op s) x = eval S α s (f x) := by
+ simp_rw [eval, ← functor_to_types.map_comp_apply, ← op_comp]; rfl
#align category_theory.eval_map CategoryTheory.eval_map
/-- Given a sheaf `S`, construct an isomorphism `S ≅ [-, S(*)]`. -/
@@ -160,12 +152,8 @@ noncomputable def equivYoneda' (S : SheafOfTypes typesGrothendieckTopology) :
where
Hom := ⟨(equivYoneda S.1 S.2).Hom⟩
inv := ⟨(equivYoneda S.1 S.2).inv⟩
- hom_inv_id' := by
- ext1
- apply (equiv_yoneda S.1 S.2).hom_inv_id
- inv_hom_id' := by
- ext1
- apply (equiv_yoneda S.1 S.2).inv_hom_id
+ hom_inv_id' := by ext1; apply (equiv_yoneda S.1 S.2).hom_inv_id
+ inv_hom_id' := by ext1; apply (equiv_yoneda S.1 S.2).inv_hom_id
#align category_theory.equiv_yoneda' CategoryTheory.equivYoneda'
theorem eval_app (S₁ S₂ : SheafOfTypes.{u} typesGrothendieckTopology) (f : S₁ ⟶ S₂) (α : Type u)
mathlib commit https://github.com/leanprover-community/mathlib/commit/e3fb84046afd187b710170887195d50bada934ee
@@ -201,7 +201,7 @@ theorem subcanonical_typesGrothendieckTopology : Sheaf.Subcanonical typesGrothen
theorem typesGrothendieckTopology_eq_canonical :
typesGrothendieckTopology.{u} = Sheaf.canonicalTopology (Type u) :=
le_antisymm subcanonical_typesGrothendieckTopology <|
- infₛ_le
+ sInf_le
⟨yoneda.obj (ULift Bool), ⟨_, rfl⟩,
GrothendieckTopology.ext <|
funext fun α =>
mathlib commit https://github.com/leanprover-community/mathlib/commit/bd9851ca476957ea4549eb19b40e7b5ade9428cc
@@ -4,7 +4,6 @@ Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.CategoryTheory.Sites.Canonical
-import Mathlib.CategoryTheory.Sites.EqualizerSheafCondition
#align_import category_theory.sites.types from "leanprover-community/mathlib"@"9f9015c645d85695581237cc761981036be8bd37"
This is the supremum of
along with some minor fixes from failures on nightly-testing as Mathlib master
is merged into it.
Note that some PRs for changes that are already compatible with the current toolchain and will be necessary have already been split out: #8380.
I am hopeful that in future we will be able to progressively merge adaptation PRs into a bump/v4.X.0
branch, so we never end up with a "big merge" like this. However one of these adaptation PRs (#8056) predates my new scheme for combined CI, and it wasn't possible to keep that PR viable in the meantime.
In particular this includes adjustments for the Lean PRs
We can get rid of all the
local macro_rules | `($x ^ $y) => `(HPow.hPow $x $y) -- Porting note: See issue [lean4#2220](https://github.com/leanprover/lean4/pull/2220)
macros across Mathlib (and in any projects that want to write natural number powers of reals).
Changes the default behaviour of simp
to (config := {decide := false})
. This makes simp
(and consequentially norm_num
) less powerful, but also more consistent, and less likely to blow up in long failures. This requires a variety of changes: changing some previously by simp
or norm_num
to decide
or rfl
, or adding (config := {decide := true})
.
This changed the behaviour of simp
so that simp [f]
will only unfold "fully applied" occurrences of f
. The old behaviour can be recovered with simp (config := { unfoldPartialApp := true })
. We may in future add a syntax for this, e.g. simp [!f]
; please provide feedback! In the meantime, we have made the following changes:
(config := { unfoldPartialApp := true })
in some places, to recover the old behaviour@[eqns]
to manually adjust the equation lemmas for a particular definition, recovering the old behaviour just for that definition. See #8371, where we do this for Function.comp
and Function.flip
.This change in Lean may require further changes down the line (e.g. adding the !f
syntax, and/or upstreaming the special treatment for Function.comp
and Function.flip
, and/or removing this special treatment). Please keep an open and skeptical mind about these changes!
Co-authored-by: leanprover-community-mathlib4-bot <leanprover-community-mathlib4-bot@users.noreply.github.com> Co-authored-by: Scott Morrison <scott.morrison@gmail.com> Co-authored-by: Eric Wieser <wieser.eric@gmail.com> Co-authored-by: Mauricio Collares <mauricio@collares.org>
@@ -191,7 +191,7 @@ theorem typesGrothendieckTopology_eq_canonical :
have : (fun _ => ULift.up true) = fun _ => ULift.up false :=
(hs PUnit fun _ => x).isSeparatedFor.ext
fun β f hf => funext fun y => hsx.elim <| S.2 hf fun _ => y
- simp at this
+ simp [Function.funext_iff] at this
#align category_theory.types_grothendieck_topology_eq_canonical CategoryTheory.typesGrothendieckTopology_eq_canonical
end CategoryTheory
I was planning to add some stuff to this file which was already over 1000 lines long.
IsSheafFor
is now in CategoryTheory/Sites/IsSheafFor
.CategoryTheory/Sites/EqualizerSheafCondition
IsSheaf
/IsSeparated
are left in the file CategoryTheory/Sites/SheafOfTypes
.@@ -4,6 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.CategoryTheory.Sites.Canonical
+import Mathlib.CategoryTheory.Sites.EqualizerSheafCondition
#align_import category_theory.sites.types from "leanprover-community/mathlib"@"9f9015c645d85695581237cc761981036be8bd37"
@@ -2,14 +2,11 @@
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-
-! This file was ported from Lean 3 source module category_theory.sites.types
-! leanprover-community/mathlib commit 9f9015c645d85695581237cc761981036be8bd37
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
-/
import Mathlib.CategoryTheory.Sites.Canonical
+#align_import category_theory.sites.types from "leanprover-community/mathlib"@"9f9015c645d85695581237cc761981036be8bd37"
+
/-!
# Grothendieck Topology and Sheaves on the Category of Types
The unported dependencies are