category_theory.grothendieck
⟷
Mathlib.CategoryTheory.Grothendieck
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
@@ -84,10 +84,10 @@ theorem ext {X Y : Grothendieck F} (f g : Hom X Y) (w_base : f.base = g.base)
by
cases f <;> cases g
congr
- dsimp at w_base
+ dsimp at w_base
induction w_base
rfl
- dsimp at w_base
+ dsimp at w_base
induction w_base
simpa using w_fiber
#align category_theory.grothendieck.ext CategoryTheory.Grothendieck.ext
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.CategoryTheory.Category.Cat
-import Mathbin.CategoryTheory.Elements
+import CategoryTheory.Category.Cat
+import CategoryTheory.Elements
#align_import category_theory.grothendieck from "leanprover-community/mathlib"@"75be6b616681ab6ca66d798ead117e75cd64f125"
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 category_theory.grothendieck
-! leanprover-community/mathlib commit 75be6b616681ab6ca66d798ead117e75cd64f125
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
-/
import Mathbin.CategoryTheory.Category.Cat
import Mathbin.CategoryTheory.Elements
+#align_import category_theory.grothendieck from "leanprover-community/mathlib"@"75be6b616681ab6ca66d798ead117e75cd64f125"
+
/-!
# The Grothendieck construction
mathlib commit https://github.com/leanprover-community/mathlib/commit/9fb8964792b4237dac6200193a0d533f1b3f7423
@@ -50,6 +50,7 @@ variable {C D : Type _} [Category C] [Category D]
variable (F : C ⥤ Cat)
+#print CategoryTheory.Grothendieck /-
/--
The Grothendieck construction (often written as `∫ F` in mathematics) for a functor `F : C ⥤ Cat`
gives a category whose
@@ -63,11 +64,13 @@ structure Grothendieck where
base : C
fiber : F.obj base
#align category_theory.grothendieck CategoryTheory.Grothendieck
+-/
namespace Grothendieck
variable {F}
+#print CategoryTheory.Grothendieck.Hom /-
/-- A morphism in the Grothendieck category `F : C ⥤ Cat` consists of
`base : X.base ⟶ Y.base` and `f.fiber : (F.map base).obj X.fiber ⟶ Y.fiber`.
-/
@@ -75,7 +78,9 @@ structure Hom (X Y : Grothendieck F) where
base : X.base ⟶ Y.base
fiber : (F.map base).obj X.fiber ⟶ Y.fiber
#align category_theory.grothendieck.hom CategoryTheory.Grothendieck.Hom
+-/
+#print CategoryTheory.Grothendieck.ext /-
@[ext]
theorem ext {X Y : Grothendieck F} (f g : Hom X Y) (w_base : f.base = g.base)
(w_fiber : eqToHom (by rw [w_base]) ≫ f.fiber = g.fiber) : f = g :=
@@ -89,7 +94,9 @@ theorem ext {X Y : Grothendieck F} (f g : Hom X Y) (w_base : f.base = g.base)
induction w_base
simpa using w_fiber
#align category_theory.grothendieck.ext CategoryTheory.Grothendieck.ext
+-/
+#print CategoryTheory.Grothendieck.id /-
/-- The identity morphism in the Grothendieck category.
-/
@[simps]
@@ -97,10 +104,12 @@ def id (X : Grothendieck F) : Hom X X where
base := 𝟙 X.base
fiber := eqToHom (by erw [CategoryTheory.Functor.map_id, functor.id_obj X.fiber])
#align category_theory.grothendieck.id CategoryTheory.Grothendieck.id
+-/
instance (X : Grothendieck F) : Inhabited (Hom X X) :=
⟨id X⟩
+#print CategoryTheory.Grothendieck.comp /-
/-- Composition of morphisms in the Grothendieck category.
-/
@[simps]
@@ -110,6 +119,7 @@ def comp {X Y Z : Grothendieck F} (f : Hom X Y) (g : Hom Y Z) : Hom X Z
fiber :=
eqToHom (by erw [functor.map_comp, functor.comp_obj]) ≫ (F.map g.base).map f.fiber ≫ g.fiber
#align category_theory.grothendieck.comp CategoryTheory.Grothendieck.comp
+-/
attribute [local simp] eq_to_hom_map
@@ -135,26 +145,32 @@ instance : Category (Grothendieck F)
simp
rfl
+#print CategoryTheory.Grothendieck.id_fiber' /-
@[simp]
theorem id_fiber' (X : Grothendieck F) :
Hom.fiber (𝟙 X) = eqToHom (by erw [CategoryTheory.Functor.map_id, functor.id_obj X.fiber]) :=
id_fiber X
#align category_theory.grothendieck.id_fiber' CategoryTheory.Grothendieck.id_fiber'
+-/
+#print CategoryTheory.Grothendieck.congr /-
theorem congr {X Y : Grothendieck F} {f g : X ⟶ Y} (h : f = g) :
f.fiber = eqToHom (by subst h) ≫ g.fiber := by subst h; dsimp; simp
#align category_theory.grothendieck.congr CategoryTheory.Grothendieck.congr
+-/
section
variable (F)
+#print CategoryTheory.Grothendieck.forget /-
/-- The forgetful functor from `grothendieck F` to the source category. -/
@[simps]
def forget : Grothendieck F ⥤ C where
obj X := X.1
map X Y f := f.1
#align category_theory.grothendieck.forget CategoryTheory.Grothendieck.forget
+-/
end
@@ -162,6 +178,7 @@ universe w
variable (G : C ⥤ Type w)
+#print CategoryTheory.Grothendieck.grothendieckTypeToCatFunctor /-
/-- Auxiliary definition for `grothendieck_Type_to_Cat`, to speed up elaboration. -/
@[simps]
def grothendieckTypeToCatFunctor : Grothendieck (G ⋙ typeToCat) ⥤ G.Elements
@@ -169,7 +186,9 @@ def grothendieckTypeToCatFunctor : Grothendieck (G ⋙ typeToCat) ⥤ G.Elements
obj X := ⟨X.1, X.2.as⟩
map X Y f := ⟨f.1, f.2.1.1⟩
#align category_theory.grothendieck.grothendieck_Type_to_Cat_functor CategoryTheory.Grothendieck.grothendieckTypeToCatFunctor
+-/
+#print CategoryTheory.Grothendieck.grothendieckTypeToCatInverse /-
/-- Auxiliary definition for `grothendieck_Type_to_Cat`, to speed up elaboration. -/
@[simps]
def grothendieckTypeToCatInverse : G.Elements ⥤ Grothendieck (G ⋙ typeToCat)
@@ -177,7 +196,9 @@ def grothendieckTypeToCatInverse : G.Elements ⥤ Grothendieck (G ⋙ typeToCat)
obj X := ⟨X.1, ⟨X.2⟩⟩
map X Y f := ⟨f.1, ⟨⟨f.2⟩⟩⟩
#align category_theory.grothendieck.grothendieck_Type_to_Cat_inverse CategoryTheory.Grothendieck.grothendieckTypeToCatInverse
+-/
+#print CategoryTheory.Grothendieck.grothendieckTypeToCat /-
/-- The Grothendieck construction applied to a functor to `Type`
(thought of as a functor to `Cat` by realising a type as a discrete category)
is the same as the 'category of elements' construction.
@@ -195,6 +216,7 @@ def grothendieckTypeToCat : Grothendieck (G ⋙ typeToCat) ≌ G.Elements
(by rintro ⟨⟩ ⟨⟩ ⟨f, e⟩; dsimp at *; subst e; ext; simp)
functor_unitIso_comp' := by rintro ⟨_, ⟨⟩⟩; dsimp; simp; rfl
#align category_theory.grothendieck.grothendieck_Type_to_Cat CategoryTheory.Grothendieck.grothendieckTypeToCat
+-/
end Grothendieck
mathlib commit https://github.com/leanprover-community/mathlib/commit/cca40788df1b8755d5baf17ab2f27dacc2e17acb
@@ -82,10 +82,10 @@ theorem ext {X Y : Grothendieck F} (f g : Hom X Y) (w_base : f.base = g.base)
by
cases f <;> cases g
congr
- dsimp at w_base
+ dsimp at w_base
induction w_base
rfl
- dsimp at w_base
+ dsimp at w_base
induction w_base
simpa using w_fiber
#align category_theory.grothendieck.ext CategoryTheory.Grothendieck.ext
mathlib commit https://github.com/leanprover-community/mathlib/commit/917c3c072e487b3cccdbfeff17e75b40e45f66cb
@@ -50,12 +50,6 @@ variable {C D : Type _} [Category C] [Category D]
variable (F : C ⥤ Cat)
-/- warning: category_theory.grothendieck -> CategoryTheory.Grothendieck is a dubious translation:
-lean 3 declaration is
- forall {C : Type.{u1}} [_inst_1 : CategoryTheory.Category.{u2, u1} C], (CategoryTheory.Functor.{u2, max u3 u4, u1, max (succ u4) u4 (succ u3)} C _inst_1 CategoryTheory.Cat.{u3, u4} CategoryTheory.Cat.category.{u3, u4}) -> Sort.{max (succ u1) (succ u4)}
-but is expected to have type
- forall {C : Type.{u1}} [_inst_1 : CategoryTheory.Category.{u2, u1} C], (CategoryTheory.Functor.{u2, max u3 u4, u1, max (succ u3) (succ u4)} C _inst_1 CategoryTheory.Cat.{u4, u3} CategoryTheory.Cat.category.{u4, u3}) -> Sort.{max (succ u1) (succ u3)}
-Case conversion may be inaccurate. Consider using '#align category_theory.grothendieck CategoryTheory.Grothendieckₓ'. -/
/--
The Grothendieck construction (often written as `∫ F` in mathematics) for a functor `F : C ⥤ Cat`
gives a category whose
@@ -74,12 +68,6 @@ namespace Grothendieck
variable {F}
-/- warning: category_theory.grothendieck.hom -> CategoryTheory.Grothendieck.Hom is a dubious translation:
-lean 3 declaration is
- forall {C : Type.{u1}} [_inst_1 : CategoryTheory.Category.{u2, u1} C] {F : CategoryTheory.Functor.{u2, max u3 u4, u1, max (succ u4) u4 (succ u3)} C _inst_1 CategoryTheory.Cat.{u3, u4} CategoryTheory.Cat.category.{u3, u4}}, (CategoryTheory.Grothendieck.{u1, u2, u3, u4} C _inst_1 F) -> (CategoryTheory.Grothendieck.{u1, u2, u3, u4} C _inst_1 F) -> Sort.{max (succ u2) (succ u3)}
-but is expected to have type
- forall {C : Type.{u1}} [_inst_1 : CategoryTheory.Category.{u2, u1} C] {F : CategoryTheory.Functor.{u2, max u3 u4, u1, max (succ u3) (succ u4)} C _inst_1 CategoryTheory.Cat.{u4, u3} CategoryTheory.Cat.category.{u4, u3}}, (CategoryTheory.Grothendieck.{u1, u2, u3, u4} C _inst_1 F) -> (CategoryTheory.Grothendieck.{u1, u2, u3, u4} C _inst_1 F) -> Sort.{max (succ u2) (succ u4)}
-Case conversion may be inaccurate. Consider using '#align category_theory.grothendieck.hom CategoryTheory.Grothendieck.Homₓ'. -/
/-- A morphism in the Grothendieck category `F : C ⥤ Cat` consists of
`base : X.base ⟶ Y.base` and `f.fiber : (F.map base).obj X.fiber ⟶ Y.fiber`.
-/
@@ -88,9 +76,6 @@ structure Hom (X Y : Grothendieck F) where
fiber : (F.map base).obj X.fiber ⟶ Y.fiber
#align category_theory.grothendieck.hom CategoryTheory.Grothendieck.Hom
-/- warning: category_theory.grothendieck.ext -> CategoryTheory.Grothendieck.ext is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align category_theory.grothendieck.ext CategoryTheory.Grothendieck.extₓ'. -/
@[ext]
theorem ext {X Y : Grothendieck F} (f g : Hom X Y) (w_base : f.base = g.base)
(w_fiber : eqToHom (by rw [w_base]) ≫ f.fiber = g.fiber) : f = g :=
@@ -105,12 +90,6 @@ theorem ext {X Y : Grothendieck F} (f g : Hom X Y) (w_base : f.base = g.base)
simpa using w_fiber
#align category_theory.grothendieck.ext CategoryTheory.Grothendieck.ext
-/- warning: category_theory.grothendieck.id -> CategoryTheory.Grothendieck.id is a dubious translation:
-lean 3 declaration is
- forall {C : Type.{u1}} [_inst_1 : CategoryTheory.Category.{u2, u1} C] {F : CategoryTheory.Functor.{u2, max u3 u4, u1, max (succ u4) u4 (succ u3)} C _inst_1 CategoryTheory.Cat.{u3, u4} CategoryTheory.Cat.category.{u3, u4}} (X : CategoryTheory.Grothendieck.{u1, u2, u3, u4} C _inst_1 F), CategoryTheory.Grothendieck.Hom.{u1, u2, u3, u4} C _inst_1 F X X
-but is expected to have type
- forall {C : Type.{u1}} [_inst_1 : CategoryTheory.Category.{u2, u1} C] {F : CategoryTheory.Functor.{u2, max u3 u4, u1, max (succ u3) (succ u4)} C _inst_1 CategoryTheory.Cat.{u4, u3} CategoryTheory.Cat.category.{u4, u3}} (X : CategoryTheory.Grothendieck.{u1, u2, u3, u4} C _inst_1 F), CategoryTheory.Grothendieck.Hom.{u1, u2, u3, u4} C _inst_1 F X X
-Case conversion may be inaccurate. Consider using '#align category_theory.grothendieck.id CategoryTheory.Grothendieck.idₓ'. -/
/-- The identity morphism in the Grothendieck category.
-/
@[simps]
@@ -122,12 +101,6 @@ def id (X : Grothendieck F) : Hom X X where
instance (X : Grothendieck F) : Inhabited (Hom X X) :=
⟨id X⟩
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-Case conversion may be inaccurate. Consider using '#align category_theory.grothendieck.comp CategoryTheory.Grothendieck.compₓ'. -/
/-- Composition of morphisms in the Grothendieck category.
-/
@[simps]
@@ -162,18 +135,12 @@ instance : Category (Grothendieck F)
simp
rfl
-/- warning: category_theory.grothendieck.id_fiber' -> CategoryTheory.Grothendieck.id_fiber' is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align category_theory.grothendieck.id_fiber' CategoryTheory.Grothendieck.id_fiber'ₓ'. -/
@[simp]
theorem id_fiber' (X : Grothendieck F) :
Hom.fiber (𝟙 X) = eqToHom (by erw [CategoryTheory.Functor.map_id, functor.id_obj X.fiber]) :=
id_fiber X
#align category_theory.grothendieck.id_fiber' CategoryTheory.Grothendieck.id_fiber'
-/- warning: category_theory.grothendieck.congr -> CategoryTheory.Grothendieck.congr is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align category_theory.grothendieck.congr CategoryTheory.Grothendieck.congrₓ'. -/
theorem congr {X Y : Grothendieck F} {f g : X ⟶ Y} (h : f = g) :
f.fiber = eqToHom (by subst h) ≫ g.fiber := by subst h; dsimp; simp
#align category_theory.grothendieck.congr CategoryTheory.Grothendieck.congr
@@ -182,12 +149,6 @@ section
variable (F)
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-Case conversion may be inaccurate. Consider using '#align category_theory.grothendieck.forget CategoryTheory.Grothendieck.forgetₓ'. -/
/-- The forgetful functor from `grothendieck F` to the source category. -/
@[simps]
def forget : Grothendieck F ⥤ C where
@@ -201,12 +162,6 @@ universe w
variable (G : C ⥤ Type w)
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/-- Auxiliary definition for `grothendieck_Type_to_Cat`, to speed up elaboration. -/
@[simps]
def grothendieckTypeToCatFunctor : Grothendieck (G ⋙ typeToCat) ⥤ G.Elements
@@ -215,12 +170,6 @@ def grothendieckTypeToCatFunctor : Grothendieck (G ⋙ typeToCat) ⥤ G.Elements
map X Y f := ⟨f.1, f.2.1.1⟩
#align category_theory.grothendieck.grothendieck_Type_to_Cat_functor CategoryTheory.Grothendieck.grothendieckTypeToCatFunctor
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-Case conversion may be inaccurate. Consider using '#align category_theory.grothendieck.grothendieck_Type_to_Cat_inverse CategoryTheory.Grothendieck.grothendieckTypeToCatInverseₓ'. -/
/-- Auxiliary definition for `grothendieck_Type_to_Cat`, to speed up elaboration. -/
@[simps]
def grothendieckTypeToCatInverse : G.Elements ⥤ Grothendieck (G ⋙ typeToCat)
@@ -229,12 +178,6 @@ def grothendieckTypeToCatInverse : G.Elements ⥤ Grothendieck (G ⋙ typeToCat)
map X Y f := ⟨f.1, ⟨⟨f.2⟩⟩⟩
#align category_theory.grothendieck.grothendieck_Type_to_Cat_inverse CategoryTheory.Grothendieck.grothendieckTypeToCatInverse
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-Case conversion may be inaccurate. Consider using '#align category_theory.grothendieck.grothendieck_Type_to_Cat CategoryTheory.Grothendieck.grothendieckTypeToCatₓ'. -/
/-- The Grothendieck construction applied to a functor to `Type`
(thought of as a functor to `Cat` by realising a type as a discrete category)
is the same as the 'category of elements' construction.
mathlib commit https://github.com/leanprover-community/mathlib/commit/917c3c072e487b3cccdbfeff17e75b40e45f66cb
@@ -175,11 +175,7 @@ theorem id_fiber' (X : Grothendieck F) :
<too large>
Case conversion may be inaccurate. Consider using '#align category_theory.grothendieck.congr CategoryTheory.Grothendieck.congrₓ'. -/
theorem congr {X Y : Grothendieck F} {f g : X ⟶ Y} (h : f = g) :
- f.fiber = eqToHom (by subst h) ≫ g.fiber :=
- by
- subst h
- dsimp
- simp
+ f.fiber = eqToHom (by subst h) ≫ g.fiber := by subst h; dsimp; simp
#align category_theory.grothendieck.congr CategoryTheory.Grothendieck.congr
section
@@ -249,32 +245,12 @@ def grothendieckTypeToCat : Grothendieck (G ⋙ typeToCat) ≌ G.Elements
Functor := grothendieckTypeToCatFunctor G
inverse := grothendieckTypeToCatInverse G
unitIso :=
- NatIso.ofComponents
- (fun X => by
- rcases X with ⟨_, ⟨⟩⟩
- exact iso.refl _)
- (by
- rintro ⟨_, ⟨⟩⟩ ⟨_, ⟨⟩⟩ ⟨base, ⟨⟨f⟩⟩⟩
- dsimp at *
- subst f
- ext
- simp)
+ NatIso.ofComponents (fun X => by rcases X with ⟨_, ⟨⟩⟩; exact iso.refl _)
+ (by rintro ⟨_, ⟨⟩⟩ ⟨_, ⟨⟩⟩ ⟨base, ⟨⟨f⟩⟩⟩; dsimp at *; subst f; ext; simp)
counitIso :=
- NatIso.ofComponents
- (fun X => by
- cases X
- exact iso.refl _)
- (by
- rintro ⟨⟩ ⟨⟩ ⟨f, e⟩
- dsimp at *
- subst e
- ext
- simp)
- functor_unitIso_comp' := by
- rintro ⟨_, ⟨⟩⟩
- dsimp
- simp
- rfl
+ NatIso.ofComponents (fun X => by cases X; exact iso.refl _)
+ (by rintro ⟨⟩ ⟨⟩ ⟨f, e⟩; dsimp at *; subst e; ext; simp)
+ functor_unitIso_comp' := by rintro ⟨_, ⟨⟩⟩; dsimp; simp; rfl
#align category_theory.grothendieck.grothendieck_Type_to_Cat CategoryTheory.Grothendieck.grothendieckTypeToCat
end Grothendieck
mathlib commit https://github.com/leanprover-community/mathlib/commit/917c3c072e487b3cccdbfeff17e75b40e45f66cb
@@ -89,10 +89,7 @@ structure Hom (X Y : Grothendieck F) where
#align category_theory.grothendieck.hom CategoryTheory.Grothendieck.Hom
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Case conversion may be inaccurate. Consider using '#align category_theory.grothendieck.ext CategoryTheory.Grothendieck.extₓ'. -/
@[ext]
theorem ext {X Y : Grothendieck F} (f g : Hom X Y) (w_base : f.base = g.base)
@@ -166,10 +163,7 @@ instance : Category (Grothendieck F)
rfl
/- warning: category_theory.grothendieck.id_fiber' -> CategoryTheory.Grothendieck.id_fiber' is a dubious translation:
-lean 3 declaration is
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Case conversion may be inaccurate. Consider using '#align category_theory.grothendieck.id_fiber' CategoryTheory.Grothendieck.id_fiber'ₓ'. -/
@[simp]
theorem id_fiber' (X : Grothendieck F) :
@@ -178,10 +172,7 @@ theorem id_fiber' (X : Grothendieck F) :
#align category_theory.grothendieck.id_fiber' CategoryTheory.Grothendieck.id_fiber'
/- warning: category_theory.grothendieck.congr -> CategoryTheory.Grothendieck.congr is a dubious translation:
-lean 3 declaration is
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Case conversion may be inaccurate. Consider using '#align category_theory.grothendieck.congr CategoryTheory.Grothendieck.congrₓ'. -/
theorem congr {X Y : Grothendieck F} {f g : X ⟶ Y} (h : f = g) :
f.fiber = eqToHom (by subst h) ≫ g.fiber :=
mathlib commit https://github.com/leanprover-community/mathlib/commit/5ec62c8106221a3f9160e4e4fcc3eed79fe213e9
@@ -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 category_theory.grothendieck
-! leanprover-community/mathlib commit 14b69e9f3c16630440a2cbd46f1ddad0d561dee7
+! leanprover-community/mathlib commit 75be6b616681ab6ca66d798ead117e75cd64f125
! Please do not edit these lines, except to modify the commit id
! if you have ported upstream changes.
-/
@@ -14,6 +14,9 @@ import Mathbin.CategoryTheory.Elements
/-!
# The Grothendieck construction
+> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
+> Any changes to this file require a corresponding PR to mathlib4.
+
Given a functor `F : C ⥤ Cat`, the objects of `grothendieck F`
consist of dependent pairs `(b, f)`, where `b : C` and `f : F.obj c`,
and a morphism `(b, f) ⟶ (b', f')` is a pair `β : b ⟶ b'` in `C`, and
mathlib commit https://github.com/leanprover-community/mathlib/commit/1a4df69ca1a9a0e5e26bfe12e2b92814216016d0
@@ -47,6 +47,12 @@ variable {C D : Type _} [Category C] [Category D]
variable (F : C ⥤ Cat)
+/- warning: category_theory.grothendieck -> CategoryTheory.Grothendieck is a dubious translation:
+lean 3 declaration is
+ forall {C : Type.{u1}} [_inst_1 : CategoryTheory.Category.{u2, u1} C], (CategoryTheory.Functor.{u2, max u3 u4, u1, max (succ u4) u4 (succ u3)} C _inst_1 CategoryTheory.Cat.{u3, u4} CategoryTheory.Cat.category.{u3, u4}) -> Sort.{max (succ u1) (succ u4)}
+but is expected to have type
+ forall {C : Type.{u1}} [_inst_1 : CategoryTheory.Category.{u2, u1} C], (CategoryTheory.Functor.{u2, max u3 u4, u1, max (succ u3) (succ u4)} C _inst_1 CategoryTheory.Cat.{u4, u3} CategoryTheory.Cat.category.{u4, u3}) -> Sort.{max (succ u1) (succ u3)}
+Case conversion may be inaccurate. Consider using '#align category_theory.grothendieck CategoryTheory.Grothendieckₓ'. -/
/--
The Grothendieck construction (often written as `∫ F` in mathematics) for a functor `F : C ⥤ Cat`
gives a category whose
@@ -65,6 +71,12 @@ namespace Grothendieck
variable {F}
+/- warning: category_theory.grothendieck.hom -> CategoryTheory.Grothendieck.Hom is a dubious translation:
+lean 3 declaration is
+ forall {C : Type.{u1}} [_inst_1 : CategoryTheory.Category.{u2, u1} C] {F : CategoryTheory.Functor.{u2, max u3 u4, u1, max (succ u4) u4 (succ u3)} C _inst_1 CategoryTheory.Cat.{u3, u4} CategoryTheory.Cat.category.{u3, u4}}, (CategoryTheory.Grothendieck.{u1, u2, u3, u4} C _inst_1 F) -> (CategoryTheory.Grothendieck.{u1, u2, u3, u4} C _inst_1 F) -> Sort.{max (succ u2) (succ u3)}
+but is expected to have type
+ forall {C : Type.{u1}} [_inst_1 : CategoryTheory.Category.{u2, u1} C] {F : CategoryTheory.Functor.{u2, max u3 u4, u1, max (succ u3) (succ u4)} C _inst_1 CategoryTheory.Cat.{u4, u3} CategoryTheory.Cat.category.{u4, u3}}, (CategoryTheory.Grothendieck.{u1, u2, u3, u4} C _inst_1 F) -> (CategoryTheory.Grothendieck.{u1, u2, u3, u4} C _inst_1 F) -> Sort.{max (succ u2) (succ u4)}
+Case conversion may be inaccurate. Consider using '#align category_theory.grothendieck.hom CategoryTheory.Grothendieck.Homₓ'. -/
/-- A morphism in the Grothendieck category `F : C ⥤ Cat` consists of
`base : X.base ⟶ Y.base` and `f.fiber : (F.map base).obj X.fiber ⟶ Y.fiber`.
-/
@@ -73,6 +85,12 @@ structure Hom (X Y : Grothendieck F) where
fiber : (F.map base).obj X.fiber ⟶ Y.fiber
#align category_theory.grothendieck.hom CategoryTheory.Grothendieck.Hom
+/- warning: category_theory.grothendieck.ext -> CategoryTheory.Grothendieck.ext is a dubious translation:
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+Case conversion may be inaccurate. Consider using '#align category_theory.grothendieck.ext CategoryTheory.Grothendieck.extₓ'. -/
@[ext]
theorem ext {X Y : Grothendieck F} (f g : Hom X Y) (w_base : f.base = g.base)
(w_fiber : eqToHom (by rw [w_base]) ≫ f.fiber = g.fiber) : f = g :=
@@ -87,6 +105,12 @@ theorem ext {X Y : Grothendieck F} (f g : Hom X Y) (w_base : f.base = g.base)
simpa using w_fiber
#align category_theory.grothendieck.ext CategoryTheory.Grothendieck.ext
+/- warning: category_theory.grothendieck.id -> CategoryTheory.Grothendieck.id is a dubious translation:
+lean 3 declaration is
+ forall {C : Type.{u1}} [_inst_1 : CategoryTheory.Category.{u2, u1} C] {F : CategoryTheory.Functor.{u2, max u3 u4, u1, max (succ u4) u4 (succ u3)} C _inst_1 CategoryTheory.Cat.{u3, u4} CategoryTheory.Cat.category.{u3, u4}} (X : CategoryTheory.Grothendieck.{u1, u2, u3, u4} C _inst_1 F), CategoryTheory.Grothendieck.Hom.{u1, u2, u3, u4} C _inst_1 F X X
+but is expected to have type
+ forall {C : Type.{u1}} [_inst_1 : CategoryTheory.Category.{u2, u1} C] {F : CategoryTheory.Functor.{u2, max u3 u4, u1, max (succ u3) (succ u4)} C _inst_1 CategoryTheory.Cat.{u4, u3} CategoryTheory.Cat.category.{u4, u3}} (X : CategoryTheory.Grothendieck.{u1, u2, u3, u4} C _inst_1 F), CategoryTheory.Grothendieck.Hom.{u1, u2, u3, u4} C _inst_1 F X X
+Case conversion may be inaccurate. Consider using '#align category_theory.grothendieck.id CategoryTheory.Grothendieck.idₓ'. -/
/-- The identity morphism in the Grothendieck category.
-/
@[simps]
@@ -98,6 +122,12 @@ def id (X : Grothendieck F) : Hom X X where
instance (X : Grothendieck F) : Inhabited (Hom X X) :=
⟨id X⟩
+/- warning: category_theory.grothendieck.comp -> CategoryTheory.Grothendieck.comp is a dubious translation:
+lean 3 declaration is
+ forall {C : Type.{u1}} [_inst_1 : CategoryTheory.Category.{u2, u1} C] {F : CategoryTheory.Functor.{u2, max u3 u4, u1, max (succ u4) u4 (succ u3)} C _inst_1 CategoryTheory.Cat.{u3, u4} CategoryTheory.Cat.category.{u3, u4}} {X : CategoryTheory.Grothendieck.{u1, u2, u3, u4} C _inst_1 F} {Y : CategoryTheory.Grothendieck.{u1, u2, u3, u4} C _inst_1 F} {Z : CategoryTheory.Grothendieck.{u1, u2, u3, u4} C _inst_1 F}, (CategoryTheory.Grothendieck.Hom.{u1, u2, u3, u4} C _inst_1 F X Y) -> (CategoryTheory.Grothendieck.Hom.{u1, u2, u3, u4} C _inst_1 F Y Z) -> (CategoryTheory.Grothendieck.Hom.{u1, u2, u3, u4} C _inst_1 F X Z)
+but is expected to have type
+ forall {C : Type.{u1}} [_inst_1 : CategoryTheory.Category.{u2, u1} C] {F : CategoryTheory.Functor.{u2, max u3 u4, u1, max (succ u3) (succ u4)} C _inst_1 CategoryTheory.Cat.{u4, u3} CategoryTheory.Cat.category.{u4, u3}} {X : CategoryTheory.Grothendieck.{u1, u2, u3, u4} C _inst_1 F} {Y : CategoryTheory.Grothendieck.{u1, u2, u3, u4} C _inst_1 F} {Z : CategoryTheory.Grothendieck.{u1, u2, u3, u4} C _inst_1 F}, (CategoryTheory.Grothendieck.Hom.{u1, u2, u3, u4} C _inst_1 F X Y) -> (CategoryTheory.Grothendieck.Hom.{u1, u2, u3, u4} C _inst_1 F Y Z) -> (CategoryTheory.Grothendieck.Hom.{u1, u2, u3, u4} C _inst_1 F X Z)
+Case conversion may be inaccurate. Consider using '#align category_theory.grothendieck.comp CategoryTheory.Grothendieck.compₓ'. -/
/-- Composition of morphisms in the Grothendieck category.
-/
@[simps]
@@ -132,12 +162,24 @@ instance : Category (Grothendieck F)
simp
rfl
+/- warning: category_theory.grothendieck.id_fiber' -> CategoryTheory.Grothendieck.id_fiber' is a dubious translation:
+lean 3 declaration is
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+Case conversion may be inaccurate. Consider using '#align category_theory.grothendieck.id_fiber' CategoryTheory.Grothendieck.id_fiber'ₓ'. -/
@[simp]
theorem id_fiber' (X : Grothendieck F) :
Hom.fiber (𝟙 X) = eqToHom (by erw [CategoryTheory.Functor.map_id, functor.id_obj X.fiber]) :=
id_fiber X
#align category_theory.grothendieck.id_fiber' CategoryTheory.Grothendieck.id_fiber'
+/- warning: category_theory.grothendieck.congr -> CategoryTheory.Grothendieck.congr is a dubious translation:
+lean 3 declaration is
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(CategoryTheory.Category.toCategoryStruct.{max u2 u1, max (succ u2) (succ u1)} CategoryTheory.Cat.{u1, u2} CategoryTheory.Cat.category.{u1, u2})) (CategoryTheory.Functor.toPrefunctor.{u3, max u2 u1, u4, max (succ u2) (succ u1)} C _inst_1 CategoryTheory.Cat.{u1, u2} CategoryTheory.Cat.category.{u1, u2} F) (CategoryTheory.Grothendieck.base.{u4, u3, u2, u1} C _inst_1 F Y))) (CategoryTheory.Cat.str.{u1, u2} (Prefunctor.obj.{succ u3, max (succ u2) (succ u1), u4, max (succ u2) (succ u1)} C (CategoryTheory.CategoryStruct.toQuiver.{u3, u4} C (CategoryTheory.Category.toCategoryStruct.{u3, u4} C _inst_1)) CategoryTheory.Cat.{u1, u2} (CategoryTheory.CategoryStruct.toQuiver.{max u2 u1, max (succ u2) (succ u1)} CategoryTheory.Cat.{u1, u2} (CategoryTheory.Category.toCategoryStruct.{max u2 u1, max (succ u2) (succ u1)} CategoryTheory.Cat.{u1, u2} CategoryTheory.Cat.category.{u1, u2})) (CategoryTheory.Functor.toPrefunctor.{u3, max u2 u1, u4, max (succ u2) (succ u1)} C _inst_1 CategoryTheory.Cat.{u1, u2} CategoryTheory.Cat.category.{u1, u2} F) (CategoryTheory.Grothendieck.base.{u4, u3, u2, u1} C _inst_1 F Y))))) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u2, u2} (CategoryTheory.Bundled.α.{u2, max u2 (succ u1)} CategoryTheory.Category.{u1, u2} (Prefunctor.obj.{succ u3, max (succ u2) (succ u1), u4, max (succ u2) (succ u1)} C (CategoryTheory.CategoryStruct.toQuiver.{u3, u4} C (CategoryTheory.Category.toCategoryStruct.{u3, u4} C _inst_1)) CategoryTheory.Cat.{u1, u2} (CategoryTheory.CategoryStruct.toQuiver.{max u2 u1, max (succ u2) (succ u1)} CategoryTheory.Cat.{u1, u2} (CategoryTheory.Category.toCategoryStruct.{max u2 u1, max (succ u2) (succ u1)} CategoryTheory.Cat.{u1, u2} CategoryTheory.Cat.category.{u1, u2})) (CategoryTheory.Functor.toPrefunctor.{u3, max u2 u1, u4, max (succ u2) (succ u1)} C _inst_1 CategoryTheory.Cat.{u1, u2} CategoryTheory.Cat.category.{u1, u2} F) (CategoryTheory.Grothendieck.base.{u4, u3, u2, u1} C _inst_1 F X))) (CategoryTheory.Cat.str.{u1, u2} (Prefunctor.obj.{succ u3, max (succ u2) (succ u1), u4, max (succ u2) (succ u1)} C (CategoryTheory.CategoryStruct.toQuiver.{u3, u4} C (CategoryTheory.Category.toCategoryStruct.{u3, u4} C _inst_1)) CategoryTheory.Cat.{u1, u2} (CategoryTheory.CategoryStruct.toQuiver.{max u2 u1, max (succ u2) (succ u1)} CategoryTheory.Cat.{u1, u2} (CategoryTheory.Category.toCategoryStruct.{max u2 u1, max (succ u2) (succ u1)} CategoryTheory.Cat.{u1, u2} CategoryTheory.Cat.category.{u1, u2})) (CategoryTheory.Functor.toPrefunctor.{u3, max u2 u1, u4, max (succ u2) (succ u1)} C _inst_1 CategoryTheory.Cat.{u1, u2} CategoryTheory.Cat.category.{u1, u2} F) (CategoryTheory.Grothendieck.base.{u4, u3, u2, u1} C _inst_1 F X))) (CategoryTheory.Bundled.α.{u2, max u2 (succ u1)} CategoryTheory.Category.{u1, u2} (Prefunctor.obj.{succ u3, max (succ u2) (succ u1), u4, max (succ u2) (succ u1)} C (CategoryTheory.CategoryStruct.toQuiver.{u3, u4} C (CategoryTheory.Category.toCategoryStruct.{u3, u4} C _inst_1)) CategoryTheory.Cat.{u1, u2} (CategoryTheory.CategoryStruct.toQuiver.{max u2 u1, max (succ u2) (succ u1)} CategoryTheory.Cat.{u1, u2} (CategoryTheory.Category.toCategoryStruct.{max u2 u1, max (succ u2) (succ u1)} CategoryTheory.Cat.{u1, u2} CategoryTheory.Cat.category.{u1, u2})) (CategoryTheory.Functor.toPrefunctor.{u3, max u2 u1, u4, max (succ u2) (succ u1)} C _inst_1 CategoryTheory.Cat.{u1, u2} CategoryTheory.Cat.category.{u1, u2} F) (CategoryTheory.Grothendieck.base.{u4, u3, u2, u1} C _inst_1 F Y))) (CategoryTheory.Cat.str.{u1, u2} (Prefunctor.obj.{succ u3, max (succ u2) (succ u1), u4, max (succ u2) (succ u1)} C (CategoryTheory.CategoryStruct.toQuiver.{u3, u4} C (CategoryTheory.Category.toCategoryStruct.{u3, u4} C _inst_1)) CategoryTheory.Cat.{u1, u2} (CategoryTheory.CategoryStruct.toQuiver.{max u2 u1, max (succ u2) (succ u1)} CategoryTheory.Cat.{u1, u2} (CategoryTheory.Category.toCategoryStruct.{max u2 u1, max (succ u2) (succ u1)} CategoryTheory.Cat.{u1, u2} CategoryTheory.Cat.category.{u1, u2})) (CategoryTheory.Functor.toPrefunctor.{u3, max u2 u1, u4, max (succ u2) (succ u1)} C _inst_1 CategoryTheory.Cat.{u1, u2} CategoryTheory.Cat.category.{u1, u2} F) (CategoryTheory.Grothendieck.base.{u4, u3, u2, u1} C _inst_1 F Y))) (Prefunctor.map.{succ u3, max (succ u2) (succ u1), u4, max (succ u2) (succ u1)} C (CategoryTheory.CategoryStruct.toQuiver.{u3, u4} C (CategoryTheory.Category.toCategoryStruct.{u3, u4} C _inst_1)) CategoryTheory.Cat.{u1, u2} (CategoryTheory.CategoryStruct.toQuiver.{max u2 u1, max (succ u2) (succ u1)} CategoryTheory.Cat.{u1, u2} (CategoryTheory.Category.toCategoryStruct.{max u2 u1, max (succ u2) (succ u1)} CategoryTheory.Cat.{u1, u2} CategoryTheory.Cat.category.{u1, u2})) (CategoryTheory.Functor.toPrefunctor.{u3, max u2 u1, u4, max (succ u2) (succ u1)} C _inst_1 CategoryTheory.Cat.{u1, u2} CategoryTheory.Cat.category.{u1, u2} F) (CategoryTheory.Grothendieck.base.{u4, u3, u2, u1} C _inst_1 F X) (CategoryTheory.Grothendieck.base.{u4, u3, u2, u1} C _inst_1 F Y) (CategoryTheory.Grothendieck.Hom.base.{u4, u3, u2, u1} C _inst_1 F X Y f))) (CategoryTheory.Grothendieck.fiber.{u4, u3, u2, u1} C _inst_1 F X))) g h)) (CategoryTheory.Grothendieck.Hom.fiber.{u4, u3, u2, u1} C _inst_1 F X Y g))
+Case conversion may be inaccurate. Consider using '#align category_theory.grothendieck.congr CategoryTheory.Grothendieck.congrₓ'. -/
theorem congr {X Y : Grothendieck F} {f g : X ⟶ Y} (h : f = g) :
f.fiber = eqToHom (by subst h) ≫ g.fiber :=
by
@@ -150,6 +192,12 @@ section
variable (F)
+/- warning: category_theory.grothendieck.forget -> CategoryTheory.Grothendieck.forget is a dubious translation:
+lean 3 declaration is
+ forall {C : Type.{u1}} [_inst_1 : CategoryTheory.Category.{u2, u1} C] (F : CategoryTheory.Functor.{u2, max u3 u4, u1, max (succ u4) u4 (succ u3)} C _inst_1 CategoryTheory.Cat.{u3, u4} CategoryTheory.Cat.category.{u3, u4}), CategoryTheory.Functor.{max u2 u3, u2, max u1 u4, u1} (CategoryTheory.Grothendieck.{u1, u2, u3, u4} C _inst_1 F) (CategoryTheory.Grothendieck.CategoryTheory.category.{u1, u2, u3, u4} C _inst_1 F) C _inst_1
+but is expected to have type
+ forall {C : Type.{u1}} [_inst_1 : CategoryTheory.Category.{u2, u1} C] (F : CategoryTheory.Functor.{u2, max u3 u4, u1, max (succ u3) (succ u4)} C _inst_1 CategoryTheory.Cat.{u4, u3} CategoryTheory.Cat.category.{u4, u3}), CategoryTheory.Functor.{max u2 u4, u2, max u3 u1, u1} (CategoryTheory.Grothendieck.{u1, u2, u3, u4} C _inst_1 F) (CategoryTheory.Grothendieck.instCategoryGrothendieck.{u1, u2, u3, u4} C _inst_1 F) C _inst_1
+Case conversion may be inaccurate. Consider using '#align category_theory.grothendieck.forget CategoryTheory.Grothendieck.forgetₓ'. -/
/-- The forgetful functor from `grothendieck F` to the source category. -/
@[simps]
def forget : Grothendieck F ⥤ C where
@@ -163,6 +211,12 @@ universe w
variable (G : C ⥤ Type w)
+/- warning: category_theory.grothendieck.grothendieck_Type_to_Cat_functor -> CategoryTheory.Grothendieck.grothendieckTypeToCatFunctor is a dubious translation:
+lean 3 declaration is
+ forall {C : Type.{u1}} [_inst_1 : CategoryTheory.Category.{u2, u1} C] (G : CategoryTheory.Functor.{u2, u3, u1, succ u3} C _inst_1 Type.{u3} CategoryTheory.types.{u3}), CategoryTheory.Functor.{max u2 u3, u2, max u1 u3, max u1 u3} (CategoryTheory.Grothendieck.{u1, u2, u3, u3} C _inst_1 (CategoryTheory.Functor.comp.{u2, u3, u3, u1, succ u3, succ u3} C _inst_1 Type.{u3} CategoryTheory.types.{u3} CategoryTheory.Cat.{u3, u3} CategoryTheory.Cat.category.{u3, u3} G CategoryTheory.typeToCat.{u3})) (CategoryTheory.Grothendieck.CategoryTheory.category.{u1, u2, u3, u3} C _inst_1 (CategoryTheory.Functor.comp.{u2, u3, u3, u1, succ u3, succ u3} C _inst_1 Type.{u3} CategoryTheory.types.{u3} CategoryTheory.Cat.{u3, u3} CategoryTheory.Cat.category.{u3, u3} G CategoryTheory.typeToCat.{u3})) (CategoryTheory.Functor.Elements.{u3, u2, u1} C _inst_1 G) (CategoryTheory.categoryOfElements.{u3, u2, u1} C _inst_1 G)
+but is expected to have type
+ forall {C : Type.{u2}} [_inst_1 : CategoryTheory.Category.{u3, u2} C] (G : CategoryTheory.Functor.{u3, u1, u2, succ u1} C _inst_1 Type.{u1} CategoryTheory.types.{u1}), CategoryTheory.Functor.{max u1 u3, u3, max u1 u2, max u1 u2} (CategoryTheory.Grothendieck.{u2, u3, u1, u1} C _inst_1 (CategoryTheory.Functor.comp.{u3, u1, u1, u2, succ u1, succ u1} C _inst_1 Type.{u1} CategoryTheory.types.{u1} CategoryTheory.Cat.{u1, u1} CategoryTheory.Cat.category.{u1, u1} G CategoryTheory.typeToCat.{u1})) (CategoryTheory.Grothendieck.instCategoryGrothendieck.{u2, u3, u1, u1} C _inst_1 (CategoryTheory.Functor.comp.{u3, u1, u1, u2, succ u1, succ u1} C _inst_1 Type.{u1} CategoryTheory.types.{u1} CategoryTheory.Cat.{u1, u1} CategoryTheory.Cat.category.{u1, u1} G CategoryTheory.typeToCat.{u1})) (CategoryTheory.Functor.Elements.{u1, u3, u2} C _inst_1 G) (CategoryTheory.categoryOfElements.{u1, u3, u2} C _inst_1 G)
+Case conversion may be inaccurate. Consider using '#align category_theory.grothendieck.grothendieck_Type_to_Cat_functor CategoryTheory.Grothendieck.grothendieckTypeToCatFunctorₓ'. -/
/-- Auxiliary definition for `grothendieck_Type_to_Cat`, to speed up elaboration. -/
@[simps]
def grothendieckTypeToCatFunctor : Grothendieck (G ⋙ typeToCat) ⥤ G.Elements
@@ -171,6 +225,12 @@ def grothendieckTypeToCatFunctor : Grothendieck (G ⋙ typeToCat) ⥤ G.Elements
map X Y f := ⟨f.1, f.2.1.1⟩
#align category_theory.grothendieck.grothendieck_Type_to_Cat_functor CategoryTheory.Grothendieck.grothendieckTypeToCatFunctor
+/- warning: category_theory.grothendieck.grothendieck_Type_to_Cat_inverse -> CategoryTheory.Grothendieck.grothendieckTypeToCatInverse is a dubious translation:
+lean 3 declaration is
+ forall {C : Type.{u1}} [_inst_1 : CategoryTheory.Category.{u2, u1} C] (G : CategoryTheory.Functor.{u2, u3, u1, succ u3} C _inst_1 Type.{u3} CategoryTheory.types.{u3}), CategoryTheory.Functor.{u2, max u2 u3, max u1 u3, max u1 u3} (CategoryTheory.Functor.Elements.{u3, u2, u1} C _inst_1 G) (CategoryTheory.categoryOfElements.{u3, u2, u1} C _inst_1 G) (CategoryTheory.Grothendieck.{u1, u2, u3, u3} C _inst_1 (CategoryTheory.Functor.comp.{u2, u3, u3, u1, succ u3, succ u3} C _inst_1 Type.{u3} CategoryTheory.types.{u3} CategoryTheory.Cat.{u3, u3} CategoryTheory.Cat.category.{u3, u3} G CategoryTheory.typeToCat.{u3})) (CategoryTheory.Grothendieck.CategoryTheory.category.{u1, u2, u3, u3} C _inst_1 (CategoryTheory.Functor.comp.{u2, u3, u3, u1, succ u3, succ u3} C _inst_1 Type.{u3} CategoryTheory.types.{u3} CategoryTheory.Cat.{u3, u3} CategoryTheory.Cat.category.{u3, u3} G CategoryTheory.typeToCat.{u3}))
+but is expected to have type
+ forall {C : Type.{u2}} [_inst_1 : CategoryTheory.Category.{u3, u2} C] (G : CategoryTheory.Functor.{u3, u1, u2, succ u1} C _inst_1 Type.{u1} CategoryTheory.types.{u1}), CategoryTheory.Functor.{u3, max u1 u3, max u1 u2, max u1 u2} (CategoryTheory.Functor.Elements.{u1, u3, u2} C _inst_1 G) (CategoryTheory.categoryOfElements.{u1, u3, u2} C _inst_1 G) (CategoryTheory.Grothendieck.{u2, u3, u1, u1} C _inst_1 (CategoryTheory.Functor.comp.{u3, u1, u1, u2, succ u1, succ u1} C _inst_1 Type.{u1} CategoryTheory.types.{u1} CategoryTheory.Cat.{u1, u1} CategoryTheory.Cat.category.{u1, u1} G CategoryTheory.typeToCat.{u1})) (CategoryTheory.Grothendieck.instCategoryGrothendieck.{u2, u3, u1, u1} C _inst_1 (CategoryTheory.Functor.comp.{u3, u1, u1, u2, succ u1, succ u1} C _inst_1 Type.{u1} CategoryTheory.types.{u1} CategoryTheory.Cat.{u1, u1} CategoryTheory.Cat.category.{u1, u1} G CategoryTheory.typeToCat.{u1}))
+Case conversion may be inaccurate. Consider using '#align category_theory.grothendieck.grothendieck_Type_to_Cat_inverse CategoryTheory.Grothendieck.grothendieckTypeToCatInverseₓ'. -/
/-- Auxiliary definition for `grothendieck_Type_to_Cat`, to speed up elaboration. -/
@[simps]
def grothendieckTypeToCatInverse : G.Elements ⥤ Grothendieck (G ⋙ typeToCat)
@@ -179,6 +239,12 @@ def grothendieckTypeToCatInverse : G.Elements ⥤ Grothendieck (G ⋙ typeToCat)
map X Y f := ⟨f.1, ⟨⟨f.2⟩⟩⟩
#align category_theory.grothendieck.grothendieck_Type_to_Cat_inverse CategoryTheory.Grothendieck.grothendieckTypeToCatInverse
+/- warning: category_theory.grothendieck.grothendieck_Type_to_Cat -> CategoryTheory.Grothendieck.grothendieckTypeToCat is a dubious translation:
+lean 3 declaration is
+ forall {C : Type.{u1}} [_inst_1 : CategoryTheory.Category.{u2, u1} C] (G : CategoryTheory.Functor.{u2, u3, u1, succ u3} C _inst_1 Type.{u3} CategoryTheory.types.{u3}), CategoryTheory.Equivalence.{max u2 u3, u2, max u1 u3, max u1 u3} (CategoryTheory.Grothendieck.{u1, u2, u3, u3} C _inst_1 (CategoryTheory.Functor.comp.{u2, u3, u3, u1, succ u3, succ u3} C _inst_1 Type.{u3} CategoryTheory.types.{u3} CategoryTheory.Cat.{u3, u3} CategoryTheory.Cat.category.{u3, u3} G CategoryTheory.typeToCat.{u3})) (CategoryTheory.Grothendieck.CategoryTheory.category.{u1, u2, u3, u3} C _inst_1 (CategoryTheory.Functor.comp.{u2, u3, u3, u1, succ u3, succ u3} C _inst_1 Type.{u3} CategoryTheory.types.{u3} CategoryTheory.Cat.{u3, u3} CategoryTheory.Cat.category.{u3, u3} G CategoryTheory.typeToCat.{u3})) (CategoryTheory.Functor.Elements.{u3, u2, u1} C _inst_1 G) (CategoryTheory.categoryOfElements.{u3, u2, u1} C _inst_1 G)
+but is expected to have type
+ forall {C : Type.{u2}} [_inst_1 : CategoryTheory.Category.{u3, u2} C] (G : CategoryTheory.Functor.{u3, u1, u2, succ u1} C _inst_1 Type.{u1} CategoryTheory.types.{u1}), CategoryTheory.Equivalence.{max u1 u3, u3, max u1 u2, max u1 u2} (CategoryTheory.Grothendieck.{u2, u3, u1, u1} C _inst_1 (CategoryTheory.Functor.comp.{u3, u1, u1, u2, succ u1, succ u1} C _inst_1 Type.{u1} CategoryTheory.types.{u1} CategoryTheory.Cat.{u1, u1} CategoryTheory.Cat.category.{u1, u1} G CategoryTheory.typeToCat.{u1})) (CategoryTheory.Functor.Elements.{u1, u3, u2} C _inst_1 G) (CategoryTheory.Grothendieck.instCategoryGrothendieck.{u2, u3, u1, u1} C _inst_1 (CategoryTheory.Functor.comp.{u3, u1, u1, u2, succ u1, succ u1} C _inst_1 Type.{u1} CategoryTheory.types.{u1} CategoryTheory.Cat.{u1, u1} CategoryTheory.Cat.category.{u1, u1} G CategoryTheory.typeToCat.{u1})) (CategoryTheory.categoryOfElements.{u1, u3, u2} C _inst_1 G)
+Case conversion may be inaccurate. Consider using '#align category_theory.grothendieck.grothendieck_Type_to_Cat CategoryTheory.Grothendieck.grothendieckTypeToCatₓ'. -/
/-- The Grothendieck construction applied to a functor to `Type`
(thought of as a functor to `Cat` by realising a type as a discrete category)
is the same as the 'category of elements' construction.
mathlib commit https://github.com/leanprover-community/mathlib/commit/bd9851ca476957ea4549eb19b40e7b5ade9428cc
@@ -51,7 +51,7 @@ gives a category whose
`base : X.base ⟶ Y.base` and
`f.fiber : (F.map base).obj X.fiber ⟶ Y.fiber`
-/
--- Porting note: no such linter yet
+-- Porting note(#5171): no such linter yet
-- @[nolint has_nonempty_instance]
structure Grothendieck where
/-- The underlying object in `C` -/
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)
@@ -41,7 +41,6 @@ universe u
namespace CategoryTheory
variable {C D : Type*} [Category C] [Category D]
-
variable (F : C ⥤ Cat)
/--
Homogenises porting notes via capitalisation and addition of whitespace.
It makes the following changes:
@@ -52,7 +52,7 @@ gives a category whose
`base : X.base ⟶ Y.base` and
`f.fiber : (F.map base).obj X.fiber ⟶ Y.fiber`
-/
--- porting note: no such linter yet
+-- Porting note: no such linter yet
-- @[nolint has_nonempty_instance]
structure Grothendieck where
/-- The underlying object in `C` -/
@@ -196,7 +196,7 @@ def grothendieckTypeToCat : Grothendieck (G ⋙ typeToCat) ≌ G.Elements where
rintro ⟨_, ⟨⟩⟩ ⟨_, ⟨⟩⟩ ⟨base, ⟨⟨f⟩⟩⟩
dsimp at *
simp
- rfl )
+ rfl)
counitIso :=
NatIso.ofComponents
(fun X => by
Type _
and Sort _
(#6499)
We remove all possible occurences of Type _
and Sort _
in favor of Type*
and Sort*
.
This has nice performance benefits.
@@ -40,7 +40,7 @@ universe u
namespace CategoryTheory
-variable {C D : Type _} [Category C] [Category D]
+variable {C D : Type*} [Category C] [Category D]
variable (F : C ⥤ Cat)
@@ -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 category_theory.grothendieck
-! leanprover-community/mathlib commit 14b69e9f3c16630440a2cbd46f1ddad0d561dee7
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
-/
import Mathlib.CategoryTheory.Category.Cat
import Mathlib.CategoryTheory.Elements
+#align_import category_theory.grothendieck from "leanprover-community/mathlib"@"14b69e9f3c16630440a2cbd46f1ddad0d561dee7"
+
/-!
# The Grothendieck construction
Co-authored-by: Komyyy <pol_tta@outlook.jp> Co-authored-by: Scott Morrison <scott.morrison@gmail.com> Co-authored-by: Scott Morrison <scott.morrison@anu.edu.au> Co-authored-by: Ruben Van de Velde <65514131+Ruben-VandeVelde@users.noreply.github.com> Co-authored-by: Mario Carneiro <di.gama@gmail.com>
@@ -114,14 +114,14 @@ instance : Category (Grothendieck F) where
id X := Grothendieck.id X
comp := @fun X Y Z f g => Grothendieck.comp f g
comp_id := @fun X Y f => by
- dsimp; ext; swap
+ dsimp; ext
· simp
· dsimp
rw [← NatIso.naturality_2 (eqToIso (F.map_id Y.base)) f.fiber]
simp
id_comp := @fun X Y f => by dsimp; ext <;> simp
assoc := @fun W X Y Z f g h => by
- dsimp; ext; swap
+ dsimp; ext
· simp
· dsimp
rw [← NatIso.naturality_2 (eqToIso (F.map_comp _ _)) f.fiber]
All dependencies are ported!