category_theory.adjunction.lifting
⟷
Mathlib.CategoryTheory.Adjunction.Lifting
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
@@ -3,10 +3,10 @@ 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.Limits.Shapes.Equalizers
-import Mathbin.CategoryTheory.Limits.Shapes.Reflexive
-import Mathbin.CategoryTheory.Monad.Adjunction
-import Mathbin.CategoryTheory.Monad.Coequalizer
+import CategoryTheory.Limits.Shapes.Equalizers
+import CategoryTheory.Limits.Shapes.Reflexive
+import CategoryTheory.Monad.Adjunction
+import CategoryTheory.Monad.Coequalizer
#align_import category_theory.adjunction.lifting from "leanprover-community/mathlib"@"1a51edf13debfcbe223fa06b1cb353b9ed9751cc"
mathlib commit https://github.com/leanprover-community/mathlib/commit/8ea5598db6caeddde6cb734aa179cc2408dbd345
@@ -2,17 +2,14 @@
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.adjunction.lifting
-! leanprover-community/mathlib commit 1a51edf13debfcbe223fa06b1cb353b9ed9751cc
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
-/
import Mathbin.CategoryTheory.Limits.Shapes.Equalizers
import Mathbin.CategoryTheory.Limits.Shapes.Reflexive
import Mathbin.CategoryTheory.Monad.Adjunction
import Mathbin.CategoryTheory.Monad.Coequalizer
+#align_import category_theory.adjunction.lifting from "leanprover-community/mathlib"@"1a51edf13debfcbe223fa06b1cb353b9ed9751cc"
+
/-!
# Adjoint lifting
mathlib commit https://github.com/leanprover-community/mathlib/commit/1a51edf13debfcbe223fa06b1cb353b9ed9751cc
@@ -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.adjunction.lifting
-! leanprover-community/mathlib commit 9bc7dfa6e50f902fb0684c9670a680459ebaed68
+! leanprover-community/mathlib commit 1a51edf13debfcbe223fa06b1cb353b9ed9751cc
! Please do not edit these lines, except to modify the commit id
! if you have ported upstream changes.
-/
@@ -16,6 +16,9 @@ import Mathbin.CategoryTheory.Monad.Coequalizer
/-!
# Adjoint lifting
+> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
+> Any changes to this file require a corresponding PR to mathlib4.
+
This file gives two constructions for building left adjoints: the adjoint triangle theorem and the
adjoint lifting theorem.
The adjoint triangle theorem says that given a functor `U : B ⥤ C` with a left adjoint `F` such
mathlib commit https://github.com/leanprover-community/mathlib/commit/8905e5ed90859939681a725b00f6063e65096d95
@@ -73,6 +73,7 @@ variable {U : B ⥤ C} {F : C ⥤ B} (R : A ⥤ B) (F' : C ⥤ A)
variable (adj₁ : F ⊣ U) (adj₂ : F' ⊣ R ⋙ U)
+#print CategoryTheory.LiftAdjoint.counitCoequalises /-
/-- To show that `ε_X` is a coequalizer for `(FUε_X, ε_FUX)`, it suffices to assume it's always a
coequalizer of something (i.e. a regular epi).
-/
@@ -91,7 +92,9 @@ def counitCoequalises [∀ X : B, RegularEpi (adj₁.counit.app X)] (X : B) :
rw [← cancel_epi (adj₁.counit.app X)]
apply hm.trans (regular_epi.desc' (adj₁.counit.app X) s.π _).2.symm
#align category_theory.lift_adjoint.counit_coequalises CategoryTheory.LiftAdjoint.counitCoequalises
+-/
+#print CategoryTheory.LiftAdjoint.otherMap /-
/-- (Implementation)
To construct the left adjoint, we use the coequalizer of `F' U ε_Y` with the composite
@@ -104,6 +107,7 @@ We will show that this coequalizer exists and that it forms the object map for a
def otherMap (X) : F'.obj (U.obj (F.obj (U.obj X))) ⟶ F'.obj (U.obj X) :=
F'.map (U.map (F.map (adj₂.Unit.app _) ≫ adj₁.counit.app _)) ≫ adj₂.counit.app _
#align category_theory.lift_adjoint.other_map CategoryTheory.LiftAdjoint.otherMap
+-/
/--
`(F'Uε_X, other_map X)` is a reflexive pair: in particular if `A` has reflexive coequalizers then
@@ -120,13 +124,16 @@ instance (X : B) :
variable [HasReflexiveCoequalizers A]
+#print CategoryTheory.LiftAdjoint.constructLeftAdjointObj /-
/-- Construct the object part of the desired left adjoint as the coequalizer of `F'Uε_Y` with
`other_map`.
-/
noncomputable def constructLeftAdjointObj (Y : B) : A :=
coequalizer (F'.map (U.map (adj₁.counit.app Y))) (otherMap _ _ adj₁ adj₂ Y)
#align category_theory.lift_adjoint.construct_left_adjoint_obj CategoryTheory.LiftAdjoint.constructLeftAdjointObj
+-/
+#print CategoryTheory.LiftAdjoint.constructLeftAdjointEquiv /-
/-- The homset equivalence which helps show that `R` is a right adjoint. -/
@[simps (config := { rhsMd := semireducible })]
noncomputable def constructLeftAdjointEquiv [∀ X : B, RegularEpi (adj₁.counit.app X)] (Y : A)
@@ -158,7 +165,9 @@ noncomputable def constructLeftAdjointEquiv [∀ X : B, RegularEpi (adj₁.couni
apply eq_comm
_ ≃ (X ⟶ R.obj Y) := (Cofork.IsColimit.homIso (counitCoequalises adj₁ X) _).symm
#align category_theory.lift_adjoint.construct_left_adjoint_equiv CategoryTheory.LiftAdjoint.constructLeftAdjointEquiv
+-/
+#print CategoryTheory.LiftAdjoint.constructLeftAdjoint /-
/-- Construct the left adjoint to `R`, with object map `construct_left_adjoint_obj`. -/
noncomputable def constructLeftAdjoint [∀ X : B, RegularEpi (adj₁.counit.app X)] : B ⥤ A :=
by
@@ -173,9 +182,11 @@ noncomputable def constructLeftAdjoint [∀ X : B, RegularEpi (adj₁.counit.app
adj₁.hom_equiv_naturality_right_symm, cofork.is_colimit.hom_iso_natural,
adj₂.hom_equiv_naturality_right, functor.comp_map]
#align category_theory.lift_adjoint.construct_left_adjoint CategoryTheory.LiftAdjoint.constructLeftAdjoint
+-/
end LiftAdjoint
+#print CategoryTheory.adjointTriangleLift /-
/-- The adjoint triangle theorem: Suppose `U : B ⥤ C` has a left adjoint `F` such that each counit
`ε_X : FUX ⟶ X` is a regular epimorphism. Then if a category `A` has coequalizers of reflexive
pairs, then a functor `R : A ⥤ B` has a left adjoint if the composite `R ⋙ U` does.
@@ -190,7 +201,9 @@ noncomputable def adjointTriangleLift {U : B ⥤ C} {F : C ⥤ B} (R : A ⥤ B)
left := LiftAdjoint.constructLeftAdjoint R _ adj₁ (Adjunction.ofRightAdjoint _)
adj := Adjunction.adjunctionOfEquivLeft _ _
#align category_theory.adjoint_triangle_lift CategoryTheory.adjointTriangleLift
+-/
+#print CategoryTheory.monadicAdjointTriangleLift /-
/-- If `R ⋙ U` has a left adjoint, the domain of `R` has reflexive coequalizers and `U` is a monadic
functor, then `R` has a left adjoint.
This is a special case of `adjoint_triangle_lift` which is often more useful in practice.
@@ -216,11 +229,13 @@ noncomputable def monadicAdjointTriangleLift (U : B ⥤ C) [MonadicRightAdjoint
exact ⟨_, _, _, _, monad.beck_algebra_coequalizer X⟩
exact adjoint_triangle_lift R' (monad.adj _)
#align category_theory.monadic_adjoint_triangle_lift CategoryTheory.monadicAdjointTriangleLift
+-/
variable {D : Type u₄}
variable [Category.{v₄} D]
+#print CategoryTheory.adjointSquareLift /-
/-- Suppose we have a commutative square of functors
Q
@@ -243,7 +258,9 @@ noncomputable def adjointSquareLift (Q : A ⥤ B) (V : B ⥤ D) (U : A ⥤ C) (R
let this.1 := adjunction.right_adjoint_of_nat_iso comm
exact adjoint_triangle_lift Q (adjunction.of_right_adjoint V)
#align category_theory.adjoint_square_lift CategoryTheory.adjointSquareLift
+-/
+#print CategoryTheory.monadicAdjointSquareLift /-
/-- Suppose we have a commutative square of functors
Q
@@ -264,6 +281,7 @@ noncomputable def monadicAdjointSquareLift (Q : A ⥤ B) (V : B ⥤ D) (U : A
let this.1 := adjunction.right_adjoint_of_nat_iso comm
exact monadic_adjoint_triangle_lift V
#align category_theory.monadic_adjoint_square_lift CategoryTheory.monadicAdjointSquareLift
+-/
end CategoryTheory
mathlib commit https://github.com/leanprover-community/mathlib/commit/9fb8964792b4237dac6200193a0d533f1b3f7423
@@ -92,8 +92,6 @@ def counitCoequalises [∀ X : B, RegularEpi (adj₁.counit.app X)] (X : B) :
apply hm.trans (regular_epi.desc' (adj₁.counit.app X) s.π _).2.symm
#align category_theory.lift_adjoint.counit_coequalises CategoryTheory.LiftAdjoint.counitCoequalises
-include adj₁ adj₂
-
/-- (Implementation)
To construct the left adjoint, we use the coequalizer of `F' U ε_Y` with the composite
mathlib commit https://github.com/leanprover-community/mathlib/commit/7e5137f579de09a059a5ce98f364a04e221aabf0
@@ -159,7 +159,6 @@ noncomputable def constructLeftAdjointEquiv [∀ X : B, RegularEpi (adj₁.couni
assoc, adj₁.counit_naturality, adj₁.counit_naturality_assoc]
apply eq_comm
_ ≃ (X ⟶ R.obj Y) := (Cofork.IsColimit.homIso (counitCoequalises adj₁ X) _).symm
-
#align category_theory.lift_adjoint.construct_left_adjoint_equiv CategoryTheory.LiftAdjoint.constructLeftAdjointEquiv
/-- Construct the left adjoint to `R`, with object map `construct_left_adjoint_obj`. -/
@@ -203,15 +202,16 @@ noncomputable def monadicAdjointTriangleLift (U : B ⥤ C) [MonadicRightAdjoint
by
let R' : A ⥤ _ := R ⋙ monad.comparison (adjunction.of_right_adjoint U)
rsuffices : is_right_adjoint R'
- · let this : is_right_adjoint (R' ⋙ (monad.comparison (adjunction.of_right_adjoint U)).inv) := by
- infer_instance
- · let this : R' ⋙ (monad.comparison (adjunction.of_right_adjoint U)).inv ≅ R :=
+ · let this.1 : is_right_adjoint (R' ⋙ (monad.comparison (adjunction.of_right_adjoint U)).inv) :=
+ by infer_instance
+ · let this.1 : R' ⋙ (monad.comparison (adjunction.of_right_adjoint U)).inv ≅ R :=
(iso_whisker_left R (monad.comparison _).asEquivalence.unitIso.symm : _) ≪≫ R.right_unitor
exact adjunction.right_adjoint_of_nat_iso this
- let this : is_right_adjoint (R' ⋙ monad.forget (adjunction.of_right_adjoint U).toMonad) :=
+ let this.1 : is_right_adjoint (R' ⋙ monad.forget (adjunction.of_right_adjoint U).toMonad) :=
adjunction.right_adjoint_of_nat_iso
(iso_whisker_left R (monad.comparison_forget (adjunction.of_right_adjoint U)).symm : _)
- let this : ∀ X, regular_epi ((monad.adj (adjunction.of_right_adjoint U).toMonad).counit.app X) :=
+ let this.1 :
+ ∀ X, regular_epi ((monad.adj (adjunction.of_right_adjoint U).toMonad).counit.app X) :=
by
intro X
simp only [monad.adj_counit]
@@ -242,7 +242,7 @@ noncomputable def adjointSquareLift (Q : A ⥤ B) (V : B ⥤ D) (U : A ⥤ C) (R
[∀ X, RegularEpi ((Adjunction.ofRightAdjoint V).counit.app X)] [HasReflexiveCoequalizers A] :
IsRightAdjoint Q :=
by
- let this := adjunction.right_adjoint_of_nat_iso comm
+ let this.1 := adjunction.right_adjoint_of_nat_iso comm
exact adjoint_triangle_lift Q (adjunction.of_right_adjoint V)
#align category_theory.adjoint_square_lift CategoryTheory.adjointSquareLift
@@ -263,7 +263,7 @@ noncomputable def monadicAdjointSquareLift (Q : A ⥤ B) (V : B ⥤ D) (U : A
(comm : U ⋙ R ≅ Q ⋙ V) [IsRightAdjoint U] [MonadicRightAdjoint V] [IsRightAdjoint R]
[HasReflexiveCoequalizers A] : IsRightAdjoint Q :=
by
- let this := adjunction.right_adjoint_of_nat_iso comm
+ let this.1 := adjunction.right_adjoint_of_nat_iso comm
exact monadic_adjoint_triangle_lift V
#align category_theory.monadic_adjoint_square_lift CategoryTheory.monadicAdjointSquareLift
mathlib commit https://github.com/leanprover-community/mathlib/commit/917c3c072e487b3cccdbfeff17e75b40e45f66cb
@@ -114,9 +114,7 @@ it has a coequalizer.
instance (X : B) :
IsReflexivePair (F'.map (U.map (adj₁.counit.app X))) (otherMap _ _ adj₁ adj₂ X) :=
IsReflexivePair.mk' (F'.map (adj₁.Unit.app (U.obj X)))
- (by
- rw [← F'.map_comp, adj₁.right_triangle_components]
- apply F'.map_id)
+ (by rw [← F'.map_comp, adj₁.right_triangle_components]; apply F'.map_id)
(by
dsimp [other_map]
rw [← F'.map_comp_assoc, U.map_comp, adj₁.unit_naturality_assoc,
mathlib commit https://github.com/leanprover-community/mathlib/commit/bd9851ca476957ea4549eb19b40e7b5ade9428cc
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)
@@ -60,14 +60,12 @@ open Category Limits
universe v₁ v₂ v₃ v₄ u₁ u₂ u₃ u₄
variable {A : Type u₁} {B : Type u₂} {C : Type u₃}
-
variable [Category.{v₁} A] [Category.{v₂} B] [Category.{v₃} C]
-- Hide implementation details in this namespace
namespace LiftAdjoint
variable {U : B ⥤ C} {F : C ⥤ B} (R : A ⥤ B) (F' : C ⥤ A)
-
variable (adj₁ : F ⊣ U) (adj₂ : F' ⊣ R ⋙ U)
/-- To show that `ε_X` is a coequalizer for `(FUε_X, ε_FUX)`, it suffices to assume it's always a
@@ -208,7 +206,6 @@ noncomputable def monadicAdjointTriangleLift (U : B ⥤ C) [MonadicRightAdjoint
#align category_theory.monadic_adjoint_triangle_lift CategoryTheory.monadicAdjointTriangleLift
variable {D : Type u₄}
-
variable [Category.{v₄} D]
/-- Suppose we have a commutative square of functors
@@ -160,10 +160,12 @@ noncomputable def constructLeftAdjoint [∀ X : B, RegularEpi (adj₁.counit.app
rw [constructLeftAdjointEquiv_apply, constructLeftAdjointEquiv_apply,
Equiv.symm_apply_eq, Subtype.ext_iff]
dsimp
- rw [Cofork.IsColimit.homIso_natural, Cofork.IsColimit.homIso_natural]
+ -- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
+ erw [Cofork.IsColimit.homIso_natural, Cofork.IsColimit.homIso_natural]
erw [adj₂.homEquiv_naturality_right]
simp_rw [Functor.comp_map]
- simp
+ -- This used to be `simp`, but we need `aesop_cat` after leanprover/lean4#2644
+ aesop_cat
#align category_theory.lift_adjoint.construct_left_adjoint CategoryTheory.LiftAdjoint.constructLeftAdjoint
end LiftAdjoint
@@ -160,12 +160,10 @@ noncomputable def constructLeftAdjoint [∀ X : B, RegularEpi (adj₁.counit.app
rw [constructLeftAdjointEquiv_apply, constructLeftAdjointEquiv_apply,
Equiv.symm_apply_eq, Subtype.ext_iff]
dsimp
- -- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
- erw [Cofork.IsColimit.homIso_natural, Cofork.IsColimit.homIso_natural]
+ rw [Cofork.IsColimit.homIso_natural, Cofork.IsColimit.homIso_natural]
erw [adj₂.homEquiv_naturality_right]
simp_rw [Functor.comp_map]
- -- This used to be `simp`, but we need `aesop_cat` after leanprover/lean4#2644
- aesop_cat
+ simp
#align category_theory.lift_adjoint.construct_left_adjoint CategoryTheory.LiftAdjoint.constructLeftAdjoint
end LiftAdjoint
@@ -160,10 +160,12 @@ noncomputable def constructLeftAdjoint [∀ X : B, RegularEpi (adj₁.counit.app
rw [constructLeftAdjointEquiv_apply, constructLeftAdjointEquiv_apply,
Equiv.symm_apply_eq, Subtype.ext_iff]
dsimp
- rw [Cofork.IsColimit.homIso_natural, Cofork.IsColimit.homIso_natural]
+ -- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
+ erw [Cofork.IsColimit.homIso_natural, Cofork.IsColimit.homIso_natural]
erw [adj₂.homEquiv_naturality_right]
simp_rw [Functor.comp_map]
- simp
+ -- This used to be `simp`, but we need `aesop_cat` after leanprover/lean4#2644
+ aesop_cat
#align category_theory.lift_adjoint.construct_left_adjoint CategoryTheory.LiftAdjoint.constructLeftAdjoint
end LiftAdjoint
@@ -2,17 +2,14 @@
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.adjunction.lifting
-! leanprover-community/mathlib commit 9bc7dfa6e50f902fb0684c9670a680459ebaed68
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
-/
import Mathlib.CategoryTheory.Limits.Shapes.Equalizers
import Mathlib.CategoryTheory.Limits.Shapes.Reflexive
import Mathlib.CategoryTheory.Monad.Adjunction
import Mathlib.CategoryTheory.Monad.Coequalizer
+#align_import category_theory.adjunction.lifting from "leanprover-community/mathlib"@"9bc7dfa6e50f902fb0684c9670a680459ebaed68"
+
/-!
# Adjoint lifting
The unported dependencies are