category_theory.closed.functor
⟷
Mathlib.CategoryTheory.Closed.Functor
The following section lists changes to this file in mathlib3 and mathlib4 that occured after the initial port. Most recent changes are shown first. Hovering over a commit will show all commits associated with the same mathlib3 commit.
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mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83
@@ -73,8 +73,8 @@ def frobeniusMorphism (h : L ⊣ F) (A : C) :
Frobenius morphism is an isomorphism.
-/
instance frobeniusMorphism_iso_of_preserves_binary_products (h : L ⊣ F) (A : C)
- [PreservesLimitsOfShape (Discrete WalkingPair) L] [Full F] [Faithful F] :
- IsIso (frobeniusMorphism F h A) :=
+ [PreservesLimitsOfShape (Discrete WalkingPair) L] [CategoryTheory.Functor.Full F]
+ [CategoryTheory.Functor.Faithful F] : IsIso (frobeniusMorphism F h A) :=
by
apply nat_iso.is_iso_of_is_iso_app _
intro B
@@ -206,7 +206,8 @@ cartesian closed.
TODO: Show the converse, that if `F` is cartesian closed and its left adjoint preserves binary
products, then it is full and faithful.
-/
-def cartesianClosedFunctorOfLeftAdjointPreservesBinaryProducts (h : L ⊣ F) [Full F] [Faithful F]
+def cartesianClosedFunctorOfLeftAdjointPreservesBinaryProducts (h : L ⊣ F)
+ [CategoryTheory.Functor.Full F] [CategoryTheory.Functor.Faithful F]
[PreservesLimitsOfShape (Discrete WalkingPair) L] : CartesianClosedFunctor F
where comparison_iso A := expComparison_iso_of_frobeniusMorphism_iso F h _
#align category_theory.cartesian_closed_functor_of_left_adjoint_preserves_binary_products CategoryTheory.cartesianClosedFunctorOfLeftAdjointPreservesBinaryProducts
mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83
@@ -183,7 +183,7 @@ at `A` is an isomorphism.
theorem frobeniusMorphism_iso_of_expComparison_iso (h : L ⊣ F) (A : C)
[i : IsIso (expComparison F A)] : IsIso (frobeniusMorphism F h A) :=
by
- rw [← frobenius_morphism_mate F h] at i
+ rw [← frobenius_morphism_mate F h] at i
exact @transfer_nat_trans_self_of_iso _ _ _ _ _ i
#align category_theory.frobenius_morphism_iso_of_exp_comparison_iso CategoryTheory.frobeniusMorphism_iso_of_expComparison_iso
-/
mathlib commit https://github.com/leanprover-community/mathlib/commit/ce64cd319bb6b3e82f31c2d38e79080d377be451
@@ -3,9 +3,9 @@ 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.Closed.Cartesian
-import Mathbin.CategoryTheory.Limits.Preserves.Shapes.BinaryProducts
-import Mathbin.CategoryTheory.Adjunction.FullyFaithful
+import CategoryTheory.Closed.Cartesian
+import CategoryTheory.Limits.Preserves.Shapes.BinaryProducts
+import CategoryTheory.Adjunction.FullyFaithful
#align_import category_theory.closed.functor from "leanprover-community/mathlib"@"cff8231f04dfa33fd8f2f45792eebd862ef30cad"
mathlib commit https://github.com/leanprover-community/mathlib/commit/8ea5598db6caeddde6cb734aa179cc2408dbd345
@@ -2,16 +2,13 @@
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.closed.functor
-! leanprover-community/mathlib commit cff8231f04dfa33fd8f2f45792eebd862ef30cad
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
-/
import Mathbin.CategoryTheory.Closed.Cartesian
import Mathbin.CategoryTheory.Limits.Preserves.Shapes.BinaryProducts
import Mathbin.CategoryTheory.Adjunction.FullyFaithful
+#align_import category_theory.closed.functor from "leanprover-community/mathlib"@"cff8231f04dfa33fd8f2f45792eebd862ef30cad"
+
/-!
# Cartesian closed functors
mathlib commit https://github.com/leanprover-community/mathlib/commit/2a0ce625dbb0ffbc7d1316597de0b25c1ec75303
@@ -54,6 +54,7 @@ variable (F : C ⥤ D) {L : D ⥤ C}
noncomputable section
+#print CategoryTheory.frobeniusMorphism /-
/-- The Frobenius morphism for an adjunction `L ⊣ F` at `A` is given by the morphism
L(FA ⨯ B) ⟶ LFA ⨯ LB ⟶ A ⨯ LB
@@ -68,7 +69,9 @@ def frobeniusMorphism (h : L ⊣ F) (A : C) :
prod.functor.obj (F.obj A) ⋙ L ⟶ L ⋙ prod.functor.obj A :=
prodComparisonNatTrans L (F.obj A) ≫ whiskerLeft _ (prod.functor.map (h.counit.app _))
#align category_theory.frobenius_morphism CategoryTheory.frobeniusMorphism
+-/
+#print CategoryTheory.frobeniusMorphism_iso_of_preserves_binary_products /-
/-- If `F` is full and faithful and has a left adjoint `L` which preserves binary products, then the
Frobenius morphism is an isomorphism.
-/
@@ -81,18 +84,22 @@ instance frobeniusMorphism_iso_of_preserves_binary_products (h : L ⊣ F) (A : C
dsimp [frobenius_morphism]
infer_instance
#align category_theory.frobenius_morphism_iso_of_preserves_binary_products CategoryTheory.frobeniusMorphism_iso_of_preserves_binary_products
+-/
variable [CartesianClosed C] [CartesianClosed D]
variable [PreservesLimitsOfShape (Discrete WalkingPair) F]
+#print CategoryTheory.expComparison /-
/-- The exponential comparison map.
`F` is a cartesian closed functor if this is an iso for all `A`.
-/
def expComparison (A : C) : exp A ⋙ F ⟶ F ⋙ exp (F.obj A) :=
transferNatTrans (exp.adjunction A) (exp.adjunction (F.obj A)) (prodComparisonNatIso F A).inv
#align category_theory.exp_comparison CategoryTheory.expComparison
+-/
+#print CategoryTheory.expComparison_ev /-
theorem expComparison_ev (A B : C) :
Limits.prod.map (𝟙 (F.obj A)) ((expComparison F A).app B) ≫ (exp.ev (F.obj A)).app (F.obj B) =
inv (prodComparison F _ _) ≫ F.map ((exp.ev _).app _) :=
@@ -101,7 +108,9 @@ theorem expComparison_ev (A B : C) :
ext
simp
#align category_theory.exp_comparison_ev CategoryTheory.expComparison_ev
+-/
+#print CategoryTheory.coev_expComparison /-
theorem coev_expComparison (A B : C) :
F.map ((exp.coev A).app B) ≫ (expComparison F A).app (A ⨯ B) =
(exp.coev _).app (F.obj B) ≫ (exp (F.obj A)).map (inv (prodComparison F A B)) :=
@@ -111,13 +120,17 @@ theorem coev_expComparison (A B : C) :
dsimp
simp
#align category_theory.coev_exp_comparison CategoryTheory.coev_expComparison
+-/
+#print CategoryTheory.uncurry_expComparison /-
theorem uncurry_expComparison (A B : C) :
CartesianClosed.uncurry ((expComparison F A).app B) =
inv (prodComparison F _ _) ≫ F.map ((exp.ev _).app _) :=
by rw [uncurry_eq, exp_comparison_ev]
#align category_theory.uncurry_exp_comparison CategoryTheory.uncurry_expComparison
+-/
+#print CategoryTheory.expComparison_whiskerLeft /-
/-- The exponential comparison map is natural in `A`. -/
theorem expComparison_whiskerLeft {A A' : C} (f : A' ⟶ A) :
expComparison F A ≫ whiskerLeft _ (pre (F.map f)) =
@@ -131,16 +144,20 @@ theorem expComparison_whiskerLeft {A A' : C} (f : A' ⟶ A) :
prod_comparison_inv_natural_assoc, ← prod_comparison_inv_natural_assoc, ← F.map_comp, ←
F.map_comp, prod_map_pre_app_comp_ev]
#align category_theory.exp_comparison_whisker_left CategoryTheory.expComparison_whiskerLeft
+-/
+#print CategoryTheory.CartesianClosedFunctor /-
/-- The functor `F` is cartesian closed (ie preserves exponentials) if each natural transformation
`exp_comparison F A` is an isomorphism
-/
class CartesianClosedFunctor where
comparison_iso : ∀ A, IsIso (expComparison F A)
#align category_theory.cartesian_closed_functor CategoryTheory.CartesianClosedFunctor
+-/
attribute [instance] cartesian_closed_functor.comparison_iso
+#print CategoryTheory.frobeniusMorphism_mate /-
theorem frobeniusMorphism_mate (h : L ⊣ F) (A : C) :
transferNatTransSelf (h.comp (exp.adjunction A)) ((exp.adjunction (F.obj A)).comp h)
(frobeniusMorphism F h A) =
@@ -159,7 +176,9 @@ theorem frobeniusMorphism_mate (h : L ⊣ F) (A : C) :
· rw [assoc, assoc, ← h.counit_naturality, ← L.map_comp_assoc, assoc, inv_prod_comparison_map_snd]
simp
#align category_theory.frobenius_morphism_mate CategoryTheory.frobeniusMorphism_mate
+-/
+#print CategoryTheory.frobeniusMorphism_iso_of_expComparison_iso /-
/--
If the exponential comparison transformation (at `A`) is an isomorphism, then the Frobenius morphism
at `A` is an isomorphism.
@@ -170,7 +189,9 @@ theorem frobeniusMorphism_iso_of_expComparison_iso (h : L ⊣ F) (A : C)
rw [← frobenius_morphism_mate F h] at i
exact @transfer_nat_trans_self_of_iso _ _ _ _ _ i
#align category_theory.frobenius_morphism_iso_of_exp_comparison_iso CategoryTheory.frobeniusMorphism_iso_of_expComparison_iso
+-/
+#print CategoryTheory.expComparison_iso_of_frobeniusMorphism_iso /-
/--
If the Frobenius morphism at `A` is an isomorphism, then the exponential comparison transformation
(at `A`) is an isomorphism.
@@ -179,7 +200,9 @@ theorem expComparison_iso_of_frobeniusMorphism_iso (h : L ⊣ F) (A : C)
[i : IsIso (frobeniusMorphism F h A)] : IsIso (expComparison F A) := by
rw [← frobenius_morphism_mate F h]; infer_instance
#align category_theory.exp_comparison_iso_of_frobenius_morphism_iso CategoryTheory.expComparison_iso_of_frobeniusMorphism_iso
+-/
+#print CategoryTheory.cartesianClosedFunctorOfLeftAdjointPreservesBinaryProducts /-
/-- If `F` is full and faithful, and has a left adjoint which preserves binary products, then it is
cartesian closed.
@@ -190,6 +213,7 @@ def cartesianClosedFunctorOfLeftAdjointPreservesBinaryProducts (h : L ⊣ F) [Fu
[PreservesLimitsOfShape (Discrete WalkingPair) L] : CartesianClosedFunctor F
where comparison_iso A := expComparison_iso_of_frobeniusMorphism_iso F h _
#align category_theory.cartesian_closed_functor_of_left_adjoint_preserves_binary_products CategoryTheory.cartesianClosedFunctorOfLeftAdjointPreservesBinaryProducts
+-/
end CategoryTheory
mathlib commit https://github.com/leanprover-community/mathlib/commit/893964fc28cefbcffc7cb784ed00a2895b4e65cf
@@ -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.closed.functor
-! leanprover-community/mathlib commit cea27692b3fdeb328a2ddba6aabf181754543184
+! leanprover-community/mathlib commit cff8231f04dfa33fd8f2f45792eebd862ef30cad
! 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.Adjunction.FullyFaithful
/-!
# Cartesian closed functors
+> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
+> Any changes to this file require a corresponding PR to mathlib4.
+
Define the exponential comparison morphisms for a functor which preserves binary products, and use
them to define a cartesian closed functor: one which (naturally) preserves exponentials.
mathlib commit https://github.com/leanprover-community/mathlib/commit/cca40788df1b8755d5baf17ab2f27dacc2e17acb
@@ -164,7 +164,7 @@ at `A` is an isomorphism.
theorem frobeniusMorphism_iso_of_expComparison_iso (h : L ⊣ F) (A : C)
[i : IsIso (expComparison F A)] : IsIso (frobeniusMorphism F h A) :=
by
- rw [← frobenius_morphism_mate F h] at i
+ rw [← frobenius_morphism_mate F h] at i
exact @transfer_nat_trans_self_of_iso _ _ _ _ _ i
#align category_theory.frobenius_morphism_iso_of_exp_comparison_iso CategoryTheory.frobeniusMorphism_iso_of_expComparison_iso
mathlib commit https://github.com/leanprover-community/mathlib/commit/917c3c072e487b3cccdbfeff17e75b40e45f66cb
@@ -173,10 +173,8 @@ If the Frobenius morphism at `A` is an isomorphism, then the exponential compari
(at `A`) is an isomorphism.
-/
theorem expComparison_iso_of_frobeniusMorphism_iso (h : L ⊣ F) (A : C)
- [i : IsIso (frobeniusMorphism F h A)] : IsIso (expComparison F A) :=
- by
- rw [← frobenius_morphism_mate F h]
- infer_instance
+ [i : IsIso (frobeniusMorphism F h A)] : IsIso (expComparison F A) := by
+ rw [← frobenius_morphism_mate F h]; infer_instance
#align category_theory.exp_comparison_iso_of_frobenius_morphism_iso CategoryTheory.expComparison_iso_of_frobeniusMorphism_iso
/-- If `F` is full and faithful, and has a left adjoint which preserves binary products, then it is
mathlib commit https://github.com/leanprover-community/mathlib/commit/bd9851ca476957ea4549eb19b40e7b5ade9428cc
These notions on functors are now Functor.Full
, Functor.Faithful
, Functor.EssSurj
, Functor.IsEquivalence
, Functor.ReflectsIsomorphisms
. Deprecated aliases are introduced for the previous names.
@@ -64,7 +64,7 @@ def frobeniusMorphism (h : L ⊣ F) (A : C) :
Frobenius morphism is an isomorphism.
-/
instance frobeniusMorphism_iso_of_preserves_binary_products (h : L ⊣ F) (A : C)
- [PreservesLimitsOfShape (Discrete WalkingPair) L] [Full F] [Faithful F] :
+ [PreservesLimitsOfShape (Discrete WalkingPair) L] [F.Full] [F.Faithful] :
IsIso (frobeniusMorphism F h A) :=
suffices ∀ (X : D), IsIso ((frobeniusMorphism F h A).app X) from NatIso.isIso_of_isIso_app _
fun B ↦ by dsimp [frobeniusMorphism]; infer_instance
@@ -174,7 +174,7 @@ cartesian closed.
TODO: Show the converse, that if `F` is cartesian closed and its left adjoint preserves binary
products, then it is full and faithful.
-/
-theorem cartesianClosedFunctorOfLeftAdjointPreservesBinaryProducts (h : L ⊣ F) [Full F] [Faithful F]
+theorem cartesianClosedFunctorOfLeftAdjointPreservesBinaryProducts (h : L ⊣ F) [F.Full] [F.Faithful]
[PreservesLimitsOfShape (Discrete WalkingPair) L] : CartesianClosedFunctor F where
comparison_iso _ := expComparison_iso_of_frobeniusMorphism_iso F h _
#align category_theory.cartesian_closed_functor_of_left_adjoint_preserves_binary_products CategoryTheory.cartesianClosedFunctorOfLeftAdjointPreservesBinaryProducts
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)
@@ -41,11 +41,8 @@ open Category Limits CartesianClosed
universe v u u'
variable {C : Type u} [Category.{v} C]
-
variable {D : Type u'} [Category.{v} D]
-
variable [HasFiniteProducts C] [HasFiniteProducts D]
-
variable (F : C ⥤ D) {L : D ⥤ C}
/-- The Frobenius morphism for an adjunction `L ⊣ F` at `A` is given by the morphism
@@ -74,7 +71,6 @@ instance frobeniusMorphism_iso_of_preserves_binary_products (h : L ⊣ F) (A : C
#align category_theory.frobenius_morphism_iso_of_preserves_binary_products CategoryTheory.frobeniusMorphism_iso_of_preserves_binary_products
variable [CartesianClosed C] [CartesianClosed D]
-
variable [PreservesLimitsOfShape (Discrete WalkingPair) F]
/-- The exponential comparison map.
@@ -123,7 +123,7 @@ theorem expComparison_whiskerLeft {A A' : C} (f : A' ⟶ A) :
/-- The functor `F` is cartesian closed (ie preserves exponentials) if each natural transformation
`exp_comparison F A` is an isomorphism
-/
-class CartesianClosedFunctor where
+class CartesianClosedFunctor : Prop where
comparison_iso : ∀ A, IsIso (expComparison F A)
#align category_theory.cartesian_closed_functor CategoryTheory.CartesianClosedFunctor
@@ -178,7 +178,7 @@ cartesian closed.
TODO: Show the converse, that if `F` is cartesian closed and its left adjoint preserves binary
products, then it is full and faithful.
-/
-def cartesianClosedFunctorOfLeftAdjointPreservesBinaryProducts (h : L ⊣ F) [Full F] [Faithful F]
+theorem cartesianClosedFunctorOfLeftAdjointPreservesBinaryProducts (h : L ⊣ F) [Full F] [Faithful F]
[PreservesLimitsOfShape (Discrete WalkingPair) L] : CartesianClosedFunctor F where
comparison_iso _ := expComparison_iso_of_frobeniusMorphism_iso F h _
#align category_theory.cartesian_closed_functor_of_left_adjoint_preserves_binary_products CategoryTheory.cartesianClosedFunctorOfLeftAdjointPreservesBinaryProducts
@@ -2,16 +2,13 @@
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.closed.functor
-! leanprover-community/mathlib commit cea27692b3fdeb328a2ddba6aabf181754543184
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
-/
import Mathlib.CategoryTheory.Closed.Cartesian
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.BinaryProducts
import Mathlib.CategoryTheory.Adjunction.FullyFaithful
+#align_import category_theory.closed.functor from "leanprover-community/mathlib"@"cea27692b3fdeb328a2ddba6aabf181754543184"
+
/-!
# Cartesian closed functors
@@ -149,7 +149,7 @@ theorem frobeniusMorphism_mate (h : L ⊣ F) (A : C) :
-- https://github.com/leanprover-community/mathlib4/issues/5164
erw [exp.ev_coev]
rw [F.map_id (A ⨯ L.obj B), comp_id]
- apply prod.hom_ext
+ ext
· rw [assoc, assoc, ← h.counit_naturality, ← L.map_comp_assoc, assoc, inv_prodComparison_map_fst]
simp
· rw [assoc, assoc, ← h.counit_naturality, ← L.map_comp_assoc, assoc, inv_prodComparison_map_snd]
@@ -142,9 +142,11 @@ theorem frobeniusMorphism_mate (h : L ⊣ F) (A : C) :
simp only [id_comp, comp_id]
rw [← L.map_comp_assoc, prod.map_id_comp, assoc]
-- Porting note: need to use `erw` here.
+ -- https://github.com/leanprover-community/mathlib4/issues/5164
erw [expComparison_ev]
rw [prod.map_id_comp, assoc, ← F.map_id, ← prodComparison_inv_natural_assoc, ← F.map_comp]
-- Porting note: need to use `erw` here.
+ -- https://github.com/leanprover-community/mathlib4/issues/5164
erw [exp.ev_coev]
rw [F.map_id (A ⨯ L.obj B), comp_id]
apply prod.hom_ext
The unported dependencies are