category_theory.monoidal.End
⟷
Mathlib.CategoryTheory.Monoidal.End
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
@@ -157,7 +157,7 @@ theorem μ_naturality₂ {m n m' n' : M} (f : m ⟶ m') (g : n ⟶ n') (X : C) :
(F.μ m n).app X ≫ (F.map (f ⊗ g)).app X :=
by
have := congr_app (F.to_lax_monoidal_functor.μ_natural f g) X
- dsimp at this
+ dsimp at this
simpa using this
#align category_theory.μ_naturality₂ CategoryTheory.μ_naturality₂
-/
@@ -216,7 +216,7 @@ theorem left_unitality_app (n : M) (X : C) :
(F.obj n).map (F.ε.app X) ≫ (F.μ (𝟙_ M) n).app X ≫ (F.map (λ_ n).Hom).app X = 𝟙 _ :=
by
have := congr_app (F.to_lax_monoidal_functor.left_unitality n) X
- dsimp at this
+ dsimp at this
simpa using this.symm
#align category_theory.left_unitality_app CategoryTheory.left_unitality_app
-/
@@ -250,7 +250,7 @@ theorem right_unitality_app (n : M) (X : C) :
F.ε.app ((F.obj n).obj X) ≫ (F.μ n (𝟙_ M)).app X ≫ (F.map (ρ_ n).Hom).app X = 𝟙 _ :=
by
have := congr_app (F.to_lax_monoidal_functor.right_unitality n) X
- dsimp at this
+ dsimp at this
simpa using this.symm
#align category_theory.right_unitality_app CategoryTheory.right_unitality_app
-/
@@ -288,7 +288,7 @@ theorem associativity_app (m₁ m₂ m₃ : M) (X : C) :
(F.μ m₂ m₃).app ((F.obj m₁).obj X) ≫ (F.μ m₁ (m₂ ⊗ m₃)).app X :=
by
have := congr_app (F.to_lax_monoidal_functor.associativity m₁ m₂ m₃) X
- dsimp at this
+ dsimp at this
simpa using this
#align category_theory.associativity_app CategoryTheory.associativity_app
-/
mathlib commit https://github.com/leanprover-community/mathlib/commit/ce64cd319bb6b3e82f31c2d38e79080d377be451
@@ -3,7 +3,7 @@ Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Andrew Yang
-/
-import Mathbin.CategoryTheory.Monoidal.Functor
+import CategoryTheory.Monoidal.Functor
#align_import category_theory.monoidal.End from "leanprover-community/mathlib"@"69c6a5a12d8a2b159f20933e60115a4f2de62b58"
mathlib commit https://github.com/leanprover-community/mathlib/commit/8ea5598db6caeddde6cb734aa179cc2408dbd345
@@ -2,14 +2,11 @@
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Andrew Yang
-
-! This file was ported from Lean 3 source module category_theory.monoidal.End
-! leanprover-community/mathlib commit 69c6a5a12d8a2b159f20933e60115a4f2de62b58
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
-/
import Mathbin.CategoryTheory.Monoidal.Functor
+#align_import category_theory.monoidal.End from "leanprover-community/mathlib"@"69c6a5a12d8a2b159f20933e60115a4f2de62b58"
+
/-!
# Endofunctors as a monoidal category.
mathlib commit https://github.com/leanprover-community/mathlib/commit/9fb8964792b4237dac6200193a0d533f1b3f7423
@@ -90,43 +90,58 @@ variable {C}
variable {M : Type _} [Category M] [MonoidalCategory M] (F : MonoidalFunctor M (C ⥤ C))
+#print CategoryTheory.μ_hom_inv_app /-
@[simp, reassoc]
theorem μ_hom_inv_app (i j : M) (X : C) : (F.μ i j).app X ≫ (F.μIso i j).inv.app X = 𝟙 _ :=
(F.μIso i j).hom_inv_id_app X
#align category_theory.μ_hom_inv_app CategoryTheory.μ_hom_inv_app
+-/
+#print CategoryTheory.μ_inv_hom_app /-
@[simp, reassoc]
theorem μ_inv_hom_app (i j : M) (X : C) : (F.μIso i j).inv.app X ≫ (F.μ i j).app X = 𝟙 _ :=
(F.μIso i j).inv_hom_id_app X
#align category_theory.μ_inv_hom_app CategoryTheory.μ_inv_hom_app
+-/
+#print CategoryTheory.ε_hom_inv_app /-
@[simp, reassoc]
theorem ε_hom_inv_app (X : C) : F.ε.app X ≫ F.εIso.inv.app X = 𝟙 _ :=
F.εIso.hom_inv_id_app X
#align category_theory.ε_hom_inv_app CategoryTheory.ε_hom_inv_app
+-/
+#print CategoryTheory.ε_inv_hom_app /-
@[simp, reassoc]
theorem ε_inv_hom_app (X : C) : F.εIso.inv.app X ≫ F.ε.app X = 𝟙 _ :=
F.εIso.inv_hom_id_app X
#align category_theory.ε_inv_hom_app CategoryTheory.ε_inv_hom_app
+-/
+#print CategoryTheory.ε_naturality /-
@[simp, reassoc]
theorem ε_naturality {X Y : C} (f : X ⟶ Y) : F.ε.app X ≫ (F.obj (𝟙_ M)).map f = f ≫ F.ε.app Y :=
(F.ε.naturality f).symm
#align category_theory.ε_naturality CategoryTheory.ε_naturality
+-/
+#print CategoryTheory.ε_inv_naturality /-
@[simp, reassoc]
theorem ε_inv_naturality {X Y : C} (f : X ⟶ Y) :
(F.obj (𝟙_ M)).map f ≫ F.εIso.inv.app Y = F.εIso.inv.app X ≫ f :=
F.εIso.inv.naturality f
#align category_theory.ε_inv_naturality CategoryTheory.ε_inv_naturality
+-/
+#print CategoryTheory.μ_naturality /-
@[simp, reassoc]
theorem μ_naturality {m n : M} {X Y : C} (f : X ⟶ Y) :
(F.obj n).map ((F.obj m).map f) ≫ (F.μ m n).app Y = (F.μ m n).app X ≫ (F.obj _).map f :=
(F.toLaxMonoidalFunctor.μ m n).naturality f
#align category_theory.μ_naturality CategoryTheory.μ_naturality
+-/
+#print CategoryTheory.μ_inv_naturality /-
-- This is a simp lemma in the reverse direction via `nat_trans.naturality`.
@[reassoc]
theorem μ_inv_naturality {m n : M} {X Y : C} (f : X ⟶ Y) :
@@ -134,8 +149,10 @@ theorem μ_inv_naturality {m n : M} {X Y : C} (f : X ⟶ Y) :
(F.obj _).map f ≫ (F.μIso m n).inv.app Y :=
((F.μIso m n).inv.naturality f).symm
#align category_theory.μ_inv_naturality CategoryTheory.μ_inv_naturality
+-/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
+#print CategoryTheory.μ_naturality₂ /-
-- This is not a simp lemma since it could be proved by the lemmas later.
@[reassoc]
theorem μ_naturality₂ {m n m' n' : M} (f : m ⟶ m') (g : n ⟶ n') (X : C) :
@@ -146,8 +163,10 @@ theorem μ_naturality₂ {m n m' n' : M} (f : m ⟶ m') (g : n ⟶ n') (X : C) :
dsimp at this
simpa using this
#align category_theory.μ_naturality₂ CategoryTheory.μ_naturality₂
+-/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
+#print CategoryTheory.μ_naturalityₗ /-
@[simp, reassoc]
theorem μ_naturalityₗ {m n m' : M} (f : m ⟶ m') (X : C) :
(F.obj n).map ((F.map f).app X) ≫ (F.μ m' n).app X =
@@ -156,8 +175,10 @@ theorem μ_naturalityₗ {m n m' : M} (f : m ⟶ m') (X : C) :
rw [← μ_naturality₂ F f (𝟙 n) X]
simp
#align category_theory.μ_naturalityₗ CategoryTheory.μ_naturalityₗ
+-/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
+#print CategoryTheory.μ_naturalityᵣ /-
@[simp, reassoc]
theorem μ_naturalityᵣ {m n n' : M} (g : n ⟶ n') (X : C) :
(F.map g).app ((F.obj m).obj X) ≫ (F.μ m n').app X =
@@ -166,8 +187,10 @@ theorem μ_naturalityᵣ {m n n' : M} (g : n ⟶ n') (X : C) :
rw [← μ_naturality₂ F (𝟙 m) g X]
simp
#align category_theory.μ_naturalityᵣ CategoryTheory.μ_naturalityᵣ
+-/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
+#print CategoryTheory.μ_inv_naturalityₗ /-
@[simp, reassoc]
theorem μ_inv_naturalityₗ {m n m' : M} (f : m ⟶ m') (X : C) :
(F.μIso m n).inv.app X ≫ (F.obj n).map ((F.map f).app X) =
@@ -176,8 +199,10 @@ theorem μ_inv_naturalityₗ {m n m' : M} (f : m ⟶ m') (X : C) :
rw [← is_iso.comp_inv_eq, category.assoc, ← is_iso.eq_inv_comp]
simp
#align category_theory.μ_inv_naturalityₗ CategoryTheory.μ_inv_naturalityₗ
+-/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
+#print CategoryTheory.μ_inv_naturalityᵣ /-
@[simp, reassoc]
theorem μ_inv_naturalityᵣ {m n n' : M} (g : n ⟶ n') (X : C) :
(F.μIso m n).inv.app X ≫ (F.map g).app ((F.obj m).obj X) =
@@ -186,7 +211,9 @@ theorem μ_inv_naturalityᵣ {m n n' : M} (g : n ⟶ n') (X : C) :
rw [← is_iso.comp_inv_eq, category.assoc, ← is_iso.eq_inv_comp]
simp
#align category_theory.μ_inv_naturalityᵣ CategoryTheory.μ_inv_naturalityᵣ
+-/
+#print CategoryTheory.left_unitality_app /-
@[reassoc]
theorem left_unitality_app (n : M) (X : C) :
(F.obj n).map (F.ε.app X) ≫ (F.μ (𝟙_ M) n).app X ≫ (F.map (λ_ n).Hom).app X = 𝟙 _ :=
@@ -195,7 +222,9 @@ theorem left_unitality_app (n : M) (X : C) :
dsimp at this
simpa using this.symm
#align category_theory.left_unitality_app CategoryTheory.left_unitality_app
+-/
+#print CategoryTheory.obj_ε_app /-
@[reassoc, simp]
theorem obj_ε_app (n : M) (X : C) :
(F.obj n).map (F.ε.app X) = (F.map (λ_ n).inv).app X ≫ (F.μIso (𝟙_ M) n).inv.app X :=
@@ -206,7 +235,9 @@ theorem obj_ε_app (n : M) (X : C) :
· simp
· ext; simpa
#align category_theory.obj_ε_app CategoryTheory.obj_ε_app
+-/
+#print CategoryTheory.obj_ε_inv_app /-
@[reassoc, simp]
theorem obj_ε_inv_app (n : M) (X : C) :
(F.obj n).map (F.εIso.inv.app X) = (F.μ (𝟙_ M) n).app X ≫ (F.map (λ_ n).Hom).app X :=
@@ -214,7 +245,9 @@ theorem obj_ε_inv_app (n : M) (X : C) :
rw [← cancel_mono ((F.obj n).map (F.ε.app X)), ← functor.map_comp]
simpa
#align category_theory.obj_ε_inv_app CategoryTheory.obj_ε_inv_app
+-/
+#print CategoryTheory.right_unitality_app /-
@[reassoc]
theorem right_unitality_app (n : M) (X : C) :
F.ε.app ((F.obj n).obj X) ≫ (F.μ n (𝟙_ M)).app X ≫ (F.map (ρ_ n).Hom).app X = 𝟙 _ :=
@@ -223,7 +256,9 @@ theorem right_unitality_app (n : M) (X : C) :
dsimp at this
simpa using this.symm
#align category_theory.right_unitality_app CategoryTheory.right_unitality_app
+-/
+#print CategoryTheory.ε_app_obj /-
@[simp]
theorem ε_app_obj (n : M) (X : C) :
F.ε.app ((F.obj n).obj X) = (F.map (ρ_ n).inv).app X ≫ (F.μIso n (𝟙_ M)).inv.app X :=
@@ -234,7 +269,9 @@ theorem ε_app_obj (n : M) (X : C) :
· simp
· ext; simpa
#align category_theory.ε_app_obj CategoryTheory.ε_app_obj
+-/
+#print CategoryTheory.ε_inv_app_obj /-
@[simp]
theorem ε_inv_app_obj (n : M) (X : C) :
F.εIso.inv.app ((F.obj n).obj X) = (F.μ n (𝟙_ M)).app X ≫ (F.map (ρ_ n).Hom).app X :=
@@ -242,9 +279,11 @@ theorem ε_inv_app_obj (n : M) (X : C) :
rw [← cancel_mono (F.ε.app ((F.obj n).obj X)), ε_inv_hom_app]
simpa
#align category_theory.ε_inv_app_obj CategoryTheory.ε_inv_app_obj
+-/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
+#print CategoryTheory.associativity_app /-
@[reassoc]
theorem associativity_app (m₁ m₂ m₃ : M) (X : C) :
(F.obj m₃).map ((F.μ m₁ m₂).app X) ≫
@@ -255,9 +294,11 @@ theorem associativity_app (m₁ m₂ m₃ : M) (X : C) :
dsimp at this
simpa using this
#align category_theory.associativity_app CategoryTheory.associativity_app
+-/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
+#print CategoryTheory.obj_μ_app /-
@[reassoc, simp]
theorem obj_μ_app (m₁ m₂ m₃ : M) (X : C) :
(F.obj m₃).map ((F.μ m₁ m₂).app X) =
@@ -271,9 +312,11 @@ theorem obj_μ_app (m₁ m₂ m₃ : M) (X : C) :
dsimp
simp
#align category_theory.obj_μ_app CategoryTheory.obj_μ_app
+-/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
+#print CategoryTheory.obj_μ_inv_app /-
@[reassoc, simp]
theorem obj_μ_inv_app (m₁ m₂ m₃ : M) (X : C) :
(F.obj m₃).map ((F.μIso m₁ m₂).inv.app X) =
@@ -289,7 +332,9 @@ theorem obj_μ_inv_app (m₁ m₂ m₃ : M) (X : C) :
· ext; simp
· ext; simpa
#align category_theory.obj_μ_inv_app CategoryTheory.obj_μ_inv_app
+-/
+#print CategoryTheory.obj_zero_map_μ_app /-
@[simp, reassoc]
theorem obj_zero_map_μ_app {m : M} {X Y : C} (f : X ⟶ (F.obj m).obj Y) :
(F.obj (𝟙_ M)).map f ≫ (F.μ m (𝟙_ M)).app _ = F.εIso.inv.app _ ≫ f ≫ (F.map (ρ_ m).inv).app _ :=
@@ -297,7 +342,9 @@ theorem obj_zero_map_μ_app {m : M} {X Y : C} (f : X ⟶ (F.obj m).obj Y) :
rw [← is_iso.inv_comp_eq, ← is_iso.comp_inv_eq]
simp
#align category_theory.obj_zero_map_μ_app CategoryTheory.obj_zero_map_μ_app
+-/
+#print CategoryTheory.obj_μ_zero_app /-
@[simp]
theorem obj_μ_zero_app (m₁ m₂ : M) (X : C) :
(F.obj m₂).map ((F.μ m₁ (𝟙_ M)).app X) =
@@ -307,18 +354,22 @@ theorem obj_μ_zero_app (m₁ m₂ : M) (X : C) :
rw [← obj_ε_inv_app_assoc, ← functor.map_comp]
congr; simp
#align category_theory.obj_μ_zero_app CategoryTheory.obj_μ_zero_app
+-/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
+#print CategoryTheory.unitOfTensorIsoUnit /-
/-- If `m ⊗ n ≅ 𝟙_M`, then `F.obj m` is a left inverse of `F.obj n`. -/
@[simps]
noncomputable def unitOfTensorIsoUnit (m n : M) (h : m ⊗ n ≅ 𝟙_ M) : F.obj m ⋙ F.obj n ≅ 𝟭 C :=
F.μIso m n ≪≫ F.toFunctor.mapIso h ≪≫ F.εIso.symm
#align category_theory.unit_of_tensor_iso_unit CategoryTheory.unitOfTensorIsoUnit
+-/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
+#print CategoryTheory.equivOfTensorIsoUnit /-
/-- If `m ⊗ n ≅ 𝟙_M` and `n ⊗ m ≅ 𝟙_M` (subject to some commuting constraints),
then `F.obj m` and `F.obj n` forms a self-equivalence of `C`. -/
@[simps]
@@ -337,6 +388,7 @@ noncomputable def equivOfTensorIsoUnit (m n : M) (h₁ : m ⊗ n ≅ 𝟙_ M) (h
unit_of_tensor_iso_unit_inv_app]
simp [← nat_trans.comp_app, ← F.to_functor.map_comp, ← H, -functor.map_comp]
#align category_theory.equiv_of_tensor_iso_unit CategoryTheory.equivOfTensorIsoUnit
+-/
end CategoryTheory
mathlib commit https://github.com/leanprover-community/mathlib/commit/cca40788df1b8755d5baf17ab2f27dacc2e17acb
@@ -143,7 +143,7 @@ theorem μ_naturality₂ {m n m' n' : M} (f : m ⟶ m') (g : n ⟶ n') (X : C) :
(F.μ m n).app X ≫ (F.map (f ⊗ g)).app X :=
by
have := congr_app (F.to_lax_monoidal_functor.μ_natural f g) X
- dsimp at this
+ dsimp at this
simpa using this
#align category_theory.μ_naturality₂ CategoryTheory.μ_naturality₂
@@ -192,7 +192,7 @@ theorem left_unitality_app (n : M) (X : C) :
(F.obj n).map (F.ε.app X) ≫ (F.μ (𝟙_ M) n).app X ≫ (F.map (λ_ n).Hom).app X = 𝟙 _ :=
by
have := congr_app (F.to_lax_monoidal_functor.left_unitality n) X
- dsimp at this
+ dsimp at this
simpa using this.symm
#align category_theory.left_unitality_app CategoryTheory.left_unitality_app
@@ -220,7 +220,7 @@ theorem right_unitality_app (n : M) (X : C) :
F.ε.app ((F.obj n).obj X) ≫ (F.μ n (𝟙_ M)).app X ≫ (F.map (ρ_ n).Hom).app X = 𝟙 _ :=
by
have := congr_app (F.to_lax_monoidal_functor.right_unitality n) X
- dsimp at this
+ dsimp at this
simpa using this.symm
#align category_theory.right_unitality_app CategoryTheory.right_unitality_app
@@ -252,7 +252,7 @@ theorem associativity_app (m₁ m₂ m₃ : M) (X : C) :
(F.μ m₂ m₃).app ((F.obj m₁).obj X) ≫ (F.μ m₁ (m₂ ⊗ m₃)).app X :=
by
have := congr_app (F.to_lax_monoidal_functor.associativity m₁ m₂ m₃) X
- dsimp at this
+ dsimp at this
simpa using this
#align category_theory.associativity_app CategoryTheory.associativity_app
@@ -305,7 +305,7 @@ theorem obj_μ_zero_app (m₁ m₂ : M) (X : C) :
(F.map (λ_ m₂).Hom).app ((F.obj m₁).obj X) ≫ (F.obj m₂).map ((F.map (ρ_ m₁).inv).app X) :=
by
rw [← obj_ε_inv_app_assoc, ← functor.map_comp]
- congr ; simp
+ congr; simp
#align category_theory.obj_μ_zero_app CategoryTheory.obj_μ_zero_app
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
mathlib commit https://github.com/leanprover-community/mathlib/commit/917c3c072e487b3cccdbfeff17e75b40e45f66cb
@@ -90,67 +90,43 @@ variable {C}
variable {M : Type _} [Category M] [MonoidalCategory M] (F : MonoidalFunctor M (C ⥤ C))
-/- warning: category_theory.μ_hom_inv_app -> CategoryTheory.μ_hom_inv_app is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align category_theory.μ_hom_inv_app CategoryTheory.μ_hom_inv_appₓ'. -/
@[simp, reassoc]
theorem μ_hom_inv_app (i j : M) (X : C) : (F.μ i j).app X ≫ (F.μIso i j).inv.app X = 𝟙 _ :=
(F.μIso i j).hom_inv_id_app X
#align category_theory.μ_hom_inv_app CategoryTheory.μ_hom_inv_app
-/- warning: category_theory.μ_inv_hom_app -> CategoryTheory.μ_inv_hom_app is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align category_theory.μ_inv_hom_app CategoryTheory.μ_inv_hom_appₓ'. -/
@[simp, reassoc]
theorem μ_inv_hom_app (i j : M) (X : C) : (F.μIso i j).inv.app X ≫ (F.μ i j).app X = 𝟙 _ :=
(F.μIso i j).inv_hom_id_app X
#align category_theory.μ_inv_hom_app CategoryTheory.μ_inv_hom_app
-/- warning: category_theory.ε_hom_inv_app -> CategoryTheory.ε_hom_inv_app is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align category_theory.ε_hom_inv_app CategoryTheory.ε_hom_inv_appₓ'. -/
@[simp, reassoc]
theorem ε_hom_inv_app (X : C) : F.ε.app X ≫ F.εIso.inv.app X = 𝟙 _ :=
F.εIso.hom_inv_id_app X
#align category_theory.ε_hom_inv_app CategoryTheory.ε_hom_inv_app
-/- warning: category_theory.ε_inv_hom_app -> CategoryTheory.ε_inv_hom_app is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align category_theory.ε_inv_hom_app CategoryTheory.ε_inv_hom_appₓ'. -/
@[simp, reassoc]
theorem ε_inv_hom_app (X : C) : F.εIso.inv.app X ≫ F.ε.app X = 𝟙 _ :=
F.εIso.inv_hom_id_app X
#align category_theory.ε_inv_hom_app CategoryTheory.ε_inv_hom_app
-/- warning: category_theory.ε_naturality -> CategoryTheory.ε_naturality is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align category_theory.ε_naturality CategoryTheory.ε_naturalityₓ'. -/
@[simp, reassoc]
theorem ε_naturality {X Y : C} (f : X ⟶ Y) : F.ε.app X ≫ (F.obj (𝟙_ M)).map f = f ≫ F.ε.app Y :=
(F.ε.naturality f).symm
#align category_theory.ε_naturality CategoryTheory.ε_naturality
-/- warning: category_theory.ε_inv_naturality -> CategoryTheory.ε_inv_naturality is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align category_theory.ε_inv_naturality CategoryTheory.ε_inv_naturalityₓ'. -/
@[simp, reassoc]
theorem ε_inv_naturality {X Y : C} (f : X ⟶ Y) :
(F.obj (𝟙_ M)).map f ≫ F.εIso.inv.app Y = F.εIso.inv.app X ≫ f :=
F.εIso.inv.naturality f
#align category_theory.ε_inv_naturality CategoryTheory.ε_inv_naturality
-/- warning: category_theory.μ_naturality -> CategoryTheory.μ_naturality is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align category_theory.μ_naturality CategoryTheory.μ_naturalityₓ'. -/
@[simp, reassoc]
theorem μ_naturality {m n : M} {X Y : C} (f : X ⟶ Y) :
(F.obj n).map ((F.obj m).map f) ≫ (F.μ m n).app Y = (F.μ m n).app X ≫ (F.obj _).map f :=
(F.toLaxMonoidalFunctor.μ m n).naturality f
#align category_theory.μ_naturality CategoryTheory.μ_naturality
-/- warning: category_theory.μ_inv_naturality -> CategoryTheory.μ_inv_naturality is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align category_theory.μ_inv_naturality CategoryTheory.μ_inv_naturalityₓ'. -/
-- This is a simp lemma in the reverse direction via `nat_trans.naturality`.
@[reassoc]
theorem μ_inv_naturality {m n : M} {X Y : C} (f : X ⟶ Y) :
@@ -159,9 +135,6 @@ theorem μ_inv_naturality {m n : M} {X Y : C} (f : X ⟶ Y) :
((F.μIso m n).inv.naturality f).symm
#align category_theory.μ_inv_naturality CategoryTheory.μ_inv_naturality
-/- warning: category_theory.μ_naturality₂ -> CategoryTheory.μ_naturality₂ is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align category_theory.μ_naturality₂ CategoryTheory.μ_naturality₂ₓ'. -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
-- This is not a simp lemma since it could be proved by the lemmas later.
@[reassoc]
@@ -174,9 +147,6 @@ theorem μ_naturality₂ {m n m' n' : M} (f : m ⟶ m') (g : n ⟶ n') (X : C) :
simpa using this
#align category_theory.μ_naturality₂ CategoryTheory.μ_naturality₂
-/- warning: category_theory.μ_naturalityₗ -> CategoryTheory.μ_naturalityₗ is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align category_theory.μ_naturalityₗ CategoryTheory.μ_naturalityₗₓ'. -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
@[simp, reassoc]
theorem μ_naturalityₗ {m n m' : M} (f : m ⟶ m') (X : C) :
@@ -187,9 +157,6 @@ theorem μ_naturalityₗ {m n m' : M} (f : m ⟶ m') (X : C) :
simp
#align category_theory.μ_naturalityₗ CategoryTheory.μ_naturalityₗ
-/- warning: category_theory.μ_naturalityᵣ -> CategoryTheory.μ_naturalityᵣ is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align category_theory.μ_naturalityᵣ CategoryTheory.μ_naturalityᵣₓ'. -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
@[simp, reassoc]
theorem μ_naturalityᵣ {m n n' : M} (g : n ⟶ n') (X : C) :
@@ -200,9 +167,6 @@ theorem μ_naturalityᵣ {m n n' : M} (g : n ⟶ n') (X : C) :
simp
#align category_theory.μ_naturalityᵣ CategoryTheory.μ_naturalityᵣ
-/- warning: category_theory.μ_inv_naturalityₗ -> CategoryTheory.μ_inv_naturalityₗ is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align category_theory.μ_inv_naturalityₗ CategoryTheory.μ_inv_naturalityₗₓ'. -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
@[simp, reassoc]
theorem μ_inv_naturalityₗ {m n m' : M} (f : m ⟶ m') (X : C) :
@@ -213,9 +177,6 @@ theorem μ_inv_naturalityₗ {m n m' : M} (f : m ⟶ m') (X : C) :
simp
#align category_theory.μ_inv_naturalityₗ CategoryTheory.μ_inv_naturalityₗ
-/- warning: category_theory.μ_inv_naturalityᵣ -> CategoryTheory.μ_inv_naturalityᵣ is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align category_theory.μ_inv_naturalityᵣ CategoryTheory.μ_inv_naturalityᵣₓ'. -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
@[simp, reassoc]
theorem μ_inv_naturalityᵣ {m n n' : M} (g : n ⟶ n') (X : C) :
@@ -226,9 +187,6 @@ theorem μ_inv_naturalityᵣ {m n n' : M} (g : n ⟶ n') (X : C) :
simp
#align category_theory.μ_inv_naturalityᵣ CategoryTheory.μ_inv_naturalityᵣ
-/- warning: category_theory.left_unitality_app -> CategoryTheory.left_unitality_app is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align category_theory.left_unitality_app CategoryTheory.left_unitality_appₓ'. -/
@[reassoc]
theorem left_unitality_app (n : M) (X : C) :
(F.obj n).map (F.ε.app X) ≫ (F.μ (𝟙_ M) n).app X ≫ (F.map (λ_ n).Hom).app X = 𝟙 _ :=
@@ -238,9 +196,6 @@ theorem left_unitality_app (n : M) (X : C) :
simpa using this.symm
#align category_theory.left_unitality_app CategoryTheory.left_unitality_app
-/- warning: category_theory.obj_ε_app -> CategoryTheory.obj_ε_app is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align category_theory.obj_ε_app CategoryTheory.obj_ε_appₓ'. -/
@[reassoc, simp]
theorem obj_ε_app (n : M) (X : C) :
(F.obj n).map (F.ε.app X) = (F.map (λ_ n).inv).app X ≫ (F.μIso (𝟙_ M) n).inv.app X :=
@@ -252,9 +207,6 @@ theorem obj_ε_app (n : M) (X : C) :
· ext; simpa
#align category_theory.obj_ε_app CategoryTheory.obj_ε_app
-/- warning: category_theory.obj_ε_inv_app -> CategoryTheory.obj_ε_inv_app is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align category_theory.obj_ε_inv_app CategoryTheory.obj_ε_inv_appₓ'. -/
@[reassoc, simp]
theorem obj_ε_inv_app (n : M) (X : C) :
(F.obj n).map (F.εIso.inv.app X) = (F.μ (𝟙_ M) n).app X ≫ (F.map (λ_ n).Hom).app X :=
@@ -263,9 +215,6 @@ theorem obj_ε_inv_app (n : M) (X : C) :
simpa
#align category_theory.obj_ε_inv_app CategoryTheory.obj_ε_inv_app
-/- warning: category_theory.right_unitality_app -> CategoryTheory.right_unitality_app is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align category_theory.right_unitality_app CategoryTheory.right_unitality_appₓ'. -/
@[reassoc]
theorem right_unitality_app (n : M) (X : C) :
F.ε.app ((F.obj n).obj X) ≫ (F.μ n (𝟙_ M)).app X ≫ (F.map (ρ_ n).Hom).app X = 𝟙 _ :=
@@ -275,9 +224,6 @@ theorem right_unitality_app (n : M) (X : C) :
simpa using this.symm
#align category_theory.right_unitality_app CategoryTheory.right_unitality_app
-/- warning: category_theory.ε_app_obj -> CategoryTheory.ε_app_obj is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align category_theory.ε_app_obj CategoryTheory.ε_app_objₓ'. -/
@[simp]
theorem ε_app_obj (n : M) (X : C) :
F.ε.app ((F.obj n).obj X) = (F.map (ρ_ n).inv).app X ≫ (F.μIso n (𝟙_ M)).inv.app X :=
@@ -289,9 +235,6 @@ theorem ε_app_obj (n : M) (X : C) :
· ext; simpa
#align category_theory.ε_app_obj CategoryTheory.ε_app_obj
-/- warning: category_theory.ε_inv_app_obj -> CategoryTheory.ε_inv_app_obj is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align category_theory.ε_inv_app_obj CategoryTheory.ε_inv_app_objₓ'. -/
@[simp]
theorem ε_inv_app_obj (n : M) (X : C) :
F.εIso.inv.app ((F.obj n).obj X) = (F.μ n (𝟙_ M)).app X ≫ (F.map (ρ_ n).Hom).app X :=
@@ -300,9 +243,6 @@ theorem ε_inv_app_obj (n : M) (X : C) :
simpa
#align category_theory.ε_inv_app_obj CategoryTheory.ε_inv_app_obj
-/- warning: category_theory.associativity_app -> CategoryTheory.associativity_app is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align category_theory.associativity_app CategoryTheory.associativity_appₓ'. -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
@[reassoc]
@@ -316,9 +256,6 @@ theorem associativity_app (m₁ m₂ m₃ : M) (X : C) :
simpa using this
#align category_theory.associativity_app CategoryTheory.associativity_app
-/- warning: category_theory.obj_μ_app -> CategoryTheory.obj_μ_app is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align category_theory.obj_μ_app CategoryTheory.obj_μ_appₓ'. -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
@[reassoc, simp]
@@ -335,9 +272,6 @@ theorem obj_μ_app (m₁ m₂ m₃ : M) (X : C) :
simp
#align category_theory.obj_μ_app CategoryTheory.obj_μ_app
-/- warning: category_theory.obj_μ_inv_app -> CategoryTheory.obj_μ_inv_app is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align category_theory.obj_μ_inv_app CategoryTheory.obj_μ_inv_appₓ'. -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
@[reassoc, simp]
@@ -356,9 +290,6 @@ theorem obj_μ_inv_app (m₁ m₂ m₃ : M) (X : C) :
· ext; simpa
#align category_theory.obj_μ_inv_app CategoryTheory.obj_μ_inv_app
-/- warning: category_theory.obj_zero_map_μ_app -> CategoryTheory.obj_zero_map_μ_app is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align category_theory.obj_zero_map_μ_app CategoryTheory.obj_zero_map_μ_appₓ'. -/
@[simp, reassoc]
theorem obj_zero_map_μ_app {m : M} {X Y : C} (f : X ⟶ (F.obj m).obj Y) :
(F.obj (𝟙_ M)).map f ≫ (F.μ m (𝟙_ M)).app _ = F.εIso.inv.app _ ≫ f ≫ (F.map (ρ_ m).inv).app _ :=
@@ -367,9 +298,6 @@ theorem obj_zero_map_μ_app {m : M} {X Y : C} (f : X ⟶ (F.obj m).obj Y) :
simp
#align category_theory.obj_zero_map_μ_app CategoryTheory.obj_zero_map_μ_app
-/- warning: category_theory.obj_μ_zero_app -> CategoryTheory.obj_μ_zero_app is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align category_theory.obj_μ_zero_app CategoryTheory.obj_μ_zero_appₓ'. -/
@[simp]
theorem obj_μ_zero_app (m₁ m₂ : M) (X : C) :
(F.obj m₂).map ((F.μ m₁ (𝟙_ M)).app X) =
@@ -380,12 +308,6 @@ theorem obj_μ_zero_app (m₁ m₂ : M) (X : C) :
congr ; simp
#align category_theory.obj_μ_zero_app CategoryTheory.obj_μ_zero_app
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/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/-- If `m ⊗ n ≅ 𝟙_M`, then `F.obj m` is a left inverse of `F.obj n`. -/
@[simps]
@@ -393,12 +315,6 @@ noncomputable def unitOfTensorIsoUnit (m n : M) (h : m ⊗ n ≅ 𝟙_ M) : F.ob
F.μIso m n ≪≫ F.toFunctor.mapIso h ≪≫ F.εIso.symm
#align category_theory.unit_of_tensor_iso_unit CategoryTheory.unitOfTensorIsoUnit
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-Case conversion may be inaccurate. Consider using '#align category_theory.equiv_of_tensor_iso_unit CategoryTheory.equivOfTensorIsoUnitₓ'. -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
mathlib commit https://github.com/leanprover-community/mathlib/commit/917c3c072e487b3cccdbfeff17e75b40e45f66cb
@@ -66,23 +66,13 @@ def tensoringRightMonoidal [MonoidalCategory.{v} C] : MonoidalFunctor C (C ⥤ C
ε := (rightUnitorNatIso C).inv
μ := fun X Y =>
{ app := fun Z => (α_ Z X Y).Hom
- naturality' := fun Z Z' f => by
- dsimp
- rw [associator_naturality]
- simp }
+ naturality' := fun Z Z' f => by dsimp; rw [associator_naturality]; simp }
μ_natural' := fun X Y X' Y' f g => by
- ext Z
- dsimp
+ ext Z; dsimp
simp only [← id_tensor_comp_tensor_id g f, id_tensor_comp, ← tensor_id, category.assoc,
associator_naturality, associator_naturality_assoc]
- associativity' := fun X Y Z => by
- ext W
- dsimp
- simp [pentagon]
- left_unitality' := fun X => by
- ext Y
- dsimp
- rw [category.id_comp, triangle, ← tensor_comp]
+ associativity' := fun X Y Z => by ext W; dsimp; simp [pentagon]
+ left_unitality' := fun X => by ext Y; dsimp; rw [category.id_comp, triangle, ← tensor_comp];
simp
right_unitality' := fun X => by
ext Y; dsimp
@@ -91,11 +81,8 @@ def tensoringRightMonoidal [MonoidalCategory.{v} C] : MonoidalFunctor C (C ⥤ C
ε_isIso := by infer_instance
μ_isIso := fun X Y =>
⟨⟨{ app := fun Z => (α_ Z X Y).inv
- naturality' := fun Z Z' f => by
- dsimp
- rw [← associator_inv_naturality]
- simp },
- by tidy⟩⟩ }
+ naturality' := fun Z Z' f => by dsimp; rw [← associator_inv_naturality]; simp }, by
+ tidy⟩⟩ }
#align category_theory.tensoring_right_monoidal CategoryTheory.tensoringRightMonoidal
-/
@@ -262,8 +249,7 @@ theorem obj_ε_app (n : M) (X : C) :
rw [← category.assoc, ← is_iso.comp_inv_eq, ← is_iso.comp_inv_eq, category.assoc]
convert left_unitality_app F n X
· simp
- · ext
- simpa
+ · ext; simpa
#align category_theory.obj_ε_app CategoryTheory.obj_ε_app
/- warning: category_theory.obj_ε_inv_app -> CategoryTheory.obj_ε_inv_app is a dubious translation:
@@ -300,8 +286,7 @@ theorem ε_app_obj (n : M) (X : C) :
rw [← category.assoc, ← is_iso.comp_inv_eq, ← is_iso.comp_inv_eq, category.assoc]
convert right_unitality_app F n X
· simp
- · ext
- simpa
+ · ext; simpa
#align category_theory.ε_app_obj CategoryTheory.ε_app_obj
/- warning: category_theory.ε_inv_app_obj -> CategoryTheory.ε_inv_app_obj is a dubious translation:
@@ -364,15 +349,11 @@ theorem obj_μ_inv_app (m₁ m₂ m₃ : M) (X : C) :
by
rw [← is_iso.inv_eq_inv]
convert obj_μ_app F m₁ m₂ m₃ X using 1
- · ext
- rw [← functor.map_comp]
- simp
+ · ext; rw [← functor.map_comp]; simp
· simp only [monoidal_functor.μ_iso_hom, category.assoc, nat_iso.inv_inv_app, is_iso.inv_comp]
congr
- · ext
- simp
- · ext
- simpa
+ · ext; simp
+ · ext; simpa
#align category_theory.obj_μ_inv_app CategoryTheory.obj_μ_inv_app
/- warning: category_theory.obj_zero_map_μ_app -> CategoryTheory.obj_zero_map_μ_app is a dubious translation:
mathlib commit https://github.com/leanprover-community/mathlib/commit/917c3c072e487b3cccdbfeff17e75b40e45f66cb
@@ -104,10 +104,7 @@ variable {C}
variable {M : Type _} [Category M] [MonoidalCategory M] (F : MonoidalFunctor M (C ⥤ C))
/- warning: category_theory.μ_hom_inv_app -> CategoryTheory.μ_hom_inv_app is a dubious translation:
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Case conversion may be inaccurate. Consider using '#align category_theory.μ_hom_inv_app CategoryTheory.μ_hom_inv_appₓ'. -/
@[simp, reassoc]
theorem μ_hom_inv_app (i j : M) (X : C) : (F.μ i j).app X ≫ (F.μIso i j).inv.app X = 𝟙 _ :=
@@ -115,10 +112,7 @@ theorem μ_hom_inv_app (i j : M) (X : C) : (F.μ i j).app X ≫ (F.μIso i j).in
#align category_theory.μ_hom_inv_app CategoryTheory.μ_hom_inv_app
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Case conversion may be inaccurate. Consider using '#align category_theory.μ_inv_hom_app CategoryTheory.μ_inv_hom_appₓ'. -/
@[simp, reassoc]
theorem μ_inv_hom_app (i j : M) (X : C) : (F.μIso i j).inv.app X ≫ (F.μ i j).app X = 𝟙 _ :=
@@ -126,10 +120,7 @@ theorem μ_inv_hom_app (i j : M) (X : C) : (F.μIso i j).inv.app X ≫ (F.μ i j
#align category_theory.μ_inv_hom_app CategoryTheory.μ_inv_hom_app
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Case conversion may be inaccurate. Consider using '#align category_theory.ε_hom_inv_app CategoryTheory.ε_hom_inv_appₓ'. -/
@[simp, reassoc]
theorem ε_hom_inv_app (X : C) : F.ε.app X ≫ F.εIso.inv.app X = 𝟙 _ :=
@@ -137,10 +128,7 @@ theorem ε_hom_inv_app (X : C) : F.ε.app X ≫ F.εIso.inv.app X = 𝟙 _ :=
#align category_theory.ε_hom_inv_app CategoryTheory.ε_hom_inv_app
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Case conversion may be inaccurate. Consider using '#align category_theory.ε_inv_hom_app CategoryTheory.ε_inv_hom_appₓ'. -/
@[simp, reassoc]
theorem ε_inv_hom_app (X : C) : F.εIso.inv.app X ≫ F.ε.app X = 𝟙 _ :=
@@ -148,10 +136,7 @@ theorem ε_inv_hom_app (X : C) : F.εIso.inv.app X ≫ F.ε.app X = 𝟙 _ :=
#align category_theory.ε_inv_hom_app CategoryTheory.ε_inv_hom_app
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Case conversion may be inaccurate. Consider using '#align category_theory.ε_naturality CategoryTheory.ε_naturalityₓ'. -/
@[simp, reassoc]
theorem ε_naturality {X Y : C} (f : X ⟶ Y) : F.ε.app X ≫ (F.obj (𝟙_ M)).map f = f ≫ F.ε.app Y :=
@@ -159,10 +144,7 @@ theorem ε_naturality {X Y : C} (f : X ⟶ Y) : F.ε.app X ≫ (F.obj (𝟙_ M))
#align category_theory.ε_naturality CategoryTheory.ε_naturality
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Case conversion may be inaccurate. Consider using '#align category_theory.ε_inv_naturality CategoryTheory.ε_inv_naturalityₓ'. -/
@[simp, reassoc]
theorem ε_inv_naturality {X Y : C} (f : X ⟶ Y) :
@@ -171,10 +153,7 @@ theorem ε_inv_naturality {X Y : C} (f : X ⟶ Y) :
#align category_theory.ε_inv_naturality CategoryTheory.ε_inv_naturality
/- warning: category_theory.μ_naturality -> CategoryTheory.μ_naturality is a dubious translation:
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Case conversion may be inaccurate. Consider using '#align category_theory.μ_naturality CategoryTheory.μ_naturalityₓ'. -/
@[simp, reassoc]
theorem μ_naturality {m n : M} {X Y : C} (f : X ⟶ Y) :
@@ -183,10 +162,7 @@ theorem μ_naturality {m n : M} {X Y : C} (f : X ⟶ Y) :
#align category_theory.μ_naturality CategoryTheory.μ_naturality
/- warning: category_theory.μ_inv_naturality -> CategoryTheory.μ_inv_naturality is a dubious translation:
-lean 3 declaration is
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Case conversion may be inaccurate. Consider using '#align category_theory.μ_inv_naturality CategoryTheory.μ_inv_naturalityₓ'. -/
-- This is a simp lemma in the reverse direction via `nat_trans.naturality`.
@[reassoc]
@@ -197,10 +173,7 @@ theorem μ_inv_naturality {m n : M} {X Y : C} (f : X ⟶ Y) :
#align category_theory.μ_inv_naturality CategoryTheory.μ_inv_naturality
/- warning: category_theory.μ_naturality₂ -> CategoryTheory.μ_naturality₂ is a dubious translation:
-lean 3 declaration is
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+<too large>
Case conversion may be inaccurate. Consider using '#align category_theory.μ_naturality₂ CategoryTheory.μ_naturality₂ₓ'. -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
-- This is not a simp lemma since it could be proved by the lemmas later.
@@ -215,10 +188,7 @@ theorem μ_naturality₂ {m n m' n' : M} (f : m ⟶ m') (g : n ⟶ n') (X : C) :
#align category_theory.μ_naturality₂ CategoryTheory.μ_naturality₂
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Case conversion may be inaccurate. Consider using '#align category_theory.μ_naturalityₗ CategoryTheory.μ_naturalityₗₓ'. -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
@[simp, reassoc]
@@ -231,10 +201,7 @@ theorem μ_naturalityₗ {m n m' : M} (f : m ⟶ m') (X : C) :
#align category_theory.μ_naturalityₗ CategoryTheory.μ_naturalityₗ
/- warning: category_theory.μ_naturalityᵣ -> CategoryTheory.μ_naturalityᵣ is a dubious translation:
-lean 3 declaration is
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Case conversion may be inaccurate. Consider using '#align category_theory.μ_naturalityᵣ CategoryTheory.μ_naturalityᵣₓ'. -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
@[simp, reassoc]
@@ -247,10 +214,7 @@ theorem μ_naturalityᵣ {m n n' : M} (g : n ⟶ n') (X : C) :
#align category_theory.μ_naturalityᵣ CategoryTheory.μ_naturalityᵣ
/- warning: category_theory.μ_inv_naturalityₗ -> CategoryTheory.μ_inv_naturalityₗ is a dubious translation:
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+<too large>
Case conversion may be inaccurate. Consider using '#align category_theory.μ_inv_naturalityₗ CategoryTheory.μ_inv_naturalityₗₓ'. -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
@[simp, reassoc]
@@ -263,10 +227,7 @@ theorem μ_inv_naturalityₗ {m n m' : M} (f : m ⟶ m') (X : C) :
#align category_theory.μ_inv_naturalityₗ CategoryTheory.μ_inv_naturalityₗ
/- warning: category_theory.μ_inv_naturalityᵣ -> CategoryTheory.μ_inv_naturalityᵣ is a dubious translation:
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Case conversion may be inaccurate. Consider using '#align category_theory.μ_inv_naturalityᵣ CategoryTheory.μ_inv_naturalityᵣₓ'. -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
@[simp, reassoc]
@@ -279,10 +240,7 @@ theorem μ_inv_naturalityᵣ {m n n' : M} (g : n ⟶ n') (X : C) :
#align category_theory.μ_inv_naturalityᵣ CategoryTheory.μ_inv_naturalityᵣ
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Case conversion may be inaccurate. Consider using '#align category_theory.left_unitality_app CategoryTheory.left_unitality_appₓ'. -/
@[reassoc]
theorem left_unitality_app (n : M) (X : C) :
@@ -294,10 +252,7 @@ theorem left_unitality_app (n : M) (X : C) :
#align category_theory.left_unitality_app CategoryTheory.left_unitality_app
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M _inst_2 _inst_3) n)) X))
+<too large>
Case conversion may be inaccurate. Consider using '#align category_theory.obj_ε_app CategoryTheory.obj_ε_appₓ'. -/
@[reassoc, simp]
theorem obj_ε_app (n : M) (X : C) :
@@ -312,10 +267,7 @@ theorem obj_ε_app (n : M) (X : C) :
#align category_theory.obj_ε_app CategoryTheory.obj_ε_app
/- warning: category_theory.obj_ε_inv_app -> CategoryTheory.obj_ε_inv_app is a dubious translation:
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+<too large>
Case conversion may be inaccurate. Consider using '#align category_theory.obj_ε_inv_app CategoryTheory.obj_ε_inv_appₓ'. -/
@[reassoc, simp]
theorem obj_ε_inv_app (n : M) (X : C) :
@@ -326,10 +278,7 @@ theorem obj_ε_inv_app (n : M) (X : C) :
#align category_theory.obj_ε_inv_app CategoryTheory.obj_ε_inv_app
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+<too large>
Case conversion may be inaccurate. Consider using '#align category_theory.right_unitality_app CategoryTheory.right_unitality_appₓ'. -/
@[reassoc]
theorem right_unitality_app (n : M) (X : C) :
@@ -341,10 +290,7 @@ theorem right_unitality_app (n : M) (X : C) :
#align category_theory.right_unitality_app CategoryTheory.right_unitality_app
/- warning: category_theory.ε_app_obj -> CategoryTheory.ε_app_obj is a dubious translation:
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+<too large>
Case conversion may be inaccurate. Consider using '#align category_theory.ε_app_obj CategoryTheory.ε_app_objₓ'. -/
@[simp]
theorem ε_app_obj (n : M) (X : C) :
@@ -359,10 +305,7 @@ theorem ε_app_obj (n : M) (X : C) :
#align category_theory.ε_app_obj CategoryTheory.ε_app_obj
/- warning: category_theory.ε_inv_app_obj -> CategoryTheory.ε_inv_app_obj is a dubious translation:
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+<too large>
Case conversion may be inaccurate. Consider using '#align category_theory.ε_inv_app_obj CategoryTheory.ε_inv_app_objₓ'. -/
@[simp]
theorem ε_inv_app_obj (n : M) (X : C) :
@@ -373,10 +316,7 @@ theorem ε_inv_app_obj (n : M) (X : C) :
#align category_theory.ε_inv_app_obj CategoryTheory.ε_inv_app_obj
/- warning: category_theory.associativity_app -> CategoryTheory.associativity_app is a dubious translation:
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+<too large>
Case conversion may be inaccurate. Consider using '#align category_theory.associativity_app CategoryTheory.associativity_appₓ'. -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
@@ -392,10 +332,7 @@ theorem associativity_app (m₁ m₂ m₃ : M) (X : C) :
#align category_theory.associativity_app CategoryTheory.associativity_app
/- warning: category_theory.obj_μ_app -> CategoryTheory.obj_μ_app is a dubious translation:
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(CategoryTheory.MonoidalCategory.tensorObj.{u2, u1} M _inst_2 _inst_3 m₁ m₂) m₃)) X))))
+<too large>
Case conversion may be inaccurate. Consider using '#align category_theory.obj_μ_app CategoryTheory.obj_μ_appₓ'. -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
@@ -414,10 +351,7 @@ theorem obj_μ_app (m₁ m₂ m₃ : M) (X : C) :
#align category_theory.obj_μ_app CategoryTheory.obj_μ_app
/- warning: category_theory.obj_μ_inv_app -> CategoryTheory.obj_μ_inv_app is a dubious translation:
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u4} C _inst_1 C _inst_1) (CategoryTheory.Functor.category.{u3, u3, u4, u4} C _inst_1 C _inst_1) (CategoryTheory.endofunctorMonoidalCategory.{u3, u4} C _inst_1) F))) m₁)) X)))))
+<too large>
Case conversion may be inaccurate. Consider using '#align category_theory.obj_μ_inv_app CategoryTheory.obj_μ_inv_appₓ'. -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
@@ -442,10 +376,7 @@ theorem obj_μ_inv_app (m₁ m₂ m₃ : M) (X : C) :
#align category_theory.obj_μ_inv_app CategoryTheory.obj_μ_inv_app
/- warning: category_theory.obj_zero_map_μ_app -> CategoryTheory.obj_zero_map_μ_app is a dubious translation:
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Case conversion may be inaccurate. Consider using '#align category_theory.obj_zero_map_μ_app CategoryTheory.obj_zero_map_μ_appₓ'. -/
@[simp, reassoc]
theorem obj_zero_map_μ_app {m : M} {X Y : C} (f : X ⟶ (F.obj m).obj Y) :
@@ -456,10 +387,7 @@ theorem obj_zero_map_μ_app {m : M} {X Y : C} (f : X ⟶ (F.obj m).obj Y) :
#align category_theory.obj_zero_map_μ_app CategoryTheory.obj_zero_map_μ_app
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Case conversion may be inaccurate. Consider using '#align category_theory.obj_μ_zero_app CategoryTheory.obj_μ_zero_appₓ'. -/
@[simp]
theorem obj_μ_zero_app (m₁ m₂ : M) (X : C) :
mathlib commit https://github.com/leanprover-community/mathlib/commit/75e7fca56381d056096ce5d05e938f63a6567828
@@ -109,7 +109,7 @@ lean 3 declaration is
but is expected to have type
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Case conversion may be inaccurate. Consider using '#align category_theory.μ_hom_inv_app CategoryTheory.μ_hom_inv_appₓ'. -/
-@[simp, reassoc.1]
+@[simp, reassoc]
theorem μ_hom_inv_app (i j : M) (X : C) : (F.μ i j).app X ≫ (F.μIso i j).inv.app X = 𝟙 _ :=
(F.μIso i j).hom_inv_id_app X
#align category_theory.μ_hom_inv_app CategoryTheory.μ_hom_inv_app
@@ -120,7 +120,7 @@ lean 3 declaration is
but is expected to have type
forall {C : Type.{u4}} [_inst_1 : CategoryTheory.Category.{u3, u4} C] {M : Type.{u1}} [_inst_2 : CategoryTheory.Category.{u2, u1} M] [_inst_3 : CategoryTheory.MonoidalCategory.{u2, u1} M _inst_2] (F : CategoryTheory.MonoidalFunctor.{u2, max u4 u3, u1, max u4 u3} M _inst_2 _inst_3 (CategoryTheory.Functor.{u3, u3, u4, u4} C _inst_1 C _inst_1) (CategoryTheory.Functor.category.{u3, u3, u4, u4} C _inst_1 C _inst_1) (CategoryTheory.endofunctorMonoidalCategory.{u3, u4} C _inst_1)) (i : M) (j : M) (X : C), Eq.{succ u3} (Quiver.Hom.{succ u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u3, u4} C (CategoryTheory.Category.toCategoryStruct.{u3, u4} C _inst_1)) (Prefunctor.obj.{succ u3, succ u3, u4, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u3, u4} C (CategoryTheory.Category.toCategoryStruct.{u3, u4} C _inst_1)) C (CategoryTheory.CategoryStruct.toQuiver.{u3, u4} C (CategoryTheory.Category.toCategoryStruct.{u3, u4} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u3, u3, u4, u4} C _inst_1 C 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F))) (CategoryTheory.MonoidalCategory.tensorObj.{u2, u1} M _inst_2 _inst_3 i j))) X))
Case conversion may be inaccurate. Consider using '#align category_theory.μ_inv_hom_app CategoryTheory.μ_inv_hom_appₓ'. -/
-@[simp, reassoc.1]
+@[simp, reassoc]
theorem μ_inv_hom_app (i j : M) (X : C) : (F.μIso i j).inv.app X ≫ (F.μ i j).app X = 𝟙 _ :=
(F.μIso i j).inv_hom_id_app X
#align category_theory.μ_inv_hom_app CategoryTheory.μ_inv_hom_app
@@ -131,7 +131,7 @@ lean 3 declaration is
but is expected to have type
forall {C : Type.{u4}} [_inst_1 : CategoryTheory.Category.{u3, u4} C] {M : Type.{u1}} [_inst_2 : CategoryTheory.Category.{u2, u1} M] [_inst_3 : CategoryTheory.MonoidalCategory.{u2, u1} M _inst_2] (F : CategoryTheory.MonoidalFunctor.{u2, max u4 u3, u1, max u4 u3} M _inst_2 _inst_3 (CategoryTheory.Functor.{u3, u3, u4, u4} C _inst_1 C _inst_1) (CategoryTheory.Functor.category.{u3, u3, u4, u4} C _inst_1 C _inst_1) (CategoryTheory.endofunctorMonoidalCategory.{u3, u4} C _inst_1)) (X : C), Eq.{succ u3} (Quiver.Hom.{succ u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u3, u4} C (CategoryTheory.Category.toCategoryStruct.{u3, u4} C _inst_1)) (Prefunctor.obj.{succ u3, succ u3, u4, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u3, u4} C (CategoryTheory.Category.toCategoryStruct.{u3, u4} C _inst_1)) C (CategoryTheory.CategoryStruct.toQuiver.{u3, u4} C (CategoryTheory.Category.toCategoryStruct.{u3, u4} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u3, u3, u4, u4} C _inst_1 C _inst_1 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Case conversion may be inaccurate. Consider using '#align category_theory.ε_hom_inv_app CategoryTheory.ε_hom_inv_appₓ'. -/
-@[simp, reassoc.1]
+@[simp, reassoc]
theorem ε_hom_inv_app (X : C) : F.ε.app X ≫ F.εIso.inv.app X = 𝟙 _ :=
F.εIso.hom_inv_id_app X
#align category_theory.ε_hom_inv_app CategoryTheory.ε_hom_inv_app
@@ -142,7 +142,7 @@ lean 3 declaration is
but is expected to have type
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Case conversion may be inaccurate. Consider using '#align category_theory.ε_inv_hom_app CategoryTheory.ε_inv_hom_appₓ'. -/
-@[simp, reassoc.1]
+@[simp, reassoc]
theorem ε_inv_hom_app (X : C) : F.εIso.inv.app X ≫ F.ε.app X = 𝟙 _ :=
F.εIso.inv_hom_id_app X
#align category_theory.ε_inv_hom_app CategoryTheory.ε_inv_hom_app
@@ -153,7 +153,7 @@ lean 3 declaration is
but is expected to have type
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Case conversion may be inaccurate. Consider using '#align category_theory.ε_naturality CategoryTheory.ε_naturalityₓ'. -/
-@[simp, reassoc.1]
+@[simp, reassoc]
theorem ε_naturality {X Y : C} (f : X ⟶ Y) : F.ε.app X ≫ (F.obj (𝟙_ M)).map f = f ≫ F.ε.app Y :=
(F.ε.naturality f).symm
#align category_theory.ε_naturality CategoryTheory.ε_naturality
@@ -164,7 +164,7 @@ lean 3 declaration is
but is expected to have type
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Case conversion may be inaccurate. Consider using '#align category_theory.ε_inv_naturality CategoryTheory.ε_inv_naturalityₓ'. -/
-@[simp, reassoc.1]
+@[simp, reassoc]
theorem ε_inv_naturality {X Y : C} (f : X ⟶ Y) :
(F.obj (𝟙_ M)).map f ≫ F.εIso.inv.app Y = F.εIso.inv.app X ≫ f :=
F.εIso.inv.naturality f
@@ -176,7 +176,7 @@ lean 3 declaration is
but is expected to have type
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u4} C _inst_1 C _inst_1 (Prefunctor.obj.{succ u2, max (succ u4) (succ u3), u1, max u4 u3} M (CategoryTheory.CategoryStruct.toQuiver.{u2, u1} M (CategoryTheory.Category.toCategoryStruct.{u2, u1} M _inst_2)) (CategoryTheory.Functor.{u3, u3, u4, u4} C _inst_1 C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max u4 u3, max u4 u3} (CategoryTheory.Functor.{u3, u3, u4, u4} C _inst_1 C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max u4 u3, max u4 u3} (CategoryTheory.Functor.{u3, u3, u4, u4} C _inst_1 C _inst_1) (CategoryTheory.Functor.category.{u3, u3, u4, u4} C _inst_1 C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u2, max u4 u3, u1, max u4 u3} M _inst_2 (CategoryTheory.Functor.{u3, u3, u4, u4} C _inst_1 C _inst_1) (CategoryTheory.Functor.category.{u3, u3, u4, u4} C _inst_1 C _inst_1) (CategoryTheory.LaxMonoidalFunctor.toFunctor.{u2, max u4 u3, u1, max u4 u3} M _inst_2 _inst_3 (CategoryTheory.Functor.{u3, u3, u4, u4} C _inst_1 C _inst_1) (CategoryTheory.Functor.category.{u3, u3, u4, u4} C _inst_1 C _inst_1) (CategoryTheory.endofunctorMonoidalCategory.{u3, u4} C _inst_1) (CategoryTheory.MonoidalFunctor.toLaxMonoidalFunctor.{u2, max u4 u3, u1, max u4 u3} M _inst_2 _inst_3 (CategoryTheory.Functor.{u3, u3, u4, u4} C _inst_1 C _inst_1) (CategoryTheory.Functor.category.{u3, u3, u4, u4} C _inst_1 C _inst_1) (CategoryTheory.endofunctorMonoidalCategory.{u3, u4} C _inst_1) F))) (CategoryTheory.MonoidalCategory.tensorObj.{u2, u1} M _inst_2 _inst_3 m n))) X Y f))
Case conversion may be inaccurate. Consider using '#align category_theory.μ_naturality CategoryTheory.μ_naturalityₓ'. -/
-@[simp, reassoc.1]
+@[simp, reassoc]
theorem μ_naturality {m n : M} {X Y : C} (f : X ⟶ Y) :
(F.obj n).map ((F.obj m).map f) ≫ (F.μ m n).app Y = (F.μ m n).app X ≫ (F.obj _).map f :=
(F.toLaxMonoidalFunctor.μ m n).naturality f
@@ -189,7 +189,7 @@ but is expected to have type
forall {C : Type.{u4}} [_inst_1 : CategoryTheory.Category.{u3, u4} C] {M : Type.{u1}} [_inst_2 : CategoryTheory.Category.{u2, u1} M] [_inst_3 : CategoryTheory.MonoidalCategory.{u2, u1} M _inst_2] (F : CategoryTheory.MonoidalFunctor.{u2, max u4 u3, u1, max u4 u3} M _inst_2 _inst_3 (CategoryTheory.Functor.{u3, u3, u4, u4} C _inst_1 C _inst_1) (CategoryTheory.Functor.category.{u3, u3, u4, u4} C _inst_1 C _inst_1) (CategoryTheory.endofunctorMonoidalCategory.{u3, u4} C _inst_1)) {m : M} {n : M} {X : C} {Y : C} (f : Quiver.Hom.{succ u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u3, u4} C (CategoryTheory.Category.toCategoryStruct.{u3, u4} C _inst_1)) X Y), Eq.{succ u3} (Quiver.Hom.{succ u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u3, u4} C (CategoryTheory.Category.toCategoryStruct.{u3, u4} C _inst_1)) (Prefunctor.obj.{succ u3, succ u3, u4, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u3, u4} C (CategoryTheory.Category.toCategoryStruct.{u3, u4} C _inst_1)) C 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(CategoryTheory.Functor.category.{u3, u3, u4, u4} C _inst_1 C _inst_1) (CategoryTheory.endofunctorMonoidalCategory.{u3, u4} C _inst_1) F m n)) Y))
Case conversion may be inaccurate. Consider using '#align category_theory.μ_inv_naturality CategoryTheory.μ_inv_naturalityₓ'. -/
-- This is a simp lemma in the reverse direction via `nat_trans.naturality`.
-@[reassoc.1]
+@[reassoc]
theorem μ_inv_naturality {m n : M} {X Y : C} (f : X ⟶ Y) :
(F.μIso m n).inv.app X ≫ (F.obj n).map ((F.obj m).map f) =
(F.obj _).map f ≫ (F.μIso m n).inv.app Y :=
@@ -204,7 +204,7 @@ but is expected to have type
Case conversion may be inaccurate. Consider using '#align category_theory.μ_naturality₂ CategoryTheory.μ_naturality₂ₓ'. -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
-- This is not a simp lemma since it could be proved by the lemmas later.
-@[reassoc.1]
+@[reassoc]
theorem μ_naturality₂ {m n m' n' : M} (f : m ⟶ m') (g : n ⟶ n') (X : C) :
(F.map g).app ((F.obj m).obj X) ≫ (F.obj n').map ((F.map f).app X) ≫ (F.μ m' n').app X =
(F.μ m n).app X ≫ (F.map (f ⊗ g)).app X :=
@@ -221,7 +221,7 @@ but is expected to have type
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Case conversion may be inaccurate. Consider using '#align category_theory.μ_naturalityₗ CategoryTheory.μ_naturalityₗₓ'. -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
-@[simp, reassoc.1]
+@[simp, reassoc]
theorem μ_naturalityₗ {m n m' : M} (f : m ⟶ m') (X : C) :
(F.obj n).map ((F.map f).app X) ≫ (F.μ m' n).app X =
(F.μ m n).app X ≫ (F.map (f ⊗ 𝟙 n)).app X :=
@@ -237,7 +237,7 @@ but is expected to have type
forall {C : Type.{u4}} [_inst_1 : CategoryTheory.Category.{u3, u4} C] {M : Type.{u1}} [_inst_2 : CategoryTheory.Category.{u2, u1} M] [_inst_3 : CategoryTheory.MonoidalCategory.{u2, u1} M _inst_2] (F : CategoryTheory.MonoidalFunctor.{u2, max u4 u3, u1, max u4 u3} M _inst_2 _inst_3 (CategoryTheory.Functor.{u3, u3, u4, u4} C _inst_1 C _inst_1) (CategoryTheory.Functor.category.{u3, u3, u4, u4} C _inst_1 C _inst_1) (CategoryTheory.endofunctorMonoidalCategory.{u3, u4} C _inst_1)) {m : M} {n : M} {n' : M} (g : Quiver.Hom.{succ u2, u1} M (CategoryTheory.CategoryStruct.toQuiver.{u2, u1} M (CategoryTheory.Category.toCategoryStruct.{u2, u1} M _inst_2)) n n') (X : C), Eq.{succ u3} (Quiver.Hom.{succ u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u3, u4} C (CategoryTheory.Category.toCategoryStruct.{u3, u4} C _inst_1)) (Prefunctor.obj.{succ u3, succ u3, u4, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u3, u4} C (CategoryTheory.Category.toCategoryStruct.{u3, u4} C _inst_1)) C 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(CategoryTheory.endofunctorMonoidalCategory.{u3, u4} C _inst_1) F))) (CategoryTheory.MonoidalCategory.tensorObj.{u2, u1} M _inst_2 _inst_3 m n) (CategoryTheory.MonoidalCategory.tensorObj.{u2, u1} M _inst_2 _inst_3 m n') (CategoryTheory.MonoidalCategory.tensorHom.{u2, u1} M _inst_2 _inst_3 m m n n' (CategoryTheory.CategoryStruct.id.{u2, u1} M (CategoryTheory.Category.toCategoryStruct.{u2, u1} M _inst_2) m) g)) X))
Case conversion may be inaccurate. Consider using '#align category_theory.μ_naturalityᵣ CategoryTheory.μ_naturalityᵣₓ'. -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
-@[simp, reassoc.1]
+@[simp, reassoc]
theorem μ_naturalityᵣ {m n n' : M} (g : n ⟶ n') (X : C) :
(F.map g).app ((F.obj m).obj X) ≫ (F.μ m n').app X =
(F.μ m n).app X ≫ (F.map (𝟙 m ⊗ g)).app X :=
@@ -253,7 +253,7 @@ but is expected to have type
forall {C : Type.{u4}} [_inst_1 : CategoryTheory.Category.{u3, u4} C] {M : Type.{u1}} [_inst_2 : CategoryTheory.Category.{u2, u1} M] [_inst_3 : CategoryTheory.MonoidalCategory.{u2, u1} M _inst_2] (F : CategoryTheory.MonoidalFunctor.{u2, max u4 u3, u1, max u4 u3} M _inst_2 _inst_3 (CategoryTheory.Functor.{u3, u3, u4, u4} C _inst_1 C _inst_1) (CategoryTheory.Functor.category.{u3, u3, u4, u4} C _inst_1 C _inst_1) (CategoryTheory.endofunctorMonoidalCategory.{u3, u4} C _inst_1)) {m : M} {n : M} {m' : M} (f : Quiver.Hom.{succ u2, u1} M (CategoryTheory.CategoryStruct.toQuiver.{u2, u1} M (CategoryTheory.Category.toCategoryStruct.{u2, u1} M _inst_2)) m m') (X : C), Eq.{succ u3} (Quiver.Hom.{succ u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u3, u4} C (CategoryTheory.Category.toCategoryStruct.{u3, u4} C _inst_1)) (Prefunctor.obj.{succ u3, succ u3, u4, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u3, u4} C (CategoryTheory.Category.toCategoryStruct.{u3, u4} C _inst_1)) C 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Case conversion may be inaccurate. Consider using '#align category_theory.μ_inv_naturalityₗ CategoryTheory.μ_inv_naturalityₗₓ'. -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
-@[simp, reassoc.1]
+@[simp, reassoc]
theorem μ_inv_naturalityₗ {m n m' : M} (f : m ⟶ m') (X : C) :
(F.μIso m n).inv.app X ≫ (F.obj n).map ((F.map f).app X) =
(F.map (f ⊗ 𝟙 n)).app X ≫ (F.μIso m' n).inv.app X :=
@@ -269,7 +269,7 @@ but is expected to have type
forall {C : Type.{u4}} [_inst_1 : CategoryTheory.Category.{u3, u4} C] {M : Type.{u1}} [_inst_2 : CategoryTheory.Category.{u2, u1} M] [_inst_3 : CategoryTheory.MonoidalCategory.{u2, u1} M _inst_2] (F : CategoryTheory.MonoidalFunctor.{u2, max u4 u3, u1, max u4 u3} M _inst_2 _inst_3 (CategoryTheory.Functor.{u3, u3, u4, u4} C _inst_1 C _inst_1) (CategoryTheory.Functor.category.{u3, u3, u4, u4} C _inst_1 C _inst_1) (CategoryTheory.endofunctorMonoidalCategory.{u3, u4} C _inst_1)) {m : M} {n : M} {n' : M} (g : Quiver.Hom.{succ u2, u1} M (CategoryTheory.CategoryStruct.toQuiver.{u2, u1} M (CategoryTheory.Category.toCategoryStruct.{u2, u1} M _inst_2)) n n') (X : C), Eq.{succ u3} (Quiver.Hom.{succ u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u3, u4} C (CategoryTheory.Category.toCategoryStruct.{u3, u4} C _inst_1)) (Prefunctor.obj.{succ u3, succ u3, u4, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u3, u4} C (CategoryTheory.Category.toCategoryStruct.{u3, u4} C _inst_1)) C 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(CategoryTheory.endofunctorMonoidalCategory.{u3, u4} C _inst_1) F m n')) X))
Case conversion may be inaccurate. Consider using '#align category_theory.μ_inv_naturalityᵣ CategoryTheory.μ_inv_naturalityᵣₓ'. -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
-@[simp, reassoc.1]
+@[simp, reassoc]
theorem μ_inv_naturalityᵣ {m n n' : M} (g : n ⟶ n') (X : C) :
(F.μIso m n).inv.app X ≫ (F.map g).app ((F.obj m).obj X) =
(F.map (𝟙 m ⊗ g)).app X ≫ (F.μIso m n').inv.app X :=
@@ -284,7 +284,7 @@ lean 3 declaration is
but is expected to have type
forall {C : Type.{u4}} [_inst_1 : CategoryTheory.Category.{u3, u4} C] {M : Type.{u1}} [_inst_2 : CategoryTheory.Category.{u2, u1} M] [_inst_3 : CategoryTheory.MonoidalCategory.{u2, u1} M _inst_2] (F : CategoryTheory.MonoidalFunctor.{u2, max u4 u3, u1, max u4 u3} M _inst_2 _inst_3 (CategoryTheory.Functor.{u3, u3, u4, u4} C _inst_1 C _inst_1) (CategoryTheory.Functor.category.{u3, u3, u4, u4} C _inst_1 C _inst_1) (CategoryTheory.endofunctorMonoidalCategory.{u3, u4} C _inst_1)) (n : M) (X : C), Eq.{succ u3} (Quiver.Hom.{succ u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u3, u4} C (CategoryTheory.Category.toCategoryStruct.{u3, u4} C _inst_1)) (Prefunctor.obj.{succ u3, succ u3, u4, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u3, u4} C (CategoryTheory.Category.toCategoryStruct.{u3, u4} C _inst_1)) C (CategoryTheory.CategoryStruct.toQuiver.{u3, u4} C (CategoryTheory.Category.toCategoryStruct.{u3, u4} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u3, u3, u4, u4} C _inst_1 C _inst_1 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u3, max u4 u3} (CategoryTheory.Functor.{u3, u3, u4, u4} C _inst_1 C _inst_1) (CategoryTheory.Functor.category.{u3, u3, u4, u4} C _inst_1 C _inst_1) (CategoryTheory.endofunctorMonoidalCategory.{u3, u4} C _inst_1))) X)))
Case conversion may be inaccurate. Consider using '#align category_theory.left_unitality_app CategoryTheory.left_unitality_appₓ'. -/
-@[reassoc.1]
+@[reassoc]
theorem left_unitality_app (n : M) (X : C) :
(F.obj n).map (F.ε.app X) ≫ (F.μ (𝟙_ M) n).app X ≫ (F.map (λ_ n).Hom).app X = 𝟙 _ :=
by
@@ -299,7 +299,7 @@ lean 3 declaration is
but is expected to have type
forall {C : Type.{u4}} [_inst_1 : CategoryTheory.Category.{u3, u4} C] {M : Type.{u1}} [_inst_2 : CategoryTheory.Category.{u2, u1} M] [_inst_3 : CategoryTheory.MonoidalCategory.{u2, u1} M _inst_2] (F : CategoryTheory.MonoidalFunctor.{u2, max u4 u3, u1, max u4 u3} M _inst_2 _inst_3 (CategoryTheory.Functor.{u3, u3, u4, u4} C _inst_1 C _inst_1) (CategoryTheory.Functor.category.{u3, u3, u4, u4} C _inst_1 C _inst_1) (CategoryTheory.endofunctorMonoidalCategory.{u3, u4} C _inst_1)) (n : M) (X : C), Eq.{succ u3} (Quiver.Hom.{succ u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u3, u4} C (CategoryTheory.Category.toCategoryStruct.{u3, u4} C _inst_1)) (Prefunctor.obj.{succ u3, succ u3, u4, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u3, u4} C (CategoryTheory.Category.toCategoryStruct.{u3, u4} C _inst_1)) C (CategoryTheory.CategoryStruct.toQuiver.{u3, u4} C (CategoryTheory.Category.toCategoryStruct.{u3, u4} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u3, u3, u4, u4} C _inst_1 C _inst_1 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Case conversion may be inaccurate. Consider using '#align category_theory.obj_ε_app CategoryTheory.obj_ε_appₓ'. -/
-@[reassoc.1, simp]
+@[reassoc, simp]
theorem obj_ε_app (n : M) (X : C) :
(F.obj n).map (F.ε.app X) = (F.map (λ_ n).inv).app X ≫ (F.μIso (𝟙_ M) n).inv.app X :=
by
@@ -317,7 +317,7 @@ lean 3 declaration is
but is expected to have type
forall {C : Type.{u4}} [_inst_1 : CategoryTheory.Category.{u3, u4} C] {M : Type.{u1}} [_inst_2 : CategoryTheory.Category.{u2, u1} M] [_inst_3 : CategoryTheory.MonoidalCategory.{u2, u1} M _inst_2] (F : CategoryTheory.MonoidalFunctor.{u2, max u4 u3, u1, max u4 u3} M _inst_2 _inst_3 (CategoryTheory.Functor.{u3, u3, u4, u4} C _inst_1 C _inst_1) (CategoryTheory.Functor.category.{u3, u3, u4, u4} C _inst_1 C _inst_1) (CategoryTheory.endofunctorMonoidalCategory.{u3, u4} C _inst_1)) (n : M) (X : C), Eq.{succ u3} (Quiver.Hom.{succ u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u3, u4} C (CategoryTheory.Category.toCategoryStruct.{u3, u4} C _inst_1)) (Prefunctor.obj.{succ u3, succ u3, u4, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u3, u4} C (CategoryTheory.Category.toCategoryStruct.{u3, u4} C _inst_1)) C (CategoryTheory.CategoryStruct.toQuiver.{u3, u4} C (CategoryTheory.Category.toCategoryStruct.{u3, u4} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u3, u3, u4, u4} C _inst_1 C _inst_1 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_inst_3 (CategoryTheory.MonoidalCategory.tensorUnit'.{u2, u1} M _inst_2 _inst_3) n) n (CategoryTheory.MonoidalCategory.leftUnitor.{u2, u1} M _inst_2 _inst_3 n))) X))
Case conversion may be inaccurate. Consider using '#align category_theory.obj_ε_inv_app CategoryTheory.obj_ε_inv_appₓ'. -/
-@[reassoc.1, simp]
+@[reassoc, simp]
theorem obj_ε_inv_app (n : M) (X : C) :
(F.obj n).map (F.εIso.inv.app X) = (F.μ (𝟙_ M) n).app X ≫ (F.map (λ_ n).Hom).app X :=
by
@@ -331,7 +331,7 @@ lean 3 declaration is
but is expected to have type
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(CategoryTheory.Iso.hom.{u2, u1} M _inst_2 (CategoryTheory.MonoidalCategory.tensorObj.{u2, u1} M _inst_2 _inst_3 n (CategoryTheory.MonoidalCategory.tensorUnit'.{u2, u1} M _inst_2 _inst_3)) n (CategoryTheory.MonoidalCategory.rightUnitor.{u2, u1} M _inst_2 _inst_3 n))) X))) (CategoryTheory.CategoryStruct.id.{u3, u4} C (CategoryTheory.Category.toCategoryStruct.{u3, u4} C _inst_1) (Prefunctor.obj.{succ u3, succ u3, u4, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u3, u4} C (CategoryTheory.Category.toCategoryStruct.{u3, u4} C _inst_1)) C (CategoryTheory.CategoryStruct.toQuiver.{u3, u4} C (CategoryTheory.Category.toCategoryStruct.{u3, u4} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u3, u3, u4, u4} C _inst_1 C _inst_1 (CategoryTheory.MonoidalCategory.tensorUnit.{max u4 u3, max u4 u3} (CategoryTheory.Functor.{u3, u3, u4, u4} C _inst_1 C _inst_1) (CategoryTheory.Functor.category.{u3, u3, u4, u4} C _inst_1 C _inst_1) (CategoryTheory.endofunctorMonoidalCategory.{u3, u4} C _inst_1))) 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Case conversion may be inaccurate. Consider using '#align category_theory.right_unitality_app CategoryTheory.right_unitality_appₓ'. -/
-@[reassoc.1]
+@[reassoc]
theorem right_unitality_app (n : M) (X : C) :
F.ε.app ((F.obj n).obj X) ≫ (F.μ n (𝟙_ M)).app X ≫ (F.map (ρ_ n).Hom).app X = 𝟙 _ :=
by
@@ -380,7 +380,7 @@ but is expected to have type
Case conversion may be inaccurate. Consider using '#align category_theory.associativity_app CategoryTheory.associativity_appₓ'. -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
-@[reassoc.1]
+@[reassoc]
theorem associativity_app (m₁ m₂ m₃ : M) (X : C) :
(F.obj m₃).map ((F.μ m₁ m₂).app X) ≫
(F.μ (m₁ ⊗ m₂) m₃).app X ≫ (F.map (α_ m₁ m₂ m₃).Hom).app X =
@@ -399,7 +399,7 @@ but is expected to have type
Case conversion may be inaccurate. Consider using '#align category_theory.obj_μ_app CategoryTheory.obj_μ_appₓ'. -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
-@[reassoc.1, simp]
+@[reassoc, simp]
theorem obj_μ_app (m₁ m₂ m₃ : M) (X : C) :
(F.obj m₃).map ((F.μ m₁ m₂).app X) =
(F.μ m₂ m₃).app ((F.obj m₁).obj X) ≫
@@ -421,7 +421,7 @@ but is expected to have type
Case conversion may be inaccurate. Consider using '#align category_theory.obj_μ_inv_app CategoryTheory.obj_μ_inv_appₓ'. -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
-@[reassoc.1, simp]
+@[reassoc, simp]
theorem obj_μ_inv_app (m₁ m₂ m₃ : M) (X : C) :
(F.obj m₃).map ((F.μIso m₁ m₂).inv.app X) =
(F.μ (m₁ ⊗ m₂) m₃).app X ≫
@@ -447,7 +447,7 @@ lean 3 declaration is
but is expected to have type
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Case conversion may be inaccurate. Consider using '#align category_theory.obj_zero_map_μ_app CategoryTheory.obj_zero_map_μ_appₓ'. -/
-@[simp, reassoc.1]
+@[simp, reassoc]
theorem obj_zero_map_μ_app {m : M} {X Y : C} (f : X ⟶ (F.obj m).obj Y) :
(F.obj (𝟙_ M)).map f ≫ (F.μ m (𝟙_ M)).app _ = F.εIso.inv.app _ ≫ f ≫ (F.map (ρ_ m).inv).app _ :=
by
mathlib commit https://github.com/leanprover-community/mathlib/commit/2196ab363eb097c008d4497125e0dde23fb36db2
@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Andrew Yang
! This file was ported from Lean 3 source module category_theory.monoidal.End
-! leanprover-community/mathlib commit 85075bccb68ab7fa49fb05db816233fb790e4fe9
+! leanprover-community/mathlib commit 69c6a5a12d8a2b159f20933e60115a4f2de62b58
! 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.Monoidal.Functor
/-!
# Endofunctors as a monoidal category.
+> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
+> Any changes to this file require a corresponding PR to mathlib4.
+
We give the monoidal category structure on `C ⥤ C`,
and show that when `C` itself is monoidal, it embeds via a monoidal functor into `C ⥤ C`.
mathlib commit https://github.com/leanprover-community/mathlib/commit/ddec54a71a0dd025c05445d467f1a2b7d586a3ba
@@ -29,6 +29,7 @@ namespace CategoryTheory
variable (C : Type u) [Category.{v} C]
+#print CategoryTheory.endofunctorMonoidalCategory /-
/-- The category of endofunctors of any category is a monoidal category,
with tensor product given by composition of functors
(and horizontal composition of natural transformations).
@@ -42,6 +43,7 @@ def endofunctorMonoidalCategory : MonoidalCategory (C ⥤ C)
leftUnitor F := Functor.leftUnitor F
rightUnitor F := Functor.rightUnitor F
#align category_theory.endofunctor_monoidal_category CategoryTheory.endofunctorMonoidalCategory
+-/
open CategoryTheory.MonoidalCategory
@@ -49,6 +51,7 @@ attribute [local instance] endofunctor_monoidal_category
attribute [local reducible] endofunctor_monoidal_category
+#print CategoryTheory.tensoringRightMonoidal /-
/-- Tensoring on the right gives a monoidal functor from `C` into endofunctors of `C`.
-/
@[simps]
@@ -91,48 +94,97 @@ def tensoringRightMonoidal [MonoidalCategory.{v} C] : MonoidalFunctor C (C ⥤ C
simp },
by tidy⟩⟩ }
#align category_theory.tensoring_right_monoidal CategoryTheory.tensoringRightMonoidal
+-/
variable {C}
variable {M : Type _} [Category M] [MonoidalCategory M] (F : MonoidalFunctor M (C ⥤ C))
+/- warning: category_theory.μ_hom_inv_app -> CategoryTheory.μ_hom_inv_app is a dubious translation:
+lean 3 declaration is
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+but is expected to have type
+ forall {C : Type.{u4}} [_inst_1 : CategoryTheory.Category.{u3, u4} C] {M : Type.{u1}} [_inst_2 : CategoryTheory.Category.{u2, u1} M] [_inst_3 : CategoryTheory.MonoidalCategory.{u2, u1} M _inst_2] (F : CategoryTheory.MonoidalFunctor.{u2, max u4 u3, u1, max u4 u3} M _inst_2 _inst_3 (CategoryTheory.Functor.{u3, u3, u4, u4} C _inst_1 C _inst_1) (CategoryTheory.Functor.category.{u3, u3, u4, u4} C _inst_1 C _inst_1) (CategoryTheory.endofunctorMonoidalCategory.{u3, u4} C _inst_1)) (i : M) (j : M) (X : C), Eq.{succ u3} (Quiver.Hom.{succ u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u3, u4} C (CategoryTheory.Category.toCategoryStruct.{u3, u4} C _inst_1)) (Prefunctor.obj.{succ u3, succ u3, u4, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u3, u4} C (CategoryTheory.Category.toCategoryStruct.{u3, u4} C _inst_1)) C (CategoryTheory.CategoryStruct.toQuiver.{u3, u4} C (CategoryTheory.Category.toCategoryStruct.{u3, u4} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u3, u3, u4, u4} C _inst_1 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+Case conversion may be inaccurate. Consider using '#align category_theory.μ_hom_inv_app CategoryTheory.μ_hom_inv_appₓ'. -/
@[simp, reassoc.1]
theorem μ_hom_inv_app (i j : M) (X : C) : (F.μ i j).app X ≫ (F.μIso i j).inv.app X = 𝟙 _ :=
(F.μIso i j).hom_inv_id_app X
#align category_theory.μ_hom_inv_app CategoryTheory.μ_hom_inv_app
+/- warning: category_theory.μ_inv_hom_app -> CategoryTheory.μ_inv_hom_app is a dubious translation:
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+but is expected to have type
+ forall {C : Type.{u4}} [_inst_1 : CategoryTheory.Category.{u3, u4} C] {M : Type.{u1}} [_inst_2 : CategoryTheory.Category.{u2, u1} M] [_inst_3 : CategoryTheory.MonoidalCategory.{u2, u1} M _inst_2] (F : CategoryTheory.MonoidalFunctor.{u2, max u4 u3, u1, max u4 u3} M _inst_2 _inst_3 (CategoryTheory.Functor.{u3, u3, u4, u4} C _inst_1 C _inst_1) (CategoryTheory.Functor.category.{u3, u3, u4, u4} C _inst_1 C _inst_1) (CategoryTheory.endofunctorMonoidalCategory.{u3, u4} C _inst_1)) (i : M) (j : M) (X : C), Eq.{succ u3} (Quiver.Hom.{succ u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u3, u4} C (CategoryTheory.Category.toCategoryStruct.{u3, u4} C _inst_1)) (Prefunctor.obj.{succ u3, succ u3, u4, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u3, u4} C (CategoryTheory.Category.toCategoryStruct.{u3, u4} C _inst_1)) C (CategoryTheory.CategoryStruct.toQuiver.{u3, u4} C (CategoryTheory.Category.toCategoryStruct.{u3, u4} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u3, u3, u4, u4} C _inst_1 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+Case conversion may be inaccurate. Consider using '#align category_theory.μ_inv_hom_app CategoryTheory.μ_inv_hom_appₓ'. -/
@[simp, reassoc.1]
theorem μ_inv_hom_app (i j : M) (X : C) : (F.μIso i j).inv.app X ≫ (F.μ i j).app X = 𝟙 _ :=
(F.μIso i j).inv_hom_id_app X
#align category_theory.μ_inv_hom_app CategoryTheory.μ_inv_hom_app
+/- warning: category_theory.ε_hom_inv_app -> CategoryTheory.ε_hom_inv_app is a dubious translation:
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+Case conversion may be inaccurate. Consider using '#align category_theory.ε_hom_inv_app CategoryTheory.ε_hom_inv_appₓ'. -/
@[simp, reassoc.1]
theorem ε_hom_inv_app (X : C) : F.ε.app X ≫ F.εIso.inv.app X = 𝟙 _ :=
F.εIso.hom_inv_id_app X
#align category_theory.ε_hom_inv_app CategoryTheory.ε_hom_inv_app
+/- warning: category_theory.ε_inv_hom_app -> CategoryTheory.ε_inv_hom_app is a dubious translation:
+lean 3 declaration is
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C _inst_1) (CategoryTheory.MonoidalFunctor.toLaxMonoidalFunctor.{u4, max u2 u1, u3, max u1 u2} M _inst_2 _inst_3 (CategoryTheory.Functor.{u1, u1, u2, u2} C _inst_1 C _inst_1) (CategoryTheory.Functor.category.{u1, u1, u2, u2} C _inst_1 C _inst_1) (CategoryTheory.endofunctorMonoidalCategory.{u1, u2} C _inst_1) F)) (CategoryTheory.MonoidalCategory.tensorUnit.{u4, u3} M _inst_2 _inst_3)) (CategoryTheory.MonoidalFunctor.εIso.{u4, max u2 u1, u3, max u1 u2} M _inst_2 _inst_3 (CategoryTheory.Functor.{u1, u1, u2, u2} C _inst_1 C _inst_1) (CategoryTheory.Functor.category.{u1, u1, u2, u2} C _inst_1 C _inst_1) (CategoryTheory.endofunctorMonoidalCategory.{u1, u2} C _inst_1) F)) X) (CategoryTheory.NatTrans.app.{u1, u1, u2, u2} C _inst_1 C _inst_1 (CategoryTheory.MonoidalCategory.tensorUnit.{max u2 u1, max u1 u2} (CategoryTheory.Functor.{u1, u1, u2, u2} C _inst_1 C _inst_1) (CategoryTheory.Functor.category.{u1, u1, u2, u2} C _inst_1 C _inst_1) (CategoryTheory.endofunctorMonoidalCategory.{u1, u2} C _inst_1)) (CategoryTheory.Functor.obj.{u4, max u2 u1, u3, max u1 u2} M _inst_2 (CategoryTheory.Functor.{u1, u1, u2, u2} C _inst_1 C _inst_1) (CategoryTheory.Functor.category.{u1, u1, u2, u2} C _inst_1 C _inst_1) (CategoryTheory.LaxMonoidalFunctor.toFunctor.{u4, max u2 u1, u3, max u1 u2} M _inst_2 _inst_3 (CategoryTheory.Functor.{u1, u1, u2, u2} C _inst_1 C _inst_1) (CategoryTheory.Functor.category.{u1, u1, u2, u2} C _inst_1 C _inst_1) (CategoryTheory.endofunctorMonoidalCategory.{u1, u2} C _inst_1) (CategoryTheory.MonoidalFunctor.toLaxMonoidalFunctor.{u4, max u2 u1, u3, max u1 u2} M _inst_2 _inst_3 (CategoryTheory.Functor.{u1, u1, u2, u2} C _inst_1 C _inst_1) (CategoryTheory.Functor.category.{u1, u1, u2, u2} C _inst_1 C _inst_1) (CategoryTheory.endofunctorMonoidalCategory.{u1, u2} C _inst_1) F)) (CategoryTheory.MonoidalCategory.tensorUnit.{u4, u3} M _inst_2 _inst_3)) (CategoryTheory.LaxMonoidalFunctor.ε.{u4, max u2 u1, u3, max u1 u2} M _inst_2 _inst_3 (CategoryTheory.Functor.{u1, u1, 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+but is expected to have type
+ forall {C : Type.{u4}} [_inst_1 : CategoryTheory.Category.{u3, u4} C] {M : Type.{u1}} [_inst_2 : CategoryTheory.Category.{u2, u1} M] [_inst_3 : CategoryTheory.MonoidalCategory.{u2, u1} M _inst_2] (F : CategoryTheory.MonoidalFunctor.{u2, max u4 u3, u1, max u4 u3} M _inst_2 _inst_3 (CategoryTheory.Functor.{u3, u3, u4, u4} C _inst_1 C _inst_1) (CategoryTheory.Functor.category.{u3, u3, u4, u4} C _inst_1 C _inst_1) (CategoryTheory.endofunctorMonoidalCategory.{u3, u4} C _inst_1)) (X : C), Eq.{succ u3} (Quiver.Hom.{succ u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u3, u4} C (CategoryTheory.Category.toCategoryStruct.{u3, u4} C _inst_1)) (Prefunctor.obj.{succ u3, succ u3, u4, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u3, u4} C (CategoryTheory.Category.toCategoryStruct.{u3, u4} C _inst_1)) C (CategoryTheory.CategoryStruct.toQuiver.{u3, u4} C (CategoryTheory.Category.toCategoryStruct.{u3, u4} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u3, u3, u4, u4} C _inst_1 C _inst_1 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+Case conversion may be inaccurate. Consider using '#align category_theory.ε_inv_hom_app CategoryTheory.ε_inv_hom_appₓ'. -/
@[simp, reassoc.1]
theorem ε_inv_hom_app (X : C) : F.εIso.inv.app X ≫ F.ε.app X = 𝟙 _ :=
F.εIso.inv_hom_id_app X
#align category_theory.ε_inv_hom_app CategoryTheory.ε_inv_hom_app
+/- warning: category_theory.ε_naturality -> CategoryTheory.ε_naturality is a dubious translation:
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+but is expected to have type
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+Case conversion may be inaccurate. Consider using '#align category_theory.ε_naturality CategoryTheory.ε_naturalityₓ'. -/
@[simp, reassoc.1]
theorem ε_naturality {X Y : C} (f : X ⟶ Y) : F.ε.app X ≫ (F.obj (𝟙_ M)).map f = f ≫ F.ε.app Y :=
(F.ε.naturality f).symm
#align category_theory.ε_naturality CategoryTheory.ε_naturality
+/- warning: category_theory.ε_inv_naturality -> CategoryTheory.ε_inv_naturality is a dubious translation:
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+but is expected to have type
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+Case conversion may be inaccurate. Consider using '#align category_theory.ε_inv_naturality CategoryTheory.ε_inv_naturalityₓ'. -/
@[simp, reassoc.1]
theorem ε_inv_naturality {X Y : C} (f : X ⟶ Y) :
(F.obj (𝟙_ M)).map f ≫ F.εIso.inv.app Y = F.εIso.inv.app X ≫ f :=
F.εIso.inv.naturality f
#align category_theory.ε_inv_naturality CategoryTheory.ε_inv_naturality
+/- warning: category_theory.μ_naturality -> CategoryTheory.μ_naturality is a dubious translation:
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+but is expected to have type
+ forall {C : Type.{u4}} [_inst_1 : CategoryTheory.Category.{u3, u4} C] {M : Type.{u1}} [_inst_2 : CategoryTheory.Category.{u2, u1} M] [_inst_3 : CategoryTheory.MonoidalCategory.{u2, u1} M _inst_2] (F : CategoryTheory.MonoidalFunctor.{u2, max u4 u3, u1, max u4 u3} M _inst_2 _inst_3 (CategoryTheory.Functor.{u3, u3, u4, u4} C _inst_1 C _inst_1) (CategoryTheory.Functor.category.{u3, u3, u4, u4} C _inst_1 C _inst_1) (CategoryTheory.endofunctorMonoidalCategory.{u3, u4} C _inst_1)) {m : M} {n : M} {X : C} {Y : C} (f : Quiver.Hom.{succ u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u3, u4} C (CategoryTheory.Category.toCategoryStruct.{u3, u4} C _inst_1)) X Y), Eq.{succ u3} (Quiver.Hom.{succ u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u3, u4} C (CategoryTheory.Category.toCategoryStruct.{u3, u4} C _inst_1)) (Prefunctor.obj.{succ u3, succ u3, u4, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u3, u4} C (CategoryTheory.Category.toCategoryStruct.{u3, u4} C _inst_1)) C 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+Case conversion may be inaccurate. Consider using '#align category_theory.μ_naturality CategoryTheory.μ_naturalityₓ'. -/
@[simp, reassoc.1]
theorem μ_naturality {m n : M} {X Y : C} (f : X ⟶ Y) :
(F.obj n).map ((F.obj m).map f) ≫ (F.μ m n).app Y = (F.μ m n).app X ≫ (F.obj _).map f :=
(F.toLaxMonoidalFunctor.μ m n).naturality f
#align category_theory.μ_naturality CategoryTheory.μ_naturality
+/- warning: category_theory.μ_inv_naturality -> CategoryTheory.μ_inv_naturality is a dubious translation:
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+but is expected to have type
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+Case conversion may be inaccurate. Consider using '#align category_theory.μ_inv_naturality CategoryTheory.μ_inv_naturalityₓ'. -/
-- This is a simp lemma in the reverse direction via `nat_trans.naturality`.
@[reassoc.1]
theorem μ_inv_naturality {m n : M} {X Y : C} (f : X ⟶ Y) :
@@ -141,6 +193,12 @@ theorem μ_inv_naturality {m n : M} {X Y : C} (f : X ⟶ Y) :
((F.μIso m n).inv.naturality f).symm
#align category_theory.μ_inv_naturality CategoryTheory.μ_inv_naturality
+/- warning: category_theory.μ_naturality₂ -> CategoryTheory.μ_naturality₂ is a dubious translation:
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+but is expected to have type
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+Case conversion may be inaccurate. Consider using '#align category_theory.μ_naturality₂ CategoryTheory.μ_naturality₂ₓ'. -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
-- This is not a simp lemma since it could be proved by the lemmas later.
@[reassoc.1]
@@ -153,6 +211,12 @@ theorem μ_naturality₂ {m n m' n' : M} (f : m ⟶ m') (g : n ⟶ n') (X : C) :
simpa using this
#align category_theory.μ_naturality₂ CategoryTheory.μ_naturality₂
+/- warning: category_theory.μ_naturalityₗ -> CategoryTheory.μ_naturalityₗ is a dubious translation:
+lean 3 declaration is
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+but is expected to have type
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+Case conversion may be inaccurate. Consider using '#align category_theory.μ_naturalityₗ CategoryTheory.μ_naturalityₗₓ'. -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
@[simp, reassoc.1]
theorem μ_naturalityₗ {m n m' : M} (f : m ⟶ m') (X : C) :
@@ -163,6 +227,12 @@ theorem μ_naturalityₗ {m n m' : M} (f : m ⟶ m') (X : C) :
simp
#align category_theory.μ_naturalityₗ CategoryTheory.μ_naturalityₗ
+/- warning: category_theory.μ_naturalityᵣ -> CategoryTheory.μ_naturalityᵣ is a dubious translation:
+lean 3 declaration is
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+but is expected to have type
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+Case conversion may be inaccurate. Consider using '#align category_theory.μ_naturalityᵣ CategoryTheory.μ_naturalityᵣₓ'. -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
@[simp, reassoc.1]
theorem μ_naturalityᵣ {m n n' : M} (g : n ⟶ n') (X : C) :
@@ -173,6 +243,12 @@ theorem μ_naturalityᵣ {m n n' : M} (g : n ⟶ n') (X : C) :
simp
#align category_theory.μ_naturalityᵣ CategoryTheory.μ_naturalityᵣ
+/- warning: category_theory.μ_inv_naturalityₗ -> CategoryTheory.μ_inv_naturalityₗ is a dubious translation:
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+Case conversion may be inaccurate. Consider using '#align category_theory.μ_inv_naturalityₗ CategoryTheory.μ_inv_naturalityₗₓ'. -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
@[simp, reassoc.1]
theorem μ_inv_naturalityₗ {m n m' : M} (f : m ⟶ m') (X : C) :
@@ -183,6 +259,12 @@ theorem μ_inv_naturalityₗ {m n m' : M} (f : m ⟶ m') (X : C) :
simp
#align category_theory.μ_inv_naturalityₗ CategoryTheory.μ_inv_naturalityₗ
+/- warning: category_theory.μ_inv_naturalityᵣ -> CategoryTheory.μ_inv_naturalityᵣ is a dubious translation:
+lean 3 declaration is
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+but is expected to have type
+ forall {C : Type.{u4}} [_inst_1 : CategoryTheory.Category.{u3, u4} C] {M : Type.{u1}} [_inst_2 : CategoryTheory.Category.{u2, u1} M] [_inst_3 : CategoryTheory.MonoidalCategory.{u2, u1} M _inst_2] (F : CategoryTheory.MonoidalFunctor.{u2, max u4 u3, u1, max u4 u3} M _inst_2 _inst_3 (CategoryTheory.Functor.{u3, u3, u4, u4} C _inst_1 C _inst_1) (CategoryTheory.Functor.category.{u3, u3, u4, u4} C _inst_1 C _inst_1) (CategoryTheory.endofunctorMonoidalCategory.{u3, u4} C _inst_1)) {m : M} {n : M} {n' : M} (g : Quiver.Hom.{succ u2, u1} M (CategoryTheory.CategoryStruct.toQuiver.{u2, u1} M (CategoryTheory.Category.toCategoryStruct.{u2, u1} M _inst_2)) n n') (X : C), Eq.{succ u3} (Quiver.Hom.{succ u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u3, u4} C (CategoryTheory.Category.toCategoryStruct.{u3, u4} C _inst_1)) (Prefunctor.obj.{succ u3, succ u3, u4, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u3, u4} C (CategoryTheory.Category.toCategoryStruct.{u3, u4} C _inst_1)) C 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+Case conversion may be inaccurate. Consider using '#align category_theory.μ_inv_naturalityᵣ CategoryTheory.μ_inv_naturalityᵣₓ'. -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
@[simp, reassoc.1]
theorem μ_inv_naturalityᵣ {m n n' : M} (g : n ⟶ n') (X : C) :
@@ -193,6 +275,12 @@ theorem μ_inv_naturalityᵣ {m n n' : M} (g : n ⟶ n') (X : C) :
simp
#align category_theory.μ_inv_naturalityᵣ CategoryTheory.μ_inv_naturalityᵣ
+/- warning: category_theory.left_unitality_app -> CategoryTheory.left_unitality_app is a dubious translation:
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+but is expected to have type
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+Case conversion may be inaccurate. Consider using '#align category_theory.left_unitality_app CategoryTheory.left_unitality_appₓ'. -/
@[reassoc.1]
theorem left_unitality_app (n : M) (X : C) :
(F.obj n).map (F.ε.app X) ≫ (F.μ (𝟙_ M) n).app X ≫ (F.map (λ_ n).Hom).app X = 𝟙 _ :=
@@ -202,6 +290,12 @@ theorem left_unitality_app (n : M) (X : C) :
simpa using this.symm
#align category_theory.left_unitality_app CategoryTheory.left_unitality_app
+/- warning: category_theory.obj_ε_app -> CategoryTheory.obj_ε_app is a dubious translation:
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+but is expected to have type
+ forall {C : Type.{u4}} [_inst_1 : CategoryTheory.Category.{u3, u4} C] {M : Type.{u1}} [_inst_2 : CategoryTheory.Category.{u2, u1} M] [_inst_3 : CategoryTheory.MonoidalCategory.{u2, u1} M _inst_2] (F : CategoryTheory.MonoidalFunctor.{u2, max u4 u3, u1, max u4 u3} M _inst_2 _inst_3 (CategoryTheory.Functor.{u3, u3, u4, u4} C _inst_1 C _inst_1) (CategoryTheory.Functor.category.{u3, u3, u4, u4} C _inst_1 C _inst_1) (CategoryTheory.endofunctorMonoidalCategory.{u3, u4} C _inst_1)) (n : M) (X : C), Eq.{succ u3} (Quiver.Hom.{succ u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u3, u4} C (CategoryTheory.Category.toCategoryStruct.{u3, u4} C _inst_1)) (Prefunctor.obj.{succ u3, succ u3, u4, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u3, u4} C (CategoryTheory.Category.toCategoryStruct.{u3, u4} C _inst_1)) C (CategoryTheory.CategoryStruct.toQuiver.{u3, u4} C (CategoryTheory.Category.toCategoryStruct.{u3, u4} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u3, u3, u4, u4} C _inst_1 C 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M _inst_2 _inst_3) n)) X))
+Case conversion may be inaccurate. Consider using '#align category_theory.obj_ε_app CategoryTheory.obj_ε_appₓ'. -/
@[reassoc.1, simp]
theorem obj_ε_app (n : M) (X : C) :
(F.obj n).map (F.ε.app X) = (F.map (λ_ n).inv).app X ≫ (F.μIso (𝟙_ M) n).inv.app X :=
@@ -214,6 +308,12 @@ theorem obj_ε_app (n : M) (X : C) :
simpa
#align category_theory.obj_ε_app CategoryTheory.obj_ε_app
+/- warning: category_theory.obj_ε_inv_app -> CategoryTheory.obj_ε_inv_app is a dubious translation:
+lean 3 declaration is
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+but is expected to have type
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+Case conversion may be inaccurate. Consider using '#align category_theory.obj_ε_inv_app CategoryTheory.obj_ε_inv_appₓ'. -/
@[reassoc.1, simp]
theorem obj_ε_inv_app (n : M) (X : C) :
(F.obj n).map (F.εIso.inv.app X) = (F.μ (𝟙_ M) n).app X ≫ (F.map (λ_ n).Hom).app X :=
@@ -222,6 +322,12 @@ theorem obj_ε_inv_app (n : M) (X : C) :
simpa
#align category_theory.obj_ε_inv_app CategoryTheory.obj_ε_inv_app
+/- warning: category_theory.right_unitality_app -> CategoryTheory.right_unitality_app is a dubious translation:
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+but is expected to have type
+ forall {C : Type.{u4}} [_inst_1 : CategoryTheory.Category.{u3, u4} C] {M : Type.{u1}} [_inst_2 : CategoryTheory.Category.{u2, u1} M] [_inst_3 : CategoryTheory.MonoidalCategory.{u2, u1} M _inst_2] (F : CategoryTheory.MonoidalFunctor.{u2, max u4 u3, u1, max u4 u3} M _inst_2 _inst_3 (CategoryTheory.Functor.{u3, u3, u4, u4} C _inst_1 C _inst_1) (CategoryTheory.Functor.category.{u3, u3, u4, u4} C _inst_1 C _inst_1) (CategoryTheory.endofunctorMonoidalCategory.{u3, u4} C _inst_1)) (n : M) (X : C), Eq.{succ u3} (Quiver.Hom.{succ u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u3, u4} C (CategoryTheory.Category.toCategoryStruct.{u3, u4} C _inst_1)) (Prefunctor.obj.{succ u3, succ u3, u4, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u3, u4} C (CategoryTheory.Category.toCategoryStruct.{u3, u4} C _inst_1)) C (CategoryTheory.CategoryStruct.toQuiver.{u3, u4} C (CategoryTheory.Category.toCategoryStruct.{u3, u4} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u3, u3, u4, u4} C _inst_1 C 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+Case conversion may be inaccurate. Consider using '#align category_theory.right_unitality_app CategoryTheory.right_unitality_appₓ'. -/
@[reassoc.1]
theorem right_unitality_app (n : M) (X : C) :
F.ε.app ((F.obj n).obj X) ≫ (F.μ n (𝟙_ M)).app X ≫ (F.map (ρ_ n).Hom).app X = 𝟙 _ :=
@@ -231,6 +337,12 @@ theorem right_unitality_app (n : M) (X : C) :
simpa using this.symm
#align category_theory.right_unitality_app CategoryTheory.right_unitality_app
+/- warning: category_theory.ε_app_obj -> CategoryTheory.ε_app_obj is a dubious translation:
+lean 3 declaration is
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+but is expected to have type
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+Case conversion may be inaccurate. Consider using '#align category_theory.ε_app_obj CategoryTheory.ε_app_objₓ'. -/
@[simp]
theorem ε_app_obj (n : M) (X : C) :
F.ε.app ((F.obj n).obj X) = (F.map (ρ_ n).inv).app X ≫ (F.μIso n (𝟙_ M)).inv.app X :=
@@ -243,6 +355,12 @@ theorem ε_app_obj (n : M) (X : C) :
simpa
#align category_theory.ε_app_obj CategoryTheory.ε_app_obj
+/- warning: category_theory.ε_inv_app_obj -> CategoryTheory.ε_inv_app_obj is a dubious translation:
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+but is expected to have type
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+Case conversion may be inaccurate. Consider using '#align category_theory.ε_inv_app_obj CategoryTheory.ε_inv_app_objₓ'. -/
@[simp]
theorem ε_inv_app_obj (n : M) (X : C) :
F.εIso.inv.app ((F.obj n).obj X) = (F.μ n (𝟙_ M)).app X ≫ (F.map (ρ_ n).Hom).app X :=
@@ -251,6 +369,12 @@ theorem ε_inv_app_obj (n : M) (X : C) :
simpa
#align category_theory.ε_inv_app_obj CategoryTheory.ε_inv_app_obj
+/- warning: category_theory.associativity_app -> CategoryTheory.associativity_app is a dubious translation:
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+but is expected to have type
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+Case conversion may be inaccurate. Consider using '#align category_theory.associativity_app CategoryTheory.associativity_appₓ'. -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
@[reassoc.1]
@@ -264,6 +388,12 @@ theorem associativity_app (m₁ m₂ m₃ : M) (X : C) :
simpa using this
#align category_theory.associativity_app CategoryTheory.associativity_app
+/- warning: category_theory.obj_μ_app -> CategoryTheory.obj_μ_app is a dubious translation:
+lean 3 declaration is
+ forall {C : Type.{u2}} [_inst_1 : CategoryTheory.Category.{u1, u2} C] {M : Type.{u3}} [_inst_2 : CategoryTheory.Category.{u4, u3} M] [_inst_3 : CategoryTheory.MonoidalCategory.{u4, u3} M _inst_2] (F : CategoryTheory.MonoidalFunctor.{u4, max u2 u1, u3, max u1 u2} M _inst_2 _inst_3 (CategoryTheory.Functor.{u1, u1, u2, u2} C _inst_1 C _inst_1) (CategoryTheory.Functor.category.{u1, u1, u2, u2} C _inst_1 C _inst_1) (CategoryTheory.endofunctorMonoidalCategory.{u1, u2} C _inst_1)) (m₁ : M) (m₂ : M) (m₃ : M) (X : C), Eq.{succ u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.obj.{u1, u1, u2, u2} C _inst_1 C _inst_1 (CategoryTheory.Functor.obj.{u4, max u2 u1, u3, max u1 u2} M _inst_2 (CategoryTheory.Functor.{u1, u1, u2, u2} C _inst_1 C _inst_1) (CategoryTheory.Functor.category.{u1, u1, u2, u2} C _inst_1 C _inst_1) (CategoryTheory.LaxMonoidalFunctor.toFunctor.{u4, max u2 u1, 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+but is expected to have type
+ forall {C : Type.{u4}} [_inst_1 : CategoryTheory.Category.{u3, u4} C] {M : Type.{u1}} [_inst_2 : CategoryTheory.Category.{u2, u1} M] [_inst_3 : CategoryTheory.MonoidalCategory.{u2, u1} M _inst_2] (F : CategoryTheory.MonoidalFunctor.{u2, max u4 u3, u1, max u4 u3} M _inst_2 _inst_3 (CategoryTheory.Functor.{u3, u3, u4, u4} C _inst_1 C _inst_1) (CategoryTheory.Functor.category.{u3, u3, u4, u4} C _inst_1 C _inst_1) (CategoryTheory.endofunctorMonoidalCategory.{u3, u4} C _inst_1)) (m₁ : M) (m₂ : M) (m₃ : M) (X : C), Eq.{succ u3} (Quiver.Hom.{succ u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u3, u4} C (CategoryTheory.Category.toCategoryStruct.{u3, u4} C _inst_1)) (Prefunctor.obj.{succ u3, succ u3, u4, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u3, u4} C (CategoryTheory.Category.toCategoryStruct.{u3, u4} C _inst_1)) C (CategoryTheory.CategoryStruct.toQuiver.{u3, u4} C (CategoryTheory.Category.toCategoryStruct.{u3, u4} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u3, u3, u4, u4} 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(CategoryTheory.MonoidalCategory.tensorObj.{u2, u1} M _inst_2 _inst_3 m₁ m₂) m₃)) X))))
+Case conversion may be inaccurate. Consider using '#align category_theory.obj_μ_app CategoryTheory.obj_μ_appₓ'. -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
@[reassoc.1, simp]
@@ -280,6 +410,12 @@ theorem obj_μ_app (m₁ m₂ m₃ : M) (X : C) :
simp
#align category_theory.obj_μ_app CategoryTheory.obj_μ_app
+/- warning: category_theory.obj_μ_inv_app -> CategoryTheory.obj_μ_inv_app is a dubious translation:
+lean 3 declaration is
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+but is expected to have type
+ forall {C : Type.{u4}} [_inst_1 : CategoryTheory.Category.{u3, u4} C] {M : Type.{u1}} [_inst_2 : CategoryTheory.Category.{u2, u1} M] [_inst_3 : CategoryTheory.MonoidalCategory.{u2, u1} M _inst_2] (F : CategoryTheory.MonoidalFunctor.{u2, max u4 u3, u1, max u4 u3} M _inst_2 _inst_3 (CategoryTheory.Functor.{u3, u3, u4, u4} C _inst_1 C _inst_1) (CategoryTheory.Functor.category.{u3, u3, u4, u4} C _inst_1 C _inst_1) (CategoryTheory.endofunctorMonoidalCategory.{u3, u4} C _inst_1)) (m₁ : M) (m₂ : M) (m₃ : M) (X : C), Eq.{succ u3} (Quiver.Hom.{succ u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u3, u4} C (CategoryTheory.Category.toCategoryStruct.{u3, u4} C _inst_1)) (Prefunctor.obj.{succ u3, succ u3, u4, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u3, u4} C (CategoryTheory.Category.toCategoryStruct.{u3, u4} C _inst_1)) C (CategoryTheory.CategoryStruct.toQuiver.{u3, u4} C (CategoryTheory.Category.toCategoryStruct.{u3, u4} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u3, u3, u4, u4} 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+Case conversion may be inaccurate. Consider using '#align category_theory.obj_μ_inv_app CategoryTheory.obj_μ_inv_appₓ'. -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
@[reassoc.1, simp]
@@ -302,6 +438,12 @@ theorem obj_μ_inv_app (m₁ m₂ m₃ : M) (X : C) :
simpa
#align category_theory.obj_μ_inv_app CategoryTheory.obj_μ_inv_app
+/- warning: category_theory.obj_zero_map_μ_app -> CategoryTheory.obj_zero_map_μ_app is a dubious translation:
+lean 3 declaration is
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+but is expected to have type
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+Case conversion may be inaccurate. Consider using '#align category_theory.obj_zero_map_μ_app CategoryTheory.obj_zero_map_μ_appₓ'. -/
@[simp, reassoc.1]
theorem obj_zero_map_μ_app {m : M} {X Y : C} (f : X ⟶ (F.obj m).obj Y) :
(F.obj (𝟙_ M)).map f ≫ (F.μ m (𝟙_ M)).app _ = F.εIso.inv.app _ ≫ f ≫ (F.map (ρ_ m).inv).app _ :=
@@ -310,6 +452,12 @@ theorem obj_zero_map_μ_app {m : M} {X Y : C} (f : X ⟶ (F.obj m).obj Y) :
simp
#align category_theory.obj_zero_map_μ_app CategoryTheory.obj_zero_map_μ_app
+/- warning: category_theory.obj_μ_zero_app -> CategoryTheory.obj_μ_zero_app is a dubious translation:
+lean 3 declaration is
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+but is expected to have type
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+Case conversion may be inaccurate. Consider using '#align category_theory.obj_μ_zero_app CategoryTheory.obj_μ_zero_appₓ'. -/
@[simp]
theorem obj_μ_zero_app (m₁ m₂ : M) (X : C) :
(F.obj m₂).map ((F.μ m₁ (𝟙_ M)).app X) =
@@ -320,6 +468,12 @@ theorem obj_μ_zero_app (m₁ m₂ : M) (X : C) :
congr ; simp
#align category_theory.obj_μ_zero_app CategoryTheory.obj_μ_zero_app
+/- warning: category_theory.unit_of_tensor_iso_unit -> CategoryTheory.unitOfTensorIsoUnit is a dubious translation:
+lean 3 declaration is
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+but is expected to have type
+ forall {C : Type.{u2}} [_inst_1 : CategoryTheory.Category.{u1, u2} C] {M : Type.{u3}} [_inst_2 : CategoryTheory.Category.{u4, u3} M] [_inst_3 : CategoryTheory.MonoidalCategory.{u4, u3} M _inst_2] (F : CategoryTheory.MonoidalFunctor.{u4, max u2 u1, u3, max u2 u1} M _inst_2 _inst_3 (CategoryTheory.Functor.{u1, u1, u2, u2} C _inst_1 C _inst_1) (CategoryTheory.Functor.category.{u1, u1, u2, u2} C _inst_1 C _inst_1) (CategoryTheory.endofunctorMonoidalCategory.{u1, u2} C _inst_1)) (m : M) (n : M), (CategoryTheory.Iso.{u4, u3} M _inst_2 (CategoryTheory.MonoidalCategory.tensorObj.{u4, u3} M _inst_2 _inst_3 m n) (CategoryTheory.MonoidalCategory.tensorUnit.{u4, u3} M _inst_2 _inst_3)) -> (CategoryTheory.Iso.{max u2 u1, max u2 u1} (CategoryTheory.Functor.{u1, u1, u2, u2} C _inst_1 C _inst_1) (CategoryTheory.Functor.category.{u1, u1, u2, u2} C _inst_1 C _inst_1) (CategoryTheory.Functor.comp.{u1, u1, u1, u2, u2, u2} C _inst_1 C _inst_1 C _inst_1 (Prefunctor.obj.{succ u4, max (succ u2) (succ u1), u3, max u2 u1} M (CategoryTheory.CategoryStruct.toQuiver.{u4, u3} M (CategoryTheory.Category.toCategoryStruct.{u4, u3} M _inst_2)) (CategoryTheory.Functor.{u1, u1, u2, u2} C _inst_1 C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max u2 u1, max u2 u1} (CategoryTheory.Functor.{u1, u1, u2, u2} C _inst_1 C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max u2 u1, max u2 u1} (CategoryTheory.Functor.{u1, u1, u2, u2} C _inst_1 C _inst_1) (CategoryTheory.Functor.category.{u1, u1, u2, u2} C _inst_1 C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u4, max u2 u1, u3, max u2 u1} M _inst_2 (CategoryTheory.Functor.{u1, u1, u2, u2} C _inst_1 C _inst_1) (CategoryTheory.Functor.category.{u1, u1, u2, u2} C _inst_1 C _inst_1) (CategoryTheory.LaxMonoidalFunctor.toFunctor.{u4, max u2 u1, u3, max u2 u1} M _inst_2 _inst_3 (CategoryTheory.Functor.{u1, u1, u2, u2} C _inst_1 C _inst_1) (CategoryTheory.Functor.category.{u1, u1, u2, u2} C _inst_1 C _inst_1) (CategoryTheory.endofunctorMonoidalCategory.{u1, u2} C _inst_1) (CategoryTheory.MonoidalFunctor.toLaxMonoidalFunctor.{u4, max u2 u1, u3, max u2 u1} M _inst_2 _inst_3 (CategoryTheory.Functor.{u1, u1, u2, u2} C _inst_1 C _inst_1) (CategoryTheory.Functor.category.{u1, u1, u2, u2} C _inst_1 C _inst_1) (CategoryTheory.endofunctorMonoidalCategory.{u1, u2} C _inst_1) F))) m) (Prefunctor.obj.{succ u4, max (succ u2) (succ u1), u3, max u2 u1} M (CategoryTheory.CategoryStruct.toQuiver.{u4, u3} M (CategoryTheory.Category.toCategoryStruct.{u4, u3} M _inst_2)) (CategoryTheory.Functor.{u1, u1, u2, u2} C _inst_1 C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max u2 u1, max u2 u1} (CategoryTheory.Functor.{u1, u1, u2, u2} C _inst_1 C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max u2 u1, max u2 u1} (CategoryTheory.Functor.{u1, u1, u2, u2} C _inst_1 C _inst_1) (CategoryTheory.Functor.category.{u1, u1, u2, u2} C _inst_1 C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u4, max u2 u1, u3, max u2 u1} M _inst_2 (CategoryTheory.Functor.{u1, u1, u2, u2} C _inst_1 C _inst_1) (CategoryTheory.Functor.category.{u1, u1, u2, u2} C _inst_1 C _inst_1) (CategoryTheory.LaxMonoidalFunctor.toFunctor.{u4, max u2 u1, u3, max u2 u1} M _inst_2 _inst_3 (CategoryTheory.Functor.{u1, u1, u2, u2} C _inst_1 C _inst_1) (CategoryTheory.Functor.category.{u1, u1, u2, u2} C _inst_1 C _inst_1) (CategoryTheory.endofunctorMonoidalCategory.{u1, u2} C _inst_1) (CategoryTheory.MonoidalFunctor.toLaxMonoidalFunctor.{u4, max u2 u1, u3, max u2 u1} M _inst_2 _inst_3 (CategoryTheory.Functor.{u1, u1, u2, u2} C _inst_1 C _inst_1) (CategoryTheory.Functor.category.{u1, u1, u2, u2} C _inst_1 C _inst_1) (CategoryTheory.endofunctorMonoidalCategory.{u1, u2} C _inst_1) F))) n)) (CategoryTheory.Functor.id.{u1, u2} C _inst_1))
+Case conversion may be inaccurate. Consider using '#align category_theory.unit_of_tensor_iso_unit CategoryTheory.unitOfTensorIsoUnitₓ'. -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/-- If `m ⊗ n ≅ 𝟙_M`, then `F.obj m` is a left inverse of `F.obj n`. -/
@[simps]
@@ -327,6 +481,12 @@ noncomputable def unitOfTensorIsoUnit (m n : M) (h : m ⊗ n ≅ 𝟙_ M) : F.ob
F.μIso m n ≪≫ F.toFunctor.mapIso h ≪≫ F.εIso.symm
#align category_theory.unit_of_tensor_iso_unit CategoryTheory.unitOfTensorIsoUnit
+/- warning: category_theory.equiv_of_tensor_iso_unit -> CategoryTheory.equivOfTensorIsoUnit is a dubious translation:
+lean 3 declaration is
+ forall {C : Type.{u2}} [_inst_1 : CategoryTheory.Category.{u1, u2} C] {M : Type.{u3}} [_inst_2 : CategoryTheory.Category.{u4, u3} M] [_inst_3 : CategoryTheory.MonoidalCategory.{u4, u3} M _inst_2], (CategoryTheory.MonoidalFunctor.{u4, max u2 u1, u3, max u1 u2} M _inst_2 _inst_3 (CategoryTheory.Functor.{u1, u1, u2, u2} C _inst_1 C _inst_1) (CategoryTheory.Functor.category.{u1, u1, u2, u2} C _inst_1 C _inst_1) (CategoryTheory.endofunctorMonoidalCategory.{u1, u2} C _inst_1)) -> (forall (m : M) (n : M) (h₁ : CategoryTheory.Iso.{u4, u3} M _inst_2 (CategoryTheory.MonoidalCategory.tensorObj.{u4, u3} M _inst_2 _inst_3 m n) (CategoryTheory.MonoidalCategory.tensorUnit.{u4, u3} M _inst_2 _inst_3)) (h₂ : CategoryTheory.Iso.{u4, u3} M _inst_2 (CategoryTheory.MonoidalCategory.tensorObj.{u4, u3} M _inst_2 _inst_3 n m) (CategoryTheory.MonoidalCategory.tensorUnit.{u4, u3} M _inst_2 _inst_3)), (Eq.{succ u4} (Quiver.Hom.{succ u4, u3} M (CategoryTheory.CategoryStruct.toQuiver.{u4, u3} M (CategoryTheory.Category.toCategoryStruct.{u4, u3} M _inst_2)) (CategoryTheory.MonoidalCategory.tensorObj.{u4, u3} M _inst_2 _inst_3 (CategoryTheory.MonoidalCategory.tensorObj.{u4, u3} M _inst_2 _inst_3 m n) m) m) (CategoryTheory.CategoryStruct.comp.{u4, u3} M (CategoryTheory.Category.toCategoryStruct.{u4, u3} M _inst_2) (CategoryTheory.MonoidalCategory.tensorObj.{u4, u3} M _inst_2 _inst_3 (CategoryTheory.MonoidalCategory.tensorObj.{u4, u3} M _inst_2 _inst_3 m n) m) (CategoryTheory.MonoidalCategory.tensorObj.{u4, u3} M _inst_2 _inst_3 (CategoryTheory.MonoidalCategory.tensorUnit.{u4, u3} M _inst_2 _inst_3) m) m (CategoryTheory.MonoidalCategory.tensorHom.{u4, u3} M _inst_2 _inst_3 (CategoryTheory.MonoidalCategory.tensorObj.{u4, u3} M _inst_2 _inst_3 m n) (CategoryTheory.MonoidalCategory.tensorUnit.{u4, u3} M _inst_2 _inst_3) m m (CategoryTheory.Iso.hom.{u4, u3} M _inst_2 (CategoryTheory.MonoidalCategory.tensorObj.{u4, u3} M _inst_2 _inst_3 m n) (CategoryTheory.MonoidalCategory.tensorUnit.{u4, u3} M _inst_2 _inst_3) h₁) (CategoryTheory.CategoryStruct.id.{u4, u3} M (CategoryTheory.Category.toCategoryStruct.{u4, u3} M _inst_2) m)) (CategoryTheory.Iso.hom.{u4, u3} M _inst_2 (CategoryTheory.MonoidalCategory.tensorObj.{u4, u3} M _inst_2 _inst_3 (CategoryTheory.MonoidalCategory.tensorUnit.{u4, u3} M _inst_2 _inst_3) m) m (CategoryTheory.MonoidalCategory.leftUnitor.{u4, u3} M _inst_2 _inst_3 m))) (CategoryTheory.CategoryStruct.comp.{u4, u3} M (CategoryTheory.Category.toCategoryStruct.{u4, u3} M _inst_2) (CategoryTheory.MonoidalCategory.tensorObj.{u4, u3} M _inst_2 _inst_3 (CategoryTheory.MonoidalCategory.tensorObj.{u4, u3} M _inst_2 _inst_3 m n) m) (CategoryTheory.MonoidalCategory.tensorObj.{u4, u3} M _inst_2 _inst_3 m (CategoryTheory.MonoidalCategory.tensorObj.{u4, u3} M _inst_2 _inst_3 n m)) m (CategoryTheory.Iso.hom.{u4, u3} M _inst_2 (CategoryTheory.MonoidalCategory.tensorObj.{u4, u3} M _inst_2 _inst_3 (CategoryTheory.MonoidalCategory.tensorObj.{u4, u3} M _inst_2 _inst_3 m n) m) (CategoryTheory.MonoidalCategory.tensorObj.{u4, u3} M _inst_2 _inst_3 m (CategoryTheory.MonoidalCategory.tensorObj.{u4, u3} M _inst_2 _inst_3 n m)) (CategoryTheory.MonoidalCategory.associator.{u4, u3} M _inst_2 _inst_3 m n m)) (CategoryTheory.CategoryStruct.comp.{u4, u3} M (CategoryTheory.Category.toCategoryStruct.{u4, u3} M _inst_2) (CategoryTheory.MonoidalCategory.tensorObj.{u4, u3} M _inst_2 _inst_3 m (CategoryTheory.MonoidalCategory.tensorObj.{u4, u3} M _inst_2 _inst_3 n m)) (CategoryTheory.MonoidalCategory.tensorObj.{u4, u3} M _inst_2 _inst_3 m (CategoryTheory.MonoidalCategory.tensorUnit.{u4, u3} M _inst_2 _inst_3)) m (CategoryTheory.MonoidalCategory.tensorHom.{u4, u3} M _inst_2 _inst_3 m m (CategoryTheory.MonoidalCategory.tensorObj.{u4, u3} M _inst_2 _inst_3 n m) (CategoryTheory.MonoidalCategory.tensorUnit.{u4, u3} M _inst_2 _inst_3) (CategoryTheory.CategoryStruct.id.{u4, u3} M (CategoryTheory.Category.toCategoryStruct.{u4, u3} M _inst_2) m) (CategoryTheory.Iso.hom.{u4, u3} M _inst_2 (CategoryTheory.MonoidalCategory.tensorObj.{u4, u3} M _inst_2 _inst_3 n m) (CategoryTheory.MonoidalCategory.tensorUnit.{u4, u3} M _inst_2 _inst_3) h₂)) (CategoryTheory.Iso.hom.{u4, u3} M _inst_2 (CategoryTheory.MonoidalCategory.tensorObj.{u4, u3} M _inst_2 _inst_3 m (CategoryTheory.MonoidalCategory.tensorUnit.{u4, u3} M _inst_2 _inst_3)) m (CategoryTheory.MonoidalCategory.rightUnitor.{u4, u3} M _inst_2 _inst_3 m))))) -> (CategoryTheory.Equivalence.{u1, u1, u2, u2} C _inst_1 C _inst_1))
+but is expected to have type
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+Case conversion may be inaccurate. Consider using '#align category_theory.equiv_of_tensor_iso_unit CategoryTheory.equivOfTensorIsoUnitₓ'. -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
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)
@@ -105,7 +105,6 @@ def tensoringRightMonoidal [MonoidalCategory.{v} C] : MonoidalFunctor C (C ⥤ C
#align category_theory.tensoring_right_monoidal CategoryTheory.tensoringRightMonoidal
variable {C}
-
variable {M : Type*} [Category M] [MonoidalCategory M] (F : MonoidalFunctor M (C ⥤ C))
@[reassoc (attr := simp)]
@@ -302,7 +302,7 @@ theorem obj_μ_zero_app (m₁ m₂ : M) (X : C) :
(F.μ (𝟙_ M) m₂).app ((F.obj m₁).obj X) ≫
(F.map (λ_ m₂).hom).app ((F.obj m₁).obj X) ≫ (F.obj m₂).map ((F.map (ρ_ m₁).inv).app X) := by
rw [← obj_ε_inv_app_assoc, ← Functor.map_comp]
- congr; simp
+ simp
#align category_theory.obj_μ_zero_app CategoryTheory.obj_μ_zero_app
/-- If `m ⊗ n ≅ 𝟙_M`, then `F.obj m` is a left inverse of `F.obj n`. -/
Homogenises porting notes via capitalisation and addition of whitespace.
It makes the following changes:
@@ -87,7 +87,7 @@ attribute [local instance] endofunctorMonoidalCategory
@[simp] theorem endofunctorMonoidalCategory_rightUnitor_inv_app (F : C ⥤ C) (X : C) :
(ρ_ F).inv.app X = 𝟙 _ := rfl
--- porting note: used `dsimp [endofunctorMonoidalCategory]` when necessary instead
+-- Porting note: used `dsimp [endofunctorMonoidalCategory]` when necessary instead
-- attribute [local reducible] endofunctorMonoidalCategory
/-- Tensoring on the right gives a monoidal functor from `C` into endofunctors of `C`.
@@ -203,7 +203,7 @@ theorem left_unitality_app (n : M) (X : C) :
simpa using this.symm
#align category_theory.left_unitality_app CategoryTheory.left_unitality_app
--- porting note: linter claims `simp can prove it`, but cnot
+-- Porting note: linter claims `simp can prove it`, but cnot
@[reassoc (attr := simp, nolint simpNF)]
theorem obj_ε_app (n : M) (X : C) :
(F.obj n).map (F.ε.app X) = (F.map (λ_ n).inv).app X ≫ (F.μIso (𝟙_ M) n).inv.app X := by
@@ -214,7 +214,7 @@ theorem obj_ε_app (n : M) (X : C) :
· simp
#align category_theory.obj_ε_app CategoryTheory.obj_ε_app
--- porting note: linter claims `simp can prove it`, but cnot
+-- Porting note: linter claims `simp can prove it`, but cnot
@[reassoc (attr := simp, nolint simpNF)]
theorem obj_ε_inv_app (n : M) (X : C) :
(F.obj n).map (F.εIso.inv.app X) = (F.μ (𝟙_ M) n).app X ≫ (F.map (λ_ n).hom).app X := by
@@ -256,7 +256,7 @@ theorem associativity_app (m₁ m₂ m₃ : M) (X : C) :
simpa using this
#align category_theory.associativity_app CategoryTheory.associativity_app
--- porting note: linter claims `simp can prove it`, but cnot
+-- Porting note: linter claims `simp can prove it`, but cnot
@[reassoc (attr := simp, nolint simpNF)]
theorem obj_μ_app (m₁ m₂ m₃ : M) (X : C) :
(F.obj m₃).map ((F.μ m₁ m₂).app X) =
@@ -267,7 +267,7 @@ theorem obj_μ_app (m₁ m₂ m₃ : M) (X : C) :
simp
#align category_theory.obj_μ_app CategoryTheory.obj_μ_app
--- porting note: linter claims `simp can prove it`, but cnot
+-- Porting note: linter claims `simp can prove it`, but cnot
@[reassoc (attr := simp, nolint simpNF)]
theorem obj_μ_inv_app (m₁ m₂ m₃ : M) (X : C) :
(F.obj m₃).map ((F.μIso m₁ m₂).inv.app X) =
@@ -90,8 +90,6 @@ attribute [local instance] endofunctorMonoidalCategory
-- porting note: used `dsimp [endofunctorMonoidalCategory]` when necessary instead
-- attribute [local reducible] endofunctorMonoidalCategory
-attribute [local simp] id_tensorHom tensorHom_id in
-
/-- Tensoring on the right gives a monoidal functor from `C` into endofunctors of `C`.
-/
@[simps!]
@@ -168,23 +166,23 @@ theorem μ_naturality₂ {m n m' n' : M} (f : m ⟶ m') (g : n ⟶ n') (X : C) :
@[reassoc (attr := simp)]
theorem μ_naturalityₗ {m n m' : M} (f : m ⟶ m') (X : C) :
(F.obj n).map ((F.map f).app X) ≫ (F.μ m' n).app X =
- (F.μ m n).app X ≫ (F.map (f ⊗ 𝟙 n)).app X := by
- rw [← μ_naturality₂ F f (𝟙 n) X]
+ (F.μ m n).app X ≫ (F.map (f ▷ n)).app X := by
+ rw [← tensorHom_id, ← μ_naturality₂ F f (𝟙 n) X]
simp
#align category_theory.μ_naturalityₗ CategoryTheory.μ_naturalityₗ
@[reassoc (attr := simp)]
theorem μ_naturalityᵣ {m n n' : M} (g : n ⟶ n') (X : C) :
(F.map g).app ((F.obj m).obj X) ≫ (F.μ m n').app X =
- (F.μ m n).app X ≫ (F.map (𝟙 m ⊗ g)).app X := by
- rw [← μ_naturality₂ F (𝟙 m) g X]
+ (F.μ m n).app X ≫ (F.map (m ◁ g)).app X := by
+ rw [← id_tensorHom, ← μ_naturality₂ F (𝟙 m) g X]
simp
#align category_theory.μ_naturalityᵣ CategoryTheory.μ_naturalityᵣ
@[reassoc (attr := simp)]
theorem μ_inv_naturalityₗ {m n m' : M} (f : m ⟶ m') (X : C) :
(F.μIso m n).inv.app X ≫ (F.obj n).map ((F.map f).app X) =
- (F.map (f ⊗ 𝟙 n)).app X ≫ (F.μIso m' n).inv.app X := by
+ (F.map (f ▷ n)).app X ≫ (F.μIso m' n).inv.app X := by
rw [← IsIso.comp_inv_eq, Category.assoc, ← IsIso.eq_inv_comp]
simp
#align category_theory.μ_inv_naturalityₗ CategoryTheory.μ_inv_naturalityₗ
@@ -192,7 +190,7 @@ theorem μ_inv_naturalityₗ {m n m' : M} (f : m ⟶ m') (X : C) :
@[reassoc (attr := simp)]
theorem μ_inv_naturalityᵣ {m n n' : M} (g : n ⟶ n') (X : C) :
(F.μIso m n).inv.app X ≫ (F.map g).app ((F.obj m).obj X) =
- (F.map (𝟙 m ⊗ g)).app X ≫ (F.μIso m n').inv.app X := by
+ (F.map (m ◁ g)).app X ≫ (F.μIso m n').inv.app X := by
rw [← IsIso.comp_inv_eq, Category.assoc, ← IsIso.eq_inv_comp]
simp
#align category_theory.μ_inv_naturalityᵣ CategoryTheory.μ_inv_naturalityᵣ
@@ -317,7 +315,7 @@ noncomputable def unitOfTensorIsoUnit (m n : M) (h : m ⊗ n ≅ 𝟙_ M) : F.ob
then `F.obj m` and `F.obj n` forms a self-equivalence of `C`. -/
@[simps]
noncomputable def equivOfTensorIsoUnit (m n : M) (h₁ : m ⊗ n ≅ 𝟙_ M) (h₂ : n ⊗ m ≅ 𝟙_ M)
- (H : (h₁.hom ⊗ 𝟙 m) ≫ (λ_ m).hom = (α_ m n m).hom ≫ (𝟙 m ⊗ h₂.hom) ≫ (ρ_ m).hom) : C ≌ C
+ (H : h₁.hom ▷ m ≫ (λ_ m).hom = (α_ m n m).hom ≫ m ◁ h₂.hom ≫ (ρ_ m).hom) : C ≌ C
where
functor := F.obj m
inverse := F.obj n
Extracted from #6307. The main reason why #6307 is so large is that many tensoring of identity morphisms that appear in mathlib should be replaced with whiskerings. This PR will leave this issue and deal with other parts. That is, we do not set id_tensorHom
and tensorHom_id
as simple lemmas at this moment, We can set them as simp lemmas locally to enable simple normal forms.
@@ -90,6 +90,8 @@ attribute [local instance] endofunctorMonoidalCategory
-- porting note: used `dsimp [endofunctorMonoidalCategory]` when necessary instead
-- attribute [local reducible] endofunctorMonoidalCategory
+attribute [local simp] id_tensorHom tensorHom_id in
+
/-- Tensoring on the right gives a monoidal functor from `C` into endofunctors of `C`.
-/
@[simps!]
@@ -97,30 +99,10 @@ def tensoringRightMonoidal [MonoidalCategory.{v} C] : MonoidalFunctor C (C ⥤ C
{ tensoringRight C with
ε := (rightUnitorNatIso C).inv
μ := fun X Y => { app := fun Z => (α_ Z X Y).hom }
- -- The proof will be automated after merging #6307.
- μ_natural_left := fun f X => by
- ext Z
- dsimp
- simp only [← id_tensor_comp_tensor_id f (𝟙 X), id_tensor_comp, ← tensor_id, Category.assoc,
- associator_naturality, associator_naturality_assoc]
- simp only [tensor_id, Category.id_comp]
- μ_natural_right := fun X g => by
- ext Z
- dsimp
- simp only [← id_tensor_comp_tensor_id (𝟙 X) g, id_tensor_comp, ← tensor_id, Category.assoc,
- associator_naturality, associator_naturality_assoc]
- simp only [tensor_id, Category.comp_id]
- associativity := fun X Y Z => by
- ext W
- simp [pentagon]
μ_isIso := fun X Y =>
-- We could avoid needing to do this explicitly by
-- constructing a partially applied analogue of `associatorNatIso`.
- ⟨⟨{ app := fun Z => (α_ Z X Y).inv
- naturality := fun Z Z' f => by
- dsimp
- rw [← associator_inv_naturality]
- simp },
+ ⟨⟨{ app := fun Z => (α_ Z X Y).inv },
by aesop_cat⟩⟩ }
#align category_theory.tensoring_right_monoidal CategoryTheory.tensoringRightMonoidal
@@ -97,11 +97,19 @@ def tensoringRightMonoidal [MonoidalCategory.{v} C] : MonoidalFunctor C (C ⥤ C
{ tensoringRight C with
ε := (rightUnitorNatIso C).inv
μ := fun X Y => { app := fun Z => (α_ Z X Y).hom }
- μ_natural := fun f g => by
+ -- The proof will be automated after merging #6307.
+ μ_natural_left := fun f X => by
ext Z
dsimp
- simp only [← id_tensor_comp_tensor_id g f, id_tensor_comp, ← tensor_id, Category.assoc,
+ simp only [← id_tensor_comp_tensor_id f (𝟙 X), id_tensor_comp, ← tensor_id, Category.assoc,
associator_naturality, associator_naturality_assoc]
+ simp only [tensor_id, Category.id_comp]
+ μ_natural_right := fun X g => by
+ ext Z
+ dsimp
+ simp only [← id_tensor_comp_tensor_id (𝟙 X) g, id_tensor_comp, ← tensor_id, Category.assoc,
+ associator_naturality, associator_naturality_assoc]
+ simp only [tensor_id, Category.comp_id]
associativity := fun X Y Z => by
ext W
simp [pentagon]
MonoidalCategory
into a Struct
class (#7279)
This matches the approach for CategoryStruct
, and allows us to use the notation within MonoidalCategory
.
It also makes it easier to induce the lawful structure along a faithful functor, as again it means by the time we are providing the proof fields, the notation is already available.
This also eliminates tensorUnit
vs tensorUnit'
, adding a custom pretty-printer to provide the unprimed version with appropriate notation.
@@ -35,7 +35,7 @@ def endofunctorMonoidalCategory : MonoidalCategory (C ⥤ C) where
whiskerLeft X _ _ F := whiskerLeft X F
whiskerRight F X := whiskerRight F X
tensorHom α β := α ◫ β
- tensorUnit' := 𝟭 C
+ tensorUnit := 𝟭 C
associator F G H := Functor.associator F G H
leftUnitor F := Functor.leftUnitor F
rightUnitor F := Functor.rightUnitor F
@@ -61,6 +61,14 @@ attribute [local instance] endofunctorMonoidalCategory
{F G H K : C ⥤ C} {α : F ⟶ G} {β : H ⟶ K} (X : C) :
(α ⊗ β).app X = β.app (F.obj X) ≫ K.map (α.app X) := rfl
+@[simp] theorem endofunctorMonoidalCategory_whiskerLeft_app
+ {F H K : C ⥤ C} {β : H ⟶ K} (X : C) :
+ (F ◁ β).app X = β.app (F.obj X) := rfl
+
+@[simp] theorem endofunctorMonoidalCategory_whiskerRight_app
+ {F G H : C ⥤ C} {α : F ⟶ G} (X : C) :
+ (α ▷ H).app X = H.map (α.app X) := rfl
+
@[simp] theorem endofunctorMonoidalCategory_associator_hom_app (F G H : C ⥤ C) (X : C) :
(α_ F G H).hom.app X = 𝟙 _ := rfl
Type _
and Sort _
(#6499)
We remove all possible occurences of Type _
and Sort _
in favor of Type*
and Sort*
.
This has nice performance benefits.
@@ -110,7 +110,7 @@ def tensoringRightMonoidal [MonoidalCategory.{v} C] : MonoidalFunctor C (C ⥤ C
variable {C}
-variable {M : Type _} [Category M] [MonoidalCategory M] (F : MonoidalFunctor M (C ⥤ C))
+variable {M : Type*} [Category M] [MonoidalCategory M] (F : MonoidalFunctor M (C ⥤ C))
@[reassoc (attr := simp)]
theorem μ_hom_inv_app (i j : M) (X : C) : (F.μ i j).app X ≫ (F.μIso i j).inv.app X = 𝟙 _ :=
@@ -32,6 +32,8 @@ with tensor product given by composition of functors
-/
def endofunctorMonoidalCategory : MonoidalCategory (C ⥤ C) where
tensorObj F G := F ⋙ G
+ whiskerLeft X _ _ F := whiskerLeft X F
+ whiskerRight F X := whiskerRight F X
tensorHom α β := α ◫ β
tensorUnit' := 𝟭 C
associator F G H := Functor.associator F G H
@@ -2,14 +2,11 @@
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Andrew Yang
-
-! This file was ported from Lean 3 source module category_theory.monoidal.End
-! leanprover-community/mathlib commit 85075bccb68ab7fa49fb05db816233fb790e4fe9
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
-/
import Mathlib.CategoryTheory.Monoidal.Functor
+#align_import category_theory.monoidal.End from "leanprover-community/mathlib"@"85075bccb68ab7fa49fb05db816233fb790e4fe9"
+
/-!
# Endofunctors as a monoidal category.
@@ -89,7 +89,7 @@ attribute [local instance] endofunctorMonoidalCategory
def tensoringRightMonoidal [MonoidalCategory.{v} C] : MonoidalFunctor C (C ⥤ C) :=
{ tensoringRight C with
ε := (rightUnitorNatIso C).inv
- μ := fun X Y => { app := fun Z => (α_ Z X Y).hom }
+ μ := fun X Y => { app := fun Z => (α_ Z X Y).hom }
μ_natural := fun f g => by
ext Z
dsimp
@@ -46,6 +46,40 @@ open CategoryTheory.MonoidalCategory
attribute [local instance] endofunctorMonoidalCategory
+@[simp] theorem endofunctorMonoidalCategory_tensorUnit_obj (X : C) :
+ (𝟙_ (C ⥤ C)).obj X = X := rfl
+
+@[simp] theorem endofunctorMonoidalCategory_tensorUnit_map {X Y : C} (f : X ⟶ Y) :
+ (𝟙_ (C ⥤ C)).map f = f := rfl
+
+@[simp] theorem endofunctorMonoidalCategory_tensorObj_obj (F G : C ⥤ C) (X : C) :
+ (F ⊗ G).obj X = G.obj (F.obj X) := rfl
+
+@[simp] theorem endofunctorMonoidalCategory_tensorObj_map (F G : C ⥤ C) {X Y : C} (f : X ⟶ Y) :
+ (F ⊗ G).map f = G.map (F.map f) := rfl
+
+@[simp] theorem endofunctorMonoidalCategory_tensorMap_app
+ {F G H K : C ⥤ C} {α : F ⟶ G} {β : H ⟶ K} (X : C) :
+ (α ⊗ β).app X = β.app (F.obj X) ≫ K.map (α.app X) := rfl
+
+@[simp] theorem endofunctorMonoidalCategory_associator_hom_app (F G H : C ⥤ C) (X : C) :
+ (α_ F G H).hom.app X = 𝟙 _ := rfl
+
+@[simp] theorem endofunctorMonoidalCategory_associator_inv_app (F G H : C ⥤ C) (X : C) :
+ (α_ F G H).inv.app X = 𝟙 _ := rfl
+
+@[simp] theorem endofunctorMonoidalCategory_leftUnitor_hom_app (F : C ⥤ C) (X : C) :
+ (λ_ F).hom.app X = 𝟙 _ := rfl
+
+@[simp] theorem endofunctorMonoidalCategory_leftUnitor_inv_app (F : C ⥤ C) (X : C) :
+ (λ_ F).inv.app X = 𝟙 _ := rfl
+
+@[simp] theorem endofunctorMonoidalCategory_rightUnitor_hom_app (F : C ⥤ C) (X : C) :
+ (ρ_ F).hom.app X = 𝟙 _ := rfl
+
+@[simp] theorem endofunctorMonoidalCategory_rightUnitor_inv_app (F : C ⥤ C) (X : C) :
+ (ρ_ F).inv.app X = 𝟙 _ := rfl
+
-- porting note: used `dsimp [endofunctorMonoidalCategory]` when necessary instead
-- attribute [local reducible] endofunctorMonoidalCategory
@@ -53,41 +87,25 @@ attribute [local instance] endofunctorMonoidalCategory
-/
@[simps!]
def tensoringRightMonoidal [MonoidalCategory.{v} C] : MonoidalFunctor C (C ⥤ C) :=
- {-- We could avoid needing to do this explicitly by
- -- constructing a partially applied analogue of `associatorNatIso`.
- tensoringRight C with
+ { tensoringRight C with
ε := (rightUnitorNatIso C).inv
- μ := fun X Y =>
- { app := fun Z => (α_ Z X Y).hom
- naturality := fun Z Z' f => by
- dsimp [endofunctorMonoidalCategory]
- rw [associator_naturality]
- simp }
+ μ := fun X Y => { app := fun Z => (α_ Z X Y).hom }
μ_natural := fun f g => by
ext Z
- dsimp [endofunctorMonoidalCategory]
+ dsimp
simp only [← id_tensor_comp_tensor_id g f, id_tensor_comp, ← tensor_id, Category.assoc,
associator_naturality, associator_naturality_assoc]
associativity := fun X Y Z => by
ext W
- dsimp [endofunctorMonoidalCategory]
simp [pentagon]
- left_unitality := fun X => by
- ext Y
- dsimp [endofunctorMonoidalCategory]
- rw [Category.id_comp, triangle, ← tensor_comp]
- aesop_cat
- right_unitality := fun X => by
- ext Y; dsimp [endofunctorMonoidalCategory]
- rw [tensor_id, Category.comp_id, rightUnitor_tensor_inv, Category.assoc,
- Iso.inv_hom_id_assoc, ← id_tensor_comp, Iso.inv_hom_id, tensor_id]
- ε_isIso := by infer_instance
μ_isIso := fun X Y =>
- ⟨⟨{ app := fun Z => (α_ Z X Y).inv
- naturality := fun Z Z' f => by
- dsimp [endofunctorMonoidalCategory]
- rw [← associator_inv_naturality]
- simp },
+ -- We could avoid needing to do this explicitly by
+ -- constructing a partially applied analogue of `associatorNatIso`.
+ ⟨⟨{ app := fun Z => (α_ Z X Y).inv
+ naturality := fun Z Z' f => by
+ dsimp
+ rw [← associator_inv_naturality]
+ simp },
by aesop_cat⟩⟩ }
#align category_theory.tensoring_right_monoidal CategoryTheory.tensoringRightMonoidal
@@ -146,7 +164,7 @@ theorem μ_naturality₂ {m n m' n' : M} (f : m ⟶ m') (g : n ⟶ n') (X : C) :
(F.map g).app ((F.obj m).obj X) ≫ (F.obj n').map ((F.map f).app X) ≫ (F.μ m' n').app X =
(F.μ m n).app X ≫ (F.map (f ⊗ g)).app X := by
have := congr_app (F.toLaxMonoidalFunctor.μ_natural f g) X
- dsimp [endofunctorMonoidalCategory] at this
+ dsimp at this
simpa using this
#align category_theory.μ_naturality₂ CategoryTheory.μ_naturality₂
@@ -186,7 +204,7 @@ theorem μ_inv_naturalityᵣ {m n n' : M} (g : n ⟶ n') (X : C) :
theorem left_unitality_app (n : M) (X : C) :
(F.obj n).map (F.ε.app X) ≫ (F.μ (𝟙_ M) n).app X ≫ (F.map (λ_ n).hom).app X = 𝟙 _ := by
have := congr_app (F.toLaxMonoidalFunctor.left_unitality n) X
- dsimp [endofunctorMonoidalCategory] at this
+ dsimp at this
simpa using this.symm
#align category_theory.left_unitality_app CategoryTheory.left_unitality_app
@@ -207,14 +225,13 @@ theorem obj_ε_inv_app (n : M) (X : C) :
(F.obj n).map (F.εIso.inv.app X) = (F.μ (𝟙_ M) n).app X ≫ (F.map (λ_ n).hom).app X := by
rw [← cancel_mono ((F.obj n).map (F.ε.app X)), ← Functor.map_comp]
simp
- rfl
#align category_theory.obj_ε_inv_app CategoryTheory.obj_ε_inv_app
@[reassoc]
theorem right_unitality_app (n : M) (X : C) :
F.ε.app ((F.obj n).obj X) ≫ (F.μ n (𝟙_ M)).app X ≫ (F.map (ρ_ n).hom).app X = 𝟙 _ := by
have := congr_app (F.toLaxMonoidalFunctor.right_unitality n) X
- dsimp [endofunctorMonoidalCategory] at this
+ dsimp at this
simpa using this.symm
#align category_theory.right_unitality_app CategoryTheory.right_unitality_app
@@ -232,7 +249,6 @@ theorem ε_inv_app_obj (n : M) (X : C) :
F.εIso.inv.app ((F.obj n).obj X) = (F.μ n (𝟙_ M)).app X ≫ (F.map (ρ_ n).hom).app X := by
rw [← cancel_mono (F.ε.app ((F.obj n).obj X)), ε_inv_hom_app]
simp
- rfl
#align category_theory.ε_inv_app_obj CategoryTheory.ε_inv_app_obj
@[reassoc]
@@ -241,7 +257,7 @@ theorem associativity_app (m₁ m₂ m₃ : M) (X : C) :
(F.μ (m₁ ⊗ m₂) m₃).app X ≫ (F.map (α_ m₁ m₂ m₃).hom).app X =
(F.μ m₂ m₃).app ((F.obj m₁).obj X) ≫ (F.μ m₁ (m₂ ⊗ m₃)).app X := by
have := congr_app (F.toLaxMonoidalFunctor.associativity m₁ m₂ m₃) X
- dsimp [endofunctorMonoidalCategory] at this
+ dsimp at this
simpa using this
#align category_theory.associativity_app CategoryTheory.associativity_app
@@ -253,9 +269,6 @@ theorem obj_μ_app (m₁ m₂ m₃ : M) (X : C) :
(F.μ m₁ (m₂ ⊗ m₃)).app X ≫
(F.map (α_ m₁ m₂ m₃).inv).app X ≫ (F.μIso (m₁ ⊗ m₂) m₃).inv.app X := by
rw [← associativity_app_assoc]
- dsimp [endofunctorMonoidalCategory]
- simp
- dsimp [endofunctorMonoidalCategory]
simp
#align category_theory.obj_μ_app CategoryTheory.obj_μ_app
@@ -277,7 +290,6 @@ theorem obj_μ_inv_app (m₁ m₂ m₃ : M) (X : C) :
simp
· refine' IsIso.inv_eq_of_hom_inv_id _
simp
- rfl
#align category_theory.obj_μ_inv_app CategoryTheory.obj_μ_inv_app
@[reassoc (attr := simp)]
by
s! (#3825)
This PR puts, with one exception, every single remaining by
that lies all by itself on its own line to the previous line, thus matching the current behaviour of start-port.sh
. The exception is when the by
begins the second or later argument to a tuple or anonymous constructor; see https://github.com/leanprover-community/mathlib4/pull/3825#discussion_r1186702599.
Essentially this is s/\n *by$/ by/g
, but with manual editing to satisfy the linter's max-100-char-line requirement. The Python style linter is also modified to catch these "isolated by
s".
@@ -290,13 +290,12 @@ theorem obj_zero_map_μ_app {m : M} {X Y : C} (f : X ⟶ (F.obj m).obj Y) :
@[simp]
theorem obj_μ_zero_app (m₁ m₂ : M) (X : C) :
- (F.μ (𝟙_ M) m₂).app ((F.obj m₁).obj X) ≫ (F.μ m₁ (𝟙_ M ⊗ m₂)).app X ≫
- (F.map (α_ m₁ (𝟙_ M) m₂).inv).app X ≫ (F.μIso (m₁ ⊗ 𝟙_ M) m₂).inv.app X =
- (F.μ (𝟙_ M) m₂).app ((F.obj m₁).obj X) ≫
- (F.map (λ_ m₂).hom).app ((F.obj m₁).obj X) ≫ (F.obj m₂).map ((F.map (ρ_ m₁).inv).app X) :=
- by
+ (F.μ (𝟙_ M) m₂).app ((F.obj m₁).obj X) ≫ (F.μ m₁ (𝟙_ M ⊗ m₂)).app X ≫
+ (F.map (α_ m₁ (𝟙_ M) m₂).inv).app X ≫ (F.μIso (m₁ ⊗ 𝟙_ M) m₂).inv.app X =
+ (F.μ (𝟙_ M) m₂).app ((F.obj m₁).obj X) ≫
+ (F.map (λ_ m₂).hom).app ((F.obj m₁).obj X) ≫ (F.obj m₂).map ((F.map (ρ_ m₁).inv).app X) := by
rw [← obj_ε_inv_app_assoc, ← Functor.map_comp]
- congr ; simp
+ congr; simp
#align category_theory.obj_μ_zero_app CategoryTheory.obj_μ_zero_app
/-- If `m ⊗ n ≅ 𝟙_M`, then `F.obj m` is a left inverse of `F.obj n`. -/
@@ -33,10 +33,9 @@ variable (C : Type u) [Category.{v} C]
with tensor product given by composition of functors
(and horizontal composition of natural transformations).
-/
-def endofunctorMonoidalCategory : MonoidalCategory (C ⥤ C)
- where
+def endofunctorMonoidalCategory : MonoidalCategory (C ⥤ C) where
tensorObj F G := F ⋙ G
- tensorHom := @fun F G F' G' α β => α ◫ β
+ tensorHom α β := α ◫ β
tensorUnit' := 𝟭 C
associator F G H := Functor.associator F G H
leftUnitor F := Functor.leftUnitor F
@@ -56,8 +55,7 @@ attribute [local instance] endofunctorMonoidalCategory
def tensoringRightMonoidal [MonoidalCategory.{v} C] : MonoidalFunctor C (C ⥤ C) :=
{-- We could avoid needing to do this explicitly by
-- constructing a partially applied analogue of `associatorNatIso`.
- tensoringRight
- C with
+ tensoringRight C with
ε := (rightUnitorNatIso C).inv
μ := fun X Y =>
{ app := fun Z => (α_ Z X Y).hom
@@ -65,7 +63,7 @@ def tensoringRightMonoidal [MonoidalCategory.{v} C] : MonoidalFunctor C (C ⥤ C
dsimp [endofunctorMonoidalCategory]
rw [associator_naturality]
simp }
- μ_natural := @fun X Y X' Y' f g => by
+ μ_natural := fun f g => by
ext Z
dsimp [endofunctorMonoidalCategory]
simp only [← id_tensor_comp_tensor_id g f, id_tensor_comp, ← tensor_id, Category.assoc,
@@ -134,7 +132,7 @@ theorem μ_naturality {m n : M} {X Y : C} (f : X ⟶ Y) :
(F.toLaxMonoidalFunctor.μ m n).naturality f
#align category_theory.μ_naturality CategoryTheory.μ_naturality
--- This is a simp lemma in the reverse direction via `nat_trans.naturality`.
+-- This is a simp lemma in the reverse direction via `NatTrans.naturality`.
@[reassoc]
theorem μ_inv_naturality {m n : M} {X Y : C} (f : X ⟶ Y) :
(F.μIso m n).inv.app X ≫ (F.obj n).map ((F.obj m).map f) =
congr!
and convert
(#2606)
congr!
, convert
, and convert_to
to control parts of the congruence algorithm, in particular transparency settings when applying congruence lemmas.congr!
now applies congruence lemmas with reducible transparency by default. This prevents it from unfolding definitions when applying congruence lemmas. It also now tries both the LHS-biased and RHS-biased simp congruence lemmas, with a configuration option to set which it should try first.HEq
congruence lemma generator that gives each hypothesis access to the proofs of previous hypotheses. This means that if you have an equality ⊢ ⟨a, x⟩ = ⟨b, y⟩
of sigma types, congr!
turns this into goals ⊢ a = b
and ⊢ a = b → HEq x y
(note that congr!
will also auto-introduce a = b
for you in the second goal). This congruence lemma generator applies to more cases than the simp congruence lemma generator does.congr!
(and hence convert
) are more careful about applying lemmas that don't force definitions to unfold. There were a number of cases in mathlib where the implementation of congr
was being abused to unfold definitions.set_option trace.congr! true
you can see what congr!
sees when it is deciding on congruence lemmas.convert_to
to do using 1
when there is no using
clause, to match its documentation.Note that congr!
is more capable than congr
at finding a way to equate left-hand sides and right-hand sides, so you will frequently need to limit its depth with a using
clause. However, there is also a new heuristic to prevent considering unlikely-to-be-provable type equalities (controlled by the typeEqs
option), which can help limit the depth automatically.
There is also a predefined configuration that you can invoke with, for example, convert (config := .unfoldSameFun) h
, that causes it to behave more like congr
, including using default transparency when unfolding.
@@ -225,9 +225,8 @@ theorem ε_app_obj (n : M) (X : C) :
F.ε.app ((F.obj n).obj X) = (F.map (ρ_ n).inv).app X ≫ (F.μIso n (𝟙_ M)).inv.app X := by
refine' Eq.trans _ (Category.id_comp _)
rw [← Category.assoc, ← IsIso.comp_inv_eq, ← IsIso.comp_inv_eq, Category.assoc]
- convert right_unitality_app F n X
- · simp
- · simp
+ convert right_unitality_app F n X using 1
+ simp
#align category_theory.ε_app_obj CategoryTheory.ε_app_obj
@[simp]
All dependencies are ported!