category_theory.sums.associator
⟷
Mathlib.CategoryTheory.Sums.Associator
The following section lists changes to this file in mathlib3 and mathlib4 that occured after the initial port. Most recent changes are shown first. Hovering over a commit will show all commits associated with the same mathlib3 commit.
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mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83
@@ -164,14 +164,15 @@ def associativity : Sum (Sum C D) E ≌ Sum C (Sum D E) :=
-/
#print CategoryTheory.sum.associatorIsEquivalence /-
-instance associatorIsEquivalence : IsEquivalence (associator C D E) :=
- (by infer_instance : IsEquivalence (associativity C D E).Functor)
+instance associatorIsEquivalence : CategoryTheory.Functor.IsEquivalence (associator C D E) :=
+ (by infer_instance : CategoryTheory.Functor.IsEquivalence (associativity C D E).Functor)
#align category_theory.sum.associator_is_equivalence CategoryTheory.sum.associatorIsEquivalence
-/
#print CategoryTheory.sum.inverseAssociatorIsEquivalence /-
-instance inverseAssociatorIsEquivalence : IsEquivalence (inverseAssociator C D E) :=
- (by infer_instance : IsEquivalence (associativity C D E).inverse)
+instance inverseAssociatorIsEquivalence :
+ CategoryTheory.Functor.IsEquivalence (inverseAssociator C D E) :=
+ (by infer_instance : CategoryTheory.Functor.IsEquivalence (associativity C D E).inverse)
#align category_theory.sum.inverse_associator_is_equivalence CategoryTheory.sum.inverseAssociatorIsEquivalence
-/
mathlib commit https://github.com/leanprover-community/mathlib/commit/ce64cd319bb6b3e82f31c2d38e79080d377be451
@@ -3,7 +3,7 @@ Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
-import Mathbin.CategoryTheory.Sums.Basic
+import CategoryTheory.Sums.Basic
#align_import category_theory.sums.associator from "leanprover-community/mathlib"@"ee05e9ce1322178f0c12004eb93c00d2c8c00ed2"
mathlib commit https://github.com/leanprover-community/mathlib/commit/8ea5598db6caeddde6cb734aa179cc2408dbd345
@@ -2,14 +2,11 @@
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-
-! This file was ported from Lean 3 source module category_theory.sums.associator
-! leanprover-community/mathlib commit ee05e9ce1322178f0c12004eb93c00d2c8c00ed2
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
-/
import Mathbin.CategoryTheory.Sums.Basic
+#align_import category_theory.sums.associator from "leanprover-community/mathlib"@"ee05e9ce1322178f0c12004eb93c00d2c8c00ed2"
+
/-!
# Associator for binary disjoint union of categories.
mathlib commit https://github.com/leanprover-community/mathlib/commit/9fb8964792b4237dac6200193a0d533f1b3f7423
@@ -48,37 +48,49 @@ def associator : Sum (Sum C D) E ⥤ Sum C (Sum D E)
#align category_theory.sum.associator CategoryTheory.sum.associator
-/
+#print CategoryTheory.sum.associator_obj_inl_inl /-
@[simp]
theorem associator_obj_inl_inl (X) : (associator C D E).obj (inl (inl X)) = inl X :=
rfl
#align category_theory.sum.associator_obj_inl_inl CategoryTheory.sum.associator_obj_inl_inl
+-/
+#print CategoryTheory.sum.associator_obj_inl_inr /-
@[simp]
theorem associator_obj_inl_inr (X) : (associator C D E).obj (inl (inr X)) = inr (inl X) :=
rfl
#align category_theory.sum.associator_obj_inl_inr CategoryTheory.sum.associator_obj_inl_inr
+-/
+#print CategoryTheory.sum.associator_obj_inr /-
@[simp]
theorem associator_obj_inr (X) : (associator C D E).obj (inr X) = inr (inr X) :=
rfl
#align category_theory.sum.associator_obj_inr CategoryTheory.sum.associator_obj_inr
+-/
+#print CategoryTheory.sum.associator_map_inl_inl /-
@[simp]
theorem associator_map_inl_inl {X Y : C} (f : inl (inl X) ⟶ inl (inl Y)) :
(associator C D E).map f = f :=
rfl
#align category_theory.sum.associator_map_inl_inl CategoryTheory.sum.associator_map_inl_inl
+-/
+#print CategoryTheory.sum.associator_map_inl_inr /-
@[simp]
theorem associator_map_inl_inr {X Y : D} (f : inl (inr X) ⟶ inl (inr Y)) :
(associator C D E).map f = f :=
rfl
#align category_theory.sum.associator_map_inl_inr CategoryTheory.sum.associator_map_inl_inr
+-/
+#print CategoryTheory.sum.associator_map_inr /-
@[simp]
theorem associator_map_inr {X Y : E} (f : inr X ⟶ inr Y) : (associator C D E).map f = f :=
rfl
#align category_theory.sum.associator_map_inr CategoryTheory.sum.associator_map_inr
+-/
#print CategoryTheory.sum.inverseAssociator /-
/-- The inverse associator functor `C ⊕ (D ⊕ E) ⥤ (C ⊕ D) ⊕ E` for sums of categories.
@@ -98,40 +110,53 @@ def inverseAssociator : Sum C (Sum D E) ⥤ Sum (Sum C D) E
#align category_theory.sum.inverse_associator CategoryTheory.sum.inverseAssociator
-/
+#print CategoryTheory.sum.inverseAssociator_obj_inl /-
@[simp]
theorem inverseAssociator_obj_inl (X) : (inverseAssociator C D E).obj (inl X) = inl (inl X) :=
rfl
#align category_theory.sum.inverse_associator_obj_inl CategoryTheory.sum.inverseAssociator_obj_inl
+-/
+#print CategoryTheory.sum.inverseAssociator_obj_inr_inl /-
@[simp]
theorem inverseAssociator_obj_inr_inl (X) :
(inverseAssociator C D E).obj (inr (inl X)) = inl (inr X) :=
rfl
#align category_theory.sum.inverse_associator_obj_inr_inl CategoryTheory.sum.inverseAssociator_obj_inr_inl
+-/
+#print CategoryTheory.sum.inverseAssociator_obj_inr_inr /-
@[simp]
theorem inverseAssociator_obj_inr_inr (X) : (inverseAssociator C D E).obj (inr (inr X)) = inr X :=
rfl
#align category_theory.sum.inverse_associator_obj_inr_inr CategoryTheory.sum.inverseAssociator_obj_inr_inr
+-/
+#print CategoryTheory.sum.inverseAssociator_map_inl /-
@[simp]
theorem inverseAssociator_map_inl {X Y : C} (f : inl X ⟶ inl Y) :
(inverseAssociator C D E).map f = f :=
rfl
#align category_theory.sum.inverse_associator_map_inl CategoryTheory.sum.inverseAssociator_map_inl
+-/
+#print CategoryTheory.sum.inverseAssociator_map_inr_inl /-
@[simp]
theorem inverseAssociator_map_inr_inl {X Y : D} (f : inr (inl X) ⟶ inr (inl Y)) :
(inverseAssociator C D E).map f = f :=
rfl
#align category_theory.sum.inverse_associator_map_inr_inl CategoryTheory.sum.inverseAssociator_map_inr_inl
+-/
+#print CategoryTheory.sum.inverseAssociator_map_inr_inr /-
@[simp]
theorem inverseAssociator_map_inr_inr {X Y : E} (f : inr (inr X) ⟶ inr (inr Y)) :
(inverseAssociator C D E).map f = f :=
rfl
#align category_theory.sum.inverse_associator_map_inr_inr CategoryTheory.sum.inverseAssociator_map_inr_inr
+-/
+#print CategoryTheory.sum.associativity /-
/-- The equivalence of categories expressing associativity of sums of categories.
-/
def associativity : Sum (Sum C D) E ≌ Sum C (Sum D E) :=
@@ -139,6 +164,7 @@ def associativity : Sum (Sum C D) E ≌ Sum C (Sum D E) :=
(NatIso.ofComponents (fun X => eqToIso (by tidy)) (by tidy))
(NatIso.ofComponents (fun X => eqToIso (by tidy)) (by tidy))
#align category_theory.sum.associativity CategoryTheory.sum.associativity
+-/
#print CategoryTheory.sum.associatorIsEquivalence /-
instance associatorIsEquivalence : IsEquivalence (associator C D E) :=
mathlib commit https://github.com/leanprover-community/mathlib/commit/917c3c072e487b3cccdbfeff17e75b40e45f66cb
@@ -48,69 +48,33 @@ def associator : Sum (Sum C D) E ⥤ Sum C (Sum D E)
#align category_theory.sum.associator CategoryTheory.sum.associator
-/
-/- warning: category_theory.sum.associator_obj_inl_inl -> CategoryTheory.sum.associator_obj_inl_inl is a dubious translation:
-lean 3 declaration is
- forall (C : Type.{u2}) [_inst_1 : CategoryTheory.Category.{u1, u2} C] (D : Type.{u2}) [_inst_2 : CategoryTheory.Category.{u1, u2} D] (E : Type.{u2}) [_inst_3 : CategoryTheory.Category.{u1, u2} E] (X : C), Eq.{succ u2} (Sum.{u2, u2} C (Sum.{u2, u2} D E)) (CategoryTheory.Functor.obj.{u1, u1, u2, u2} (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (CategoryTheory.sum.{u1, u2} (Sum.{u2, u2} C D) (CategoryTheory.sum.{u1, u2} C _inst_1 D _inst_2) E _inst_3) (Sum.{u2, u2} C (Sum.{u2, u2} D E)) (CategoryTheory.sum.{u1, u2} C _inst_1 (Sum.{u2, u2} D E) (CategoryTheory.sum.{u1, u2} D _inst_2 E _inst_3)) (CategoryTheory.sum.associator.{u1, u2} C _inst_1 D _inst_2 E _inst_3) (Sum.inl.{u2, u2} (Sum.{u2, u2} C D) E (Sum.inl.{u2, u2} C D X))) (Sum.inl.{u2, u2} C (Sum.{u2, u2} D E) X)
-but is expected to have type
- forall (C : Type.{u2}) [_inst_1 : CategoryTheory.Category.{u1, u2} C] (D : Type.{u2}) [_inst_2 : CategoryTheory.Category.{u1, u2} D] (E : Type.{u2}) [_inst_3 : CategoryTheory.Category.{u1, u2} E] (X : C), Eq.{succ u2} (Sum.{u2, u2} C (Sum.{u2, u2} D E)) (Prefunctor.obj.{succ u1, succ u1, u2, u2} (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (CategoryTheory.Category.toCategoryStruct.{u1, u2} (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (CategoryTheory.sum.{u1, u2} (Sum.{u2, u2} C D) (CategoryTheory.sum.{u1, u2} C _inst_1 D _inst_2) E _inst_3))) (Sum.{u2, u2} C (Sum.{u2, u2} D E)) (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} (Sum.{u2, u2} C (Sum.{u2, u2} D E)) (CategoryTheory.Category.toCategoryStruct.{u1, u2} (Sum.{u2, u2} C (Sum.{u2, u2} D E)) (CategoryTheory.sum.{u1, u2} C _inst_1 (Sum.{u2, u2} D E) (CategoryTheory.sum.{u1, u2} D _inst_2 E _inst_3)))) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u2, u2} (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (CategoryTheory.sum.{u1, u2} (Sum.{u2, u2} C D) (CategoryTheory.sum.{u1, u2} C _inst_1 D _inst_2) E _inst_3) (Sum.{u2, u2} C (Sum.{u2, u2} D E)) (CategoryTheory.sum.{u1, u2} C _inst_1 (Sum.{u2, u2} D E) (CategoryTheory.sum.{u1, u2} D _inst_2 E _inst_3)) (CategoryTheory.sum.associator.{u1, u2} C _inst_1 D _inst_2 E _inst_3)) (Sum.inl.{u2, u2} (Sum.{u2, u2} C D) E (Sum.inl.{u2, u2} C D X))) (Sum.inl.{u2, u2} C (Sum.{u2, u2} D E) X)
-Case conversion may be inaccurate. Consider using '#align category_theory.sum.associator_obj_inl_inl CategoryTheory.sum.associator_obj_inl_inlₓ'. -/
@[simp]
theorem associator_obj_inl_inl (X) : (associator C D E).obj (inl (inl X)) = inl X :=
rfl
#align category_theory.sum.associator_obj_inl_inl CategoryTheory.sum.associator_obj_inl_inl
-/- warning: category_theory.sum.associator_obj_inl_inr -> CategoryTheory.sum.associator_obj_inl_inr is a dubious translation:
-lean 3 declaration is
- forall (C : Type.{u2}) [_inst_1 : CategoryTheory.Category.{u1, u2} C] (D : Type.{u2}) [_inst_2 : CategoryTheory.Category.{u1, u2} D] (E : Type.{u2}) [_inst_3 : CategoryTheory.Category.{u1, u2} E] (X : D), Eq.{succ u2} (Sum.{u2, u2} C (Sum.{u2, u2} D E)) (CategoryTheory.Functor.obj.{u1, u1, u2, u2} (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (CategoryTheory.sum.{u1, u2} (Sum.{u2, u2} C D) (CategoryTheory.sum.{u1, u2} C _inst_1 D _inst_2) E _inst_3) (Sum.{u2, u2} C (Sum.{u2, u2} D E)) (CategoryTheory.sum.{u1, u2} C _inst_1 (Sum.{u2, u2} D E) (CategoryTheory.sum.{u1, u2} D _inst_2 E _inst_3)) (CategoryTheory.sum.associator.{u1, u2} C _inst_1 D _inst_2 E _inst_3) (Sum.inl.{u2, u2} (Sum.{u2, u2} C D) E (Sum.inr.{u2, u2} C D X))) (Sum.inr.{u2, u2} C (Sum.{u2, u2} D E) (Sum.inl.{u2, u2} D E X))
-but is expected to have type
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@[simp]
theorem associator_obj_inl_inr (X) : (associator C D E).obj (inl (inr X)) = inr (inl X) :=
rfl
#align category_theory.sum.associator_obj_inl_inr CategoryTheory.sum.associator_obj_inl_inr
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@[simp]
theorem associator_obj_inr (X) : (associator C D E).obj (inr X) = inr (inr X) :=
rfl
#align category_theory.sum.associator_obj_inr CategoryTheory.sum.associator_obj_inr
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@[simp]
theorem associator_map_inl_inl {X Y : C} (f : inl (inl X) ⟶ inl (inl Y)) :
(associator C D E).map f = f :=
rfl
#align category_theory.sum.associator_map_inl_inl CategoryTheory.sum.associator_map_inl_inl
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@[simp]
theorem associator_map_inl_inr {X Y : D} (f : inl (inr X) ⟶ inl (inr Y)) :
(associator C D E).map f = f :=
rfl
#align category_theory.sum.associator_map_inl_inr CategoryTheory.sum.associator_map_inl_inr
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@[simp]
theorem associator_map_inr {X Y : E} (f : inr X ⟶ inr Y) : (associator C D E).map f = f :=
rfl
@@ -134,82 +98,40 @@ def inverseAssociator : Sum C (Sum D E) ⥤ Sum (Sum C D) E
#align category_theory.sum.inverse_associator CategoryTheory.sum.inverseAssociator
-/
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@[simp]
theorem inverseAssociator_obj_inl (X) : (inverseAssociator C D E).obj (inl X) = inl (inl X) :=
rfl
#align category_theory.sum.inverse_associator_obj_inl CategoryTheory.sum.inverseAssociator_obj_inl
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@[simp]
theorem inverseAssociator_obj_inr_inl (X) :
(inverseAssociator C D E).obj (inr (inl X)) = inl (inr X) :=
rfl
#align category_theory.sum.inverse_associator_obj_inr_inl CategoryTheory.sum.inverseAssociator_obj_inr_inl
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@[simp]
theorem inverseAssociator_obj_inr_inr (X) : (inverseAssociator C D E).obj (inr (inr X)) = inr X :=
rfl
#align category_theory.sum.inverse_associator_obj_inr_inr CategoryTheory.sum.inverseAssociator_obj_inr_inr
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@[simp]
theorem inverseAssociator_map_inl {X Y : C} (f : inl X ⟶ inl Y) :
(inverseAssociator C D E).map f = f :=
rfl
#align category_theory.sum.inverse_associator_map_inl CategoryTheory.sum.inverseAssociator_map_inl
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-Case conversion may be inaccurate. Consider using '#align category_theory.sum.inverse_associator_map_inr_inl CategoryTheory.sum.inverseAssociator_map_inr_inlₓ'. -/
@[simp]
theorem inverseAssociator_map_inr_inl {X Y : D} (f : inr (inl X) ⟶ inr (inl Y)) :
(inverseAssociator C D E).map f = f :=
rfl
#align category_theory.sum.inverse_associator_map_inr_inl CategoryTheory.sum.inverseAssociator_map_inr_inl
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-Case conversion may be inaccurate. Consider using '#align category_theory.sum.inverse_associator_map_inr_inr CategoryTheory.sum.inverseAssociator_map_inr_inrₓ'. -/
@[simp]
theorem inverseAssociator_map_inr_inr {X Y : E} (f : inr (inr X) ⟶ inr (inr Y)) :
(inverseAssociator C D E).map f = f :=
rfl
#align category_theory.sum.inverse_associator_map_inr_inr CategoryTheory.sum.inverseAssociator_map_inr_inr
-/- warning: category_theory.sum.associativity -> CategoryTheory.sum.associativity 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] (D : Type.{u2}) [_inst_2 : CategoryTheory.Category.{u1, u2} D] (E : Type.{u2}) [_inst_3 : CategoryTheory.Category.{u1, u2} E], CategoryTheory.Equivalence.{u1, u1, u2, u2} (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (Sum.{u2, u2} C (Sum.{u2, u2} D E)) (CategoryTheory.sum.{u1, u2} (Sum.{u2, u2} C D) (CategoryTheory.sum.{u1, u2} C _inst_1 D _inst_2) E _inst_3) (CategoryTheory.sum.{u1, u2} C _inst_1 (Sum.{u2, u2} D E) (CategoryTheory.sum.{u1, u2} D _inst_2 E _inst_3))
-Case conversion may be inaccurate. Consider using '#align category_theory.sum.associativity CategoryTheory.sum.associativityₓ'. -/
/-- The equivalence of categories expressing associativity of sums of categories.
-/
def associativity : Sum (Sum C D) E ≌ Sum C (Sum D E) :=
mathlib commit https://github.com/leanprover-community/mathlib/commit/195fcd60ff2bfe392543bceb0ec2adcdb472db4c
@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
! This file was ported from Lean 3 source module category_theory.sums.associator
-! leanprover-community/mathlib commit 590f43db91071eb3134fef935ec9d7cd2a3bd4ce
+! leanprover-community/mathlib commit ee05e9ce1322178f0c12004eb93c00d2c8c00ed2
! 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.Sums.Basic
/-!
# Associator for binary disjoint union of categories.
+> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
+> Any changes to this file require a corresponding PR to mathlib4.
+
The associator functor `((C ⊕ D) ⊕ E) ⥤ (C ⊕ (D ⊕ E))` and its inverse form an equivalence.
-/
mathlib commit https://github.com/leanprover-community/mathlib/commit/9da1b3534b65d9661eb8f42443598a92bbb49211
@@ -27,6 +27,7 @@ namespace CategoryTheory.sum
variable (C : Type u) [Category.{v} C] (D : Type u) [Category.{v} D] (E : Type u) [Category.{v} E]
+#print CategoryTheory.sum.associator /-
/-- The associator functor `(C ⊕ D) ⊕ E ⥤ C ⊕ (D ⊕ E)` for sums of categories.
-/
def associator : Sum (Sum C D) E ⥤ Sum C (Sum D E)
@@ -42,39 +43,77 @@ def associator : Sum (Sum C D) E ⥤ Sum C (Sum D E)
| inl (inr X), inl (inr Y), f => f
| inr X, inr Y, f => f
#align category_theory.sum.associator CategoryTheory.sum.associator
+-/
+/- warning: category_theory.sum.associator_obj_inl_inl -> CategoryTheory.sum.associator_obj_inl_inl is a dubious translation:
+lean 3 declaration is
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+Case conversion may be inaccurate. Consider using '#align category_theory.sum.associator_obj_inl_inl CategoryTheory.sum.associator_obj_inl_inlₓ'. -/
@[simp]
theorem associator_obj_inl_inl (X) : (associator C D E).obj (inl (inl X)) = inl X :=
rfl
#align category_theory.sum.associator_obj_inl_inl CategoryTheory.sum.associator_obj_inl_inl
+/- warning: category_theory.sum.associator_obj_inl_inr -> CategoryTheory.sum.associator_obj_inl_inr is a dubious translation:
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+Case conversion may be inaccurate. Consider using '#align category_theory.sum.associator_obj_inl_inr CategoryTheory.sum.associator_obj_inl_inrₓ'. -/
@[simp]
theorem associator_obj_inl_inr (X) : (associator C D E).obj (inl (inr X)) = inr (inl X) :=
rfl
#align category_theory.sum.associator_obj_inl_inr CategoryTheory.sum.associator_obj_inl_inr
+/- warning: category_theory.sum.associator_obj_inr -> CategoryTheory.sum.associator_obj_inr 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.sum.associator_obj_inr CategoryTheory.sum.associator_obj_inrₓ'. -/
@[simp]
theorem associator_obj_inr (X) : (associator C D E).obj (inr X) = inr (inr X) :=
rfl
#align category_theory.sum.associator_obj_inr CategoryTheory.sum.associator_obj_inr
+/- warning: category_theory.sum.associator_map_inl_inl -> CategoryTheory.sum.associator_map_inl_inl is a dubious translation:
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+ forall (C : Type.{u2}) [_inst_1 : CategoryTheory.Category.{u1, u2} C] (D : Type.{u2}) [_inst_2 : CategoryTheory.Category.{u1, u2} D] (E : Type.{u2}) [_inst_3 : CategoryTheory.Category.{u1, u2} E] {X : C} {Y : C} (f : Quiver.Hom.{succ u1, u2} (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (CategoryTheory.Category.toCategoryStruct.{u1, u2} (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (CategoryTheory.sum.{u1, u2} (Sum.{u2, u2} C D) (CategoryTheory.sum.{u1, u2} C _inst_1 D _inst_2) E _inst_3))) (Sum.inl.{u2, u2} (Sum.{u2, u2} C D) E (Sum.inl.{u2, u2} C D X)) (Sum.inl.{u2, u2} (Sum.{u2, u2} C D) E (Sum.inl.{u2, u2} C D Y))), Eq.{succ u1} (Quiver.Hom.{succ u1, u2} (Sum.{u2, u2} C (Sum.{u2, u2} D E)) (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} (Sum.{u2, u2} C (Sum.{u2, u2} D E)) (CategoryTheory.Category.toCategoryStruct.{u1, u2} (Sum.{u2, u2} C (Sum.{u2, u2} D E)) (CategoryTheory.sum.{u1, u2} C _inst_1 (Sum.{u2, u2} D E) (CategoryTheory.sum.{u1, u2} D _inst_2 E _inst_3)))) (CategoryTheory.Functor.obj.{u1, u1, u2, u2} (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (CategoryTheory.sum.{u1, u2} (Sum.{u2, u2} C D) (CategoryTheory.sum.{u1, u2} C _inst_1 D _inst_2) E _inst_3) (Sum.{u2, u2} C (Sum.{u2, u2} D E)) (CategoryTheory.sum.{u1, u2} C _inst_1 (Sum.{u2, u2} D E) (CategoryTheory.sum.{u1, u2} D _inst_2 E _inst_3)) (CategoryTheory.sum.associator.{u1, u2} C _inst_1 D _inst_2 E _inst_3) (Sum.inl.{u2, u2} (Sum.{u2, u2} C D) E (Sum.inl.{u2, u2} C D X))) (CategoryTheory.Functor.obj.{u1, u1, u2, u2} (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (CategoryTheory.sum.{u1, u2} (Sum.{u2, u2} C D) (CategoryTheory.sum.{u1, u2} C _inst_1 D _inst_2) E _inst_3) (Sum.{u2, u2} C (Sum.{u2, u2} D E)) (CategoryTheory.sum.{u1, u2} C _inst_1 (Sum.{u2, u2} D E) (CategoryTheory.sum.{u1, u2} D _inst_2 E _inst_3)) (CategoryTheory.sum.associator.{u1, u2} C _inst_1 D _inst_2 E _inst_3) (Sum.inl.{u2, u2} (Sum.{u2, u2} C D) E (Sum.inl.{u2, u2} C D Y)))) (CategoryTheory.Functor.map.{u1, u1, u2, u2} (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (CategoryTheory.sum.{u1, u2} (Sum.{u2, u2} C D) (CategoryTheory.sum.{u1, u2} C _inst_1 D _inst_2) E _inst_3) (Sum.{u2, u2} C (Sum.{u2, u2} D E)) (CategoryTheory.sum.{u1, u2} C _inst_1 (Sum.{u2, u2} D E) (CategoryTheory.sum.{u1, u2} D _inst_2 E _inst_3)) (CategoryTheory.sum.associator.{u1, u2} C _inst_1 D _inst_2 E _inst_3) (Sum.inl.{u2, u2} (Sum.{u2, u2} C D) E (Sum.inl.{u2, u2} C D X)) (Sum.inl.{u2, u2} (Sum.{u2, u2} C D) E (Sum.inl.{u2, u2} C D Y)) f) f
+but is expected to have type
+ forall (C : Type.{u2}) [_inst_1 : CategoryTheory.Category.{u1, u2} C] (D : Type.{u2}) [_inst_2 : CategoryTheory.Category.{u1, u2} D] (E : Type.{u2}) [_inst_3 : CategoryTheory.Category.{u1, u2} E] {X : C} {Y : C} (f : Quiver.Hom.{succ u1, u2} (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (CategoryTheory.Category.toCategoryStruct.{u1, u2} (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (CategoryTheory.sum.{u1, u2} (Sum.{u2, u2} C D) (CategoryTheory.sum.{u1, u2} C _inst_1 D _inst_2) E _inst_3))) (Sum.inl.{u2, u2} (Sum.{u2, u2} C D) E (Sum.inl.{u2, u2} C D X)) (Sum.inl.{u2, u2} (Sum.{u2, u2} C D) E (Sum.inl.{u2, u2} C D Y))), Eq.{succ u1} (Quiver.Hom.{succ u1, u2} (Sum.{u2, u2} C (Sum.{u2, u2} D E)) (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} (Sum.{u2, u2} C (Sum.{u2, u2} D E)) (CategoryTheory.Category.toCategoryStruct.{u1, u2} (Sum.{u2, u2} C (Sum.{u2, u2} D E)) (CategoryTheory.sum.{u1, u2} C _inst_1 (Sum.{u2, u2} D E) (CategoryTheory.sum.{u1, u2} D _inst_2 E _inst_3)))) (Prefunctor.obj.{succ u1, succ u1, u2, u2} (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (CategoryTheory.Category.toCategoryStruct.{u1, u2} (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (CategoryTheory.sum.{u1, u2} (Sum.{u2, u2} C D) (CategoryTheory.sum.{u1, u2} C _inst_1 D _inst_2) E _inst_3))) (Sum.{u2, u2} C (Sum.{u2, u2} D E)) (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} (Sum.{u2, u2} C (Sum.{u2, u2} D E)) (CategoryTheory.Category.toCategoryStruct.{u1, u2} (Sum.{u2, u2} C (Sum.{u2, u2} D E)) (CategoryTheory.sum.{u1, u2} C _inst_1 (Sum.{u2, u2} D E) (CategoryTheory.sum.{u1, u2} D _inst_2 E _inst_3)))) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u2, u2} (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (CategoryTheory.sum.{u1, u2} (Sum.{u2, u2} C D) (CategoryTheory.sum.{u1, u2} C _inst_1 D _inst_2) E _inst_3) (Sum.{u2, u2} C (Sum.{u2, u2} D E)) (CategoryTheory.sum.{u1, u2} C _inst_1 (Sum.{u2, u2} D E) (CategoryTheory.sum.{u1, u2} D _inst_2 E _inst_3)) (CategoryTheory.sum.associator.{u1, u2} C _inst_1 D _inst_2 E _inst_3)) (Sum.inl.{u2, u2} (Sum.{u2, u2} C D) E (Sum.inl.{u2, u2} C D X))) (Prefunctor.obj.{succ u1, succ u1, u2, u2} (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (CategoryTheory.Category.toCategoryStruct.{u1, u2} (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (CategoryTheory.sum.{u1, u2} (Sum.{u2, u2} C D) (CategoryTheory.sum.{u1, u2} C _inst_1 D _inst_2) E _inst_3))) (Sum.{u2, u2} C (Sum.{u2, u2} D E)) (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} (Sum.{u2, u2} C (Sum.{u2, u2} D E)) (CategoryTheory.Category.toCategoryStruct.{u1, u2} (Sum.{u2, u2} C (Sum.{u2, u2} D E)) (CategoryTheory.sum.{u1, u2} C _inst_1 (Sum.{u2, u2} D E) (CategoryTheory.sum.{u1, u2} D _inst_2 E _inst_3)))) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u2, u2} (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (CategoryTheory.sum.{u1, u2} (Sum.{u2, u2} C D) (CategoryTheory.sum.{u1, u2} C _inst_1 D _inst_2) E _inst_3) (Sum.{u2, u2} C (Sum.{u2, u2} D E)) (CategoryTheory.sum.{u1, u2} C _inst_1 (Sum.{u2, u2} D E) (CategoryTheory.sum.{u1, u2} D _inst_2 E _inst_3)) (CategoryTheory.sum.associator.{u1, u2} C _inst_1 D _inst_2 E _inst_3)) (Sum.inl.{u2, u2} (Sum.{u2, u2} C D) E (Sum.inl.{u2, u2} C D Y)))) (Prefunctor.map.{succ u1, succ u1, u2, u2} (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (CategoryTheory.Category.toCategoryStruct.{u1, u2} (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (CategoryTheory.sum.{u1, u2} (Sum.{u2, u2} C D) (CategoryTheory.sum.{u1, u2} C _inst_1 D _inst_2) E _inst_3))) (Sum.{u2, u2} C (Sum.{u2, u2} D E)) (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} (Sum.{u2, u2} C (Sum.{u2, u2} D E)) (CategoryTheory.Category.toCategoryStruct.{u1, u2} (Sum.{u2, u2} C (Sum.{u2, u2} D E)) (CategoryTheory.sum.{u1, u2} C _inst_1 (Sum.{u2, u2} D E) (CategoryTheory.sum.{u1, u2} D _inst_2 E _inst_3)))) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u2, u2} (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (CategoryTheory.sum.{u1, u2} (Sum.{u2, u2} C D) (CategoryTheory.sum.{u1, u2} C _inst_1 D _inst_2) E _inst_3) (Sum.{u2, u2} C (Sum.{u2, u2} D E)) (CategoryTheory.sum.{u1, u2} C _inst_1 (Sum.{u2, u2} D E) (CategoryTheory.sum.{u1, u2} D _inst_2 E _inst_3)) (CategoryTheory.sum.associator.{u1, u2} C _inst_1 D _inst_2 E _inst_3)) (Sum.inl.{u2, u2} (Sum.{u2, u2} C D) E (Sum.inl.{u2, u2} C D X)) (Sum.inl.{u2, u2} (Sum.{u2, u2} C D) E (Sum.inl.{u2, u2} C D Y)) f) f
+Case conversion may be inaccurate. Consider using '#align category_theory.sum.associator_map_inl_inl CategoryTheory.sum.associator_map_inl_inlₓ'. -/
@[simp]
theorem associator_map_inl_inl {X Y : C} (f : inl (inl X) ⟶ inl (inl Y)) :
(associator C D E).map f = f :=
rfl
#align category_theory.sum.associator_map_inl_inl CategoryTheory.sum.associator_map_inl_inl
+/- warning: category_theory.sum.associator_map_inl_inr -> CategoryTheory.sum.associator_map_inl_inr is a dubious translation:
+lean 3 declaration is
+ forall (C : Type.{u2}) [_inst_1 : CategoryTheory.Category.{u1, u2} C] (D : Type.{u2}) [_inst_2 : CategoryTheory.Category.{u1, u2} D] (E : Type.{u2}) [_inst_3 : CategoryTheory.Category.{u1, u2} E] {X : D} {Y : D} (f : Quiver.Hom.{succ u1, u2} (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (CategoryTheory.Category.toCategoryStruct.{u1, u2} (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (CategoryTheory.sum.{u1, u2} (Sum.{u2, u2} C D) (CategoryTheory.sum.{u1, u2} C _inst_1 D _inst_2) E _inst_3))) (Sum.inl.{u2, u2} (Sum.{u2, u2} C D) E (Sum.inr.{u2, u2} C D X)) (Sum.inl.{u2, u2} (Sum.{u2, u2} C D) E (Sum.inr.{u2, u2} C D Y))), Eq.{succ u1} (Quiver.Hom.{succ u1, u2} (Sum.{u2, u2} C (Sum.{u2, u2} D E)) (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} (Sum.{u2, u2} C (Sum.{u2, u2} D E)) (CategoryTheory.Category.toCategoryStruct.{u1, u2} (Sum.{u2, u2} C (Sum.{u2, u2} D E)) (CategoryTheory.sum.{u1, u2} C _inst_1 (Sum.{u2, u2} D E) (CategoryTheory.sum.{u1, u2} D _inst_2 E _inst_3)))) (CategoryTheory.Functor.obj.{u1, u1, u2, u2} (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (CategoryTheory.sum.{u1, u2} (Sum.{u2, u2} C D) (CategoryTheory.sum.{u1, u2} C _inst_1 D _inst_2) E _inst_3) (Sum.{u2, u2} C (Sum.{u2, u2} D E)) (CategoryTheory.sum.{u1, u2} C _inst_1 (Sum.{u2, u2} D E) (CategoryTheory.sum.{u1, u2} D _inst_2 E _inst_3)) (CategoryTheory.sum.associator.{u1, u2} C _inst_1 D _inst_2 E _inst_3) (Sum.inl.{u2, u2} (Sum.{u2, u2} C D) E (Sum.inr.{u2, u2} C D X))) (CategoryTheory.Functor.obj.{u1, u1, u2, u2} (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (CategoryTheory.sum.{u1, u2} (Sum.{u2, u2} C D) (CategoryTheory.sum.{u1, u2} C _inst_1 D _inst_2) E _inst_3) (Sum.{u2, u2} C (Sum.{u2, u2} D E)) (CategoryTheory.sum.{u1, u2} C _inst_1 (Sum.{u2, u2} D E) (CategoryTheory.sum.{u1, u2} D _inst_2 E _inst_3)) (CategoryTheory.sum.associator.{u1, u2} C _inst_1 D _inst_2 E _inst_3) (Sum.inl.{u2, u2} (Sum.{u2, u2} C D) E (Sum.inr.{u2, u2} C D Y)))) (CategoryTheory.Functor.map.{u1, u1, u2, u2} (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (CategoryTheory.sum.{u1, u2} (Sum.{u2, u2} C D) (CategoryTheory.sum.{u1, u2} C _inst_1 D _inst_2) E _inst_3) (Sum.{u2, u2} C (Sum.{u2, u2} D E)) (CategoryTheory.sum.{u1, u2} C _inst_1 (Sum.{u2, u2} D E) (CategoryTheory.sum.{u1, u2} D _inst_2 E _inst_3)) (CategoryTheory.sum.associator.{u1, u2} C _inst_1 D _inst_2 E _inst_3) (Sum.inl.{u2, u2} (Sum.{u2, u2} C D) E (Sum.inr.{u2, u2} C D X)) (Sum.inl.{u2, u2} (Sum.{u2, u2} C D) E (Sum.inr.{u2, u2} C D Y)) f) f
+but is expected to have type
+ forall (C : Type.{u2}) [_inst_1 : CategoryTheory.Category.{u1, u2} C] (D : Type.{u2}) [_inst_2 : CategoryTheory.Category.{u1, u2} D] (E : Type.{u2}) [_inst_3 : CategoryTheory.Category.{u1, u2} E] {X : D} {Y : D} (f : Quiver.Hom.{succ u1, u2} (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (CategoryTheory.Category.toCategoryStruct.{u1, u2} (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (CategoryTheory.sum.{u1, u2} (Sum.{u2, u2} C D) (CategoryTheory.sum.{u1, u2} C _inst_1 D _inst_2) E _inst_3))) (Sum.inl.{u2, u2} (Sum.{u2, u2} C D) E (Sum.inr.{u2, u2} C D X)) (Sum.inl.{u2, u2} (Sum.{u2, u2} C D) E (Sum.inr.{u2, u2} C D Y))), Eq.{succ u1} (Quiver.Hom.{succ u1, u2} (Sum.{u2, u2} C (Sum.{u2, u2} D E)) (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} (Sum.{u2, u2} C (Sum.{u2, u2} D E)) (CategoryTheory.Category.toCategoryStruct.{u1, u2} (Sum.{u2, u2} C (Sum.{u2, u2} D E)) (CategoryTheory.sum.{u1, u2} C _inst_1 (Sum.{u2, u2} D E) (CategoryTheory.sum.{u1, u2} D _inst_2 E _inst_3)))) (Prefunctor.obj.{succ u1, succ u1, u2, u2} (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (CategoryTheory.Category.toCategoryStruct.{u1, u2} (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (CategoryTheory.sum.{u1, u2} (Sum.{u2, u2} C D) (CategoryTheory.sum.{u1, u2} C _inst_1 D _inst_2) E _inst_3))) (Sum.{u2, u2} C (Sum.{u2, u2} D E)) (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} (Sum.{u2, u2} C (Sum.{u2, u2} D E)) (CategoryTheory.Category.toCategoryStruct.{u1, u2} (Sum.{u2, u2} C (Sum.{u2, u2} D E)) (CategoryTheory.sum.{u1, u2} C _inst_1 (Sum.{u2, u2} D E) (CategoryTheory.sum.{u1, u2} D _inst_2 E _inst_3)))) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u2, u2} (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (CategoryTheory.sum.{u1, u2} (Sum.{u2, u2} C D) (CategoryTheory.sum.{u1, u2} C _inst_1 D _inst_2) E _inst_3) (Sum.{u2, u2} C (Sum.{u2, u2} D E)) (CategoryTheory.sum.{u1, u2} C _inst_1 (Sum.{u2, u2} D E) (CategoryTheory.sum.{u1, u2} D _inst_2 E _inst_3)) (CategoryTheory.sum.associator.{u1, u2} C _inst_1 D _inst_2 E _inst_3)) (Sum.inl.{u2, u2} (Sum.{u2, u2} C D) E (Sum.inr.{u2, u2} C D X))) (Prefunctor.obj.{succ u1, succ u1, u2, u2} (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (CategoryTheory.Category.toCategoryStruct.{u1, u2} (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (CategoryTheory.sum.{u1, u2} (Sum.{u2, u2} C D) (CategoryTheory.sum.{u1, u2} C _inst_1 D _inst_2) E _inst_3))) (Sum.{u2, u2} C (Sum.{u2, u2} D E)) (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} (Sum.{u2, u2} C (Sum.{u2, u2} D E)) (CategoryTheory.Category.toCategoryStruct.{u1, u2} (Sum.{u2, u2} C (Sum.{u2, u2} D E)) (CategoryTheory.sum.{u1, u2} C _inst_1 (Sum.{u2, u2} D E) (CategoryTheory.sum.{u1, u2} D _inst_2 E _inst_3)))) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u2, u2} (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (CategoryTheory.sum.{u1, u2} (Sum.{u2, u2} C D) (CategoryTheory.sum.{u1, u2} C _inst_1 D _inst_2) E _inst_3) (Sum.{u2, u2} C (Sum.{u2, u2} D E)) (CategoryTheory.sum.{u1, u2} C _inst_1 (Sum.{u2, u2} D E) (CategoryTheory.sum.{u1, u2} D _inst_2 E _inst_3)) (CategoryTheory.sum.associator.{u1, u2} C _inst_1 D _inst_2 E _inst_3)) (Sum.inl.{u2, u2} (Sum.{u2, u2} C D) E (Sum.inr.{u2, u2} C D Y)))) (Prefunctor.map.{succ u1, succ u1, u2, u2} (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (CategoryTheory.Category.toCategoryStruct.{u1, u2} (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (CategoryTheory.sum.{u1, u2} (Sum.{u2, u2} C D) (CategoryTheory.sum.{u1, u2} C _inst_1 D _inst_2) E _inst_3))) (Sum.{u2, u2} C (Sum.{u2, u2} D E)) (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} (Sum.{u2, u2} C (Sum.{u2, u2} D E)) (CategoryTheory.Category.toCategoryStruct.{u1, u2} (Sum.{u2, u2} C (Sum.{u2, u2} D E)) (CategoryTheory.sum.{u1, u2} C _inst_1 (Sum.{u2, u2} D E) (CategoryTheory.sum.{u1, u2} D _inst_2 E _inst_3)))) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u2, u2} (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (CategoryTheory.sum.{u1, u2} (Sum.{u2, u2} C D) (CategoryTheory.sum.{u1, u2} C _inst_1 D _inst_2) E _inst_3) (Sum.{u2, u2} C (Sum.{u2, u2} D E)) (CategoryTheory.sum.{u1, u2} C _inst_1 (Sum.{u2, u2} D E) (CategoryTheory.sum.{u1, u2} D _inst_2 E _inst_3)) (CategoryTheory.sum.associator.{u1, u2} C _inst_1 D _inst_2 E _inst_3)) (Sum.inl.{u2, u2} (Sum.{u2, u2} C D) E (Sum.inr.{u2, u2} C D X)) (Sum.inl.{u2, u2} (Sum.{u2, u2} C D) E (Sum.inr.{u2, u2} C D Y)) f) f
+Case conversion may be inaccurate. Consider using '#align category_theory.sum.associator_map_inl_inr CategoryTheory.sum.associator_map_inl_inrₓ'. -/
@[simp]
theorem associator_map_inl_inr {X Y : D} (f : inl (inr X) ⟶ inl (inr Y)) :
(associator C D E).map f = f :=
rfl
#align category_theory.sum.associator_map_inl_inr CategoryTheory.sum.associator_map_inl_inr
+/- warning: category_theory.sum.associator_map_inr -> CategoryTheory.sum.associator_map_inr is a dubious translation:
+lean 3 declaration is
+ forall (C : Type.{u2}) [_inst_1 : CategoryTheory.Category.{u1, u2} C] (D : Type.{u2}) [_inst_2 : CategoryTheory.Category.{u1, u2} D] (E : Type.{u2}) [_inst_3 : CategoryTheory.Category.{u1, u2} E] {X : E} {Y : E} (f : Quiver.Hom.{succ u1, u2} (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (CategoryTheory.Category.toCategoryStruct.{u1, u2} (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (CategoryTheory.sum.{u1, u2} (Sum.{u2, u2} C D) (CategoryTheory.sum.{u1, u2} C _inst_1 D _inst_2) E _inst_3))) (Sum.inr.{u2, u2} (Sum.{u2, u2} C D) E X) (Sum.inr.{u2, u2} (Sum.{u2, u2} C D) E Y)), Eq.{succ u1} (Quiver.Hom.{succ u1, u2} (Sum.{u2, u2} C (Sum.{u2, u2} D E)) (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} (Sum.{u2, u2} C (Sum.{u2, u2} D E)) (CategoryTheory.Category.toCategoryStruct.{u1, u2} (Sum.{u2, u2} C (Sum.{u2, u2} D E)) (CategoryTheory.sum.{u1, u2} C _inst_1 (Sum.{u2, u2} D E) (CategoryTheory.sum.{u1, u2} D _inst_2 E _inst_3)))) (CategoryTheory.Functor.obj.{u1, u1, u2, u2} (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (CategoryTheory.sum.{u1, u2} (Sum.{u2, u2} C D) (CategoryTheory.sum.{u1, u2} C _inst_1 D _inst_2) E _inst_3) (Sum.{u2, u2} C (Sum.{u2, u2} D E)) (CategoryTheory.sum.{u1, u2} C _inst_1 (Sum.{u2, u2} D E) (CategoryTheory.sum.{u1, u2} D _inst_2 E _inst_3)) (CategoryTheory.sum.associator.{u1, u2} C _inst_1 D _inst_2 E _inst_3) (Sum.inr.{u2, u2} (Sum.{u2, u2} C D) E X)) (CategoryTheory.Functor.obj.{u1, u1, u2, u2} (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (CategoryTheory.sum.{u1, u2} (Sum.{u2, u2} C D) (CategoryTheory.sum.{u1, u2} C _inst_1 D _inst_2) E _inst_3) (Sum.{u2, u2} C (Sum.{u2, u2} D E)) (CategoryTheory.sum.{u1, u2} C _inst_1 (Sum.{u2, u2} D E) (CategoryTheory.sum.{u1, u2} D _inst_2 E _inst_3)) (CategoryTheory.sum.associator.{u1, u2} C _inst_1 D _inst_2 E _inst_3) (Sum.inr.{u2, u2} (Sum.{u2, u2} C D) E Y))) (CategoryTheory.Functor.map.{u1, u1, u2, u2} (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (CategoryTheory.sum.{u1, u2} (Sum.{u2, u2} C D) (CategoryTheory.sum.{u1, u2} C _inst_1 D _inst_2) E _inst_3) (Sum.{u2, u2} C (Sum.{u2, u2} D E)) (CategoryTheory.sum.{u1, u2} C _inst_1 (Sum.{u2, u2} D E) (CategoryTheory.sum.{u1, u2} D _inst_2 E _inst_3)) (CategoryTheory.sum.associator.{u1, u2} C _inst_1 D _inst_2 E _inst_3) (Sum.inr.{u2, u2} (Sum.{u2, u2} C D) E X) (Sum.inr.{u2, u2} (Sum.{u2, u2} C D) E Y) f) f
+but is expected to have type
+ forall (C : Type.{u2}) [_inst_1 : CategoryTheory.Category.{u1, u2} C] (D : Type.{u2}) [_inst_2 : CategoryTheory.Category.{u1, u2} D] (E : Type.{u2}) [_inst_3 : CategoryTheory.Category.{u1, u2} E] {X : E} {Y : E} (f : Quiver.Hom.{succ u1, u2} (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (CategoryTheory.Category.toCategoryStruct.{u1, u2} (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (CategoryTheory.sum.{u1, u2} (Sum.{u2, u2} C D) (CategoryTheory.sum.{u1, u2} C _inst_1 D _inst_2) E _inst_3))) (Sum.inr.{u2, u2} (Sum.{u2, u2} C D) E X) (Sum.inr.{u2, u2} (Sum.{u2, u2} C D) E Y)), Eq.{succ u1} (Quiver.Hom.{succ u1, u2} (Sum.{u2, u2} C (Sum.{u2, u2} D E)) (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} (Sum.{u2, u2} C (Sum.{u2, u2} D E)) (CategoryTheory.Category.toCategoryStruct.{u1, u2} (Sum.{u2, u2} C (Sum.{u2, u2} D E)) (CategoryTheory.sum.{u1, u2} C _inst_1 (Sum.{u2, u2} D E) (CategoryTheory.sum.{u1, u2} D _inst_2 E _inst_3)))) (Prefunctor.obj.{succ u1, succ u1, u2, u2} (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (CategoryTheory.Category.toCategoryStruct.{u1, u2} (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (CategoryTheory.sum.{u1, u2} (Sum.{u2, u2} C D) (CategoryTheory.sum.{u1, u2} C _inst_1 D _inst_2) E _inst_3))) (Sum.{u2, u2} C (Sum.{u2, u2} D E)) (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} (Sum.{u2, u2} C (Sum.{u2, u2} D E)) (CategoryTheory.Category.toCategoryStruct.{u1, u2} (Sum.{u2, u2} C (Sum.{u2, u2} D E)) (CategoryTheory.sum.{u1, u2} C _inst_1 (Sum.{u2, u2} D E) (CategoryTheory.sum.{u1, u2} D _inst_2 E _inst_3)))) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u2, u2} (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (CategoryTheory.sum.{u1, u2} (Sum.{u2, u2} C D) (CategoryTheory.sum.{u1, u2} C _inst_1 D _inst_2) E _inst_3) (Sum.{u2, u2} C (Sum.{u2, u2} D E)) (CategoryTheory.sum.{u1, u2} C _inst_1 (Sum.{u2, u2} D E) (CategoryTheory.sum.{u1, u2} D _inst_2 E _inst_3)) (CategoryTheory.sum.associator.{u1, u2} C _inst_1 D _inst_2 E _inst_3)) (Sum.inr.{u2, u2} (Sum.{u2, u2} C D) E X)) (Prefunctor.obj.{succ u1, succ u1, u2, u2} (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (CategoryTheory.Category.toCategoryStruct.{u1, u2} (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (CategoryTheory.sum.{u1, u2} (Sum.{u2, u2} C D) (CategoryTheory.sum.{u1, u2} C _inst_1 D _inst_2) E _inst_3))) (Sum.{u2, u2} C (Sum.{u2, u2} D E)) (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} (Sum.{u2, u2} C (Sum.{u2, u2} D E)) (CategoryTheory.Category.toCategoryStruct.{u1, u2} (Sum.{u2, u2} C (Sum.{u2, u2} D E)) (CategoryTheory.sum.{u1, u2} C _inst_1 (Sum.{u2, u2} D E) (CategoryTheory.sum.{u1, u2} D _inst_2 E _inst_3)))) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u2, u2} (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (CategoryTheory.sum.{u1, u2} (Sum.{u2, u2} C D) (CategoryTheory.sum.{u1, u2} C _inst_1 D _inst_2) E _inst_3) (Sum.{u2, u2} C (Sum.{u2, u2} D E)) (CategoryTheory.sum.{u1, u2} C _inst_1 (Sum.{u2, u2} D E) (CategoryTheory.sum.{u1, u2} D _inst_2 E _inst_3)) (CategoryTheory.sum.associator.{u1, u2} C _inst_1 D _inst_2 E _inst_3)) (Sum.inr.{u2, u2} (Sum.{u2, u2} C D) E Y))) (Prefunctor.map.{succ u1, succ u1, u2, u2} (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (CategoryTheory.Category.toCategoryStruct.{u1, u2} (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (CategoryTheory.sum.{u1, u2} (Sum.{u2, u2} C D) (CategoryTheory.sum.{u1, u2} C _inst_1 D _inst_2) E _inst_3))) (Sum.{u2, u2} C (Sum.{u2, u2} D E)) (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} (Sum.{u2, u2} C (Sum.{u2, u2} D E)) (CategoryTheory.Category.toCategoryStruct.{u1, u2} (Sum.{u2, u2} C (Sum.{u2, u2} D E)) (CategoryTheory.sum.{u1, u2} C _inst_1 (Sum.{u2, u2} D E) (CategoryTheory.sum.{u1, u2} D _inst_2 E _inst_3)))) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u2, u2} (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (CategoryTheory.sum.{u1, u2} (Sum.{u2, u2} C D) (CategoryTheory.sum.{u1, u2} C _inst_1 D _inst_2) E _inst_3) (Sum.{u2, u2} C (Sum.{u2, u2} D E)) (CategoryTheory.sum.{u1, u2} C _inst_1 (Sum.{u2, u2} D E) (CategoryTheory.sum.{u1, u2} D _inst_2 E _inst_3)) (CategoryTheory.sum.associator.{u1, u2} C _inst_1 D _inst_2 E _inst_3)) (Sum.inr.{u2, u2} (Sum.{u2, u2} C D) E X) (Sum.inr.{u2, u2} (Sum.{u2, u2} C D) E Y) f) f
+Case conversion may be inaccurate. Consider using '#align category_theory.sum.associator_map_inr CategoryTheory.sum.associator_map_inrₓ'. -/
@[simp]
theorem associator_map_inr {X Y : E} (f : inr X ⟶ inr Y) : (associator C D E).map f = f :=
rfl
#align category_theory.sum.associator_map_inr CategoryTheory.sum.associator_map_inr
+#print CategoryTheory.sum.inverseAssociator /-
/-- The inverse associator functor `C ⊕ (D ⊕ E) ⥤ (C ⊕ D) ⊕ E` for sums of categories.
-/
def inverseAssociator : Sum C (Sum D E) ⥤ Sum (Sum C D) E
@@ -90,41 +129,84 @@ def inverseAssociator : Sum C (Sum D E) ⥤ Sum (Sum C D) E
| inr (inl X), inr (inl Y), f => f
| inr (inr X), inr (inr Y), f => f
#align category_theory.sum.inverse_associator CategoryTheory.sum.inverseAssociator
+-/
+/- warning: category_theory.sum.inverse_associator_obj_inl -> CategoryTheory.sum.inverseAssociator_obj_inl is a dubious translation:
+lean 3 declaration is
+ forall (C : Type.{u2}) [_inst_1 : CategoryTheory.Category.{u1, u2} C] (D : Type.{u2}) [_inst_2 : CategoryTheory.Category.{u1, u2} D] (E : Type.{u2}) [_inst_3 : CategoryTheory.Category.{u1, u2} E] (X : C), Eq.{succ u2} (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (CategoryTheory.Functor.obj.{u1, u1, u2, u2} (Sum.{u2, u2} C (Sum.{u2, u2} D E)) (CategoryTheory.sum.{u1, u2} C _inst_1 (Sum.{u2, u2} D E) (CategoryTheory.sum.{u1, u2} D _inst_2 E _inst_3)) (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (CategoryTheory.sum.{u1, u2} (Sum.{u2, u2} C D) (CategoryTheory.sum.{u1, u2} C _inst_1 D _inst_2) E _inst_3) (CategoryTheory.sum.inverseAssociator.{u1, u2} C _inst_1 D _inst_2 E _inst_3) (Sum.inl.{u2, u2} C (Sum.{u2, u2} D E) X)) (Sum.inl.{u2, u2} (Sum.{u2, u2} C D) E (Sum.inl.{u2, u2} C D X))
+but is expected to have type
+ forall (C : Type.{u2}) [_inst_1 : CategoryTheory.Category.{u1, u2} C] (D : Type.{u2}) [_inst_2 : CategoryTheory.Category.{u1, u2} D] (E : Type.{u2}) [_inst_3 : CategoryTheory.Category.{u1, u2} E] (X : C), Eq.{succ u2} (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (Prefunctor.obj.{succ u1, succ u1, u2, u2} (Sum.{u2, u2} C (Sum.{u2, u2} D E)) (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} (Sum.{u2, u2} C (Sum.{u2, u2} D E)) (CategoryTheory.Category.toCategoryStruct.{u1, u2} (Sum.{u2, u2} C (Sum.{u2, u2} D E)) (CategoryTheory.sum.{u1, u2} C _inst_1 (Sum.{u2, u2} D E) (CategoryTheory.sum.{u1, u2} D _inst_2 E _inst_3)))) (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (CategoryTheory.Category.toCategoryStruct.{u1, u2} (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (CategoryTheory.sum.{u1, u2} (Sum.{u2, u2} C D) (CategoryTheory.sum.{u1, u2} C _inst_1 D _inst_2) E _inst_3))) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u2, u2} (Sum.{u2, u2} C (Sum.{u2, u2} D E)) (CategoryTheory.sum.{u1, u2} C _inst_1 (Sum.{u2, u2} D E) (CategoryTheory.sum.{u1, u2} D _inst_2 E _inst_3)) (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (CategoryTheory.sum.{u1, u2} (Sum.{u2, u2} C D) (CategoryTheory.sum.{u1, u2} C _inst_1 D _inst_2) E _inst_3) (CategoryTheory.sum.inverseAssociator.{u1, u2} C _inst_1 D _inst_2 E _inst_3)) (Sum.inl.{u2, u2} C (Sum.{u2, u2} D E) X)) (Sum.inl.{u2, u2} (Sum.{u2, u2} C D) E (Sum.inl.{u2, u2} C D X))
+Case conversion may be inaccurate. Consider using '#align category_theory.sum.inverse_associator_obj_inl CategoryTheory.sum.inverseAssociator_obj_inlₓ'. -/
@[simp]
theorem inverseAssociator_obj_inl (X) : (inverseAssociator C D E).obj (inl X) = inl (inl X) :=
rfl
#align category_theory.sum.inverse_associator_obj_inl CategoryTheory.sum.inverseAssociator_obj_inl
+/- warning: category_theory.sum.inverse_associator_obj_inr_inl -> CategoryTheory.sum.inverseAssociator_obj_inr_inl is a dubious translation:
+lean 3 declaration is
+ forall (C : Type.{u2}) [_inst_1 : CategoryTheory.Category.{u1, u2} C] (D : Type.{u2}) [_inst_2 : CategoryTheory.Category.{u1, u2} D] (E : Type.{u2}) [_inst_3 : CategoryTheory.Category.{u1, u2} E] (X : D), Eq.{succ u2} (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (CategoryTheory.Functor.obj.{u1, u1, u2, u2} (Sum.{u2, u2} C (Sum.{u2, u2} D E)) (CategoryTheory.sum.{u1, u2} C _inst_1 (Sum.{u2, u2} D E) (CategoryTheory.sum.{u1, u2} D _inst_2 E _inst_3)) (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (CategoryTheory.sum.{u1, u2} (Sum.{u2, u2} C D) (CategoryTheory.sum.{u1, u2} C _inst_1 D _inst_2) E _inst_3) (CategoryTheory.sum.inverseAssociator.{u1, u2} C _inst_1 D _inst_2 E _inst_3) (Sum.inr.{u2, u2} C (Sum.{u2, u2} D E) (Sum.inl.{u2, u2} D E X))) (Sum.inl.{u2, u2} (Sum.{u2, u2} C D) E (Sum.inr.{u2, u2} C D X))
+but is expected to have type
+ forall (C : Type.{u2}) [_inst_1 : CategoryTheory.Category.{u1, u2} C] (D : Type.{u2}) [_inst_2 : CategoryTheory.Category.{u1, u2} D] (E : Type.{u2}) [_inst_3 : CategoryTheory.Category.{u1, u2} E] (X : D), Eq.{succ u2} (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (Prefunctor.obj.{succ u1, succ u1, u2, u2} (Sum.{u2, u2} C (Sum.{u2, u2} D E)) (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} (Sum.{u2, u2} C (Sum.{u2, u2} D E)) (CategoryTheory.Category.toCategoryStruct.{u1, u2} (Sum.{u2, u2} C (Sum.{u2, u2} D E)) (CategoryTheory.sum.{u1, u2} C _inst_1 (Sum.{u2, u2} D E) (CategoryTheory.sum.{u1, u2} D _inst_2 E _inst_3)))) (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (CategoryTheory.Category.toCategoryStruct.{u1, u2} (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (CategoryTheory.sum.{u1, u2} (Sum.{u2, u2} C D) (CategoryTheory.sum.{u1, u2} C _inst_1 D _inst_2) E _inst_3))) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u2, u2} (Sum.{u2, u2} C (Sum.{u2, u2} D E)) (CategoryTheory.sum.{u1, u2} C _inst_1 (Sum.{u2, u2} D E) (CategoryTheory.sum.{u1, u2} D _inst_2 E _inst_3)) (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (CategoryTheory.sum.{u1, u2} (Sum.{u2, u2} C D) (CategoryTheory.sum.{u1, u2} C _inst_1 D _inst_2) E _inst_3) (CategoryTheory.sum.inverseAssociator.{u1, u2} C _inst_1 D _inst_2 E _inst_3)) (Sum.inr.{u2, u2} C (Sum.{u2, u2} D E) (Sum.inl.{u2, u2} D E X))) (Sum.inl.{u2, u2} (Sum.{u2, u2} C D) E (Sum.inr.{u2, u2} C D X))
+Case conversion may be inaccurate. Consider using '#align category_theory.sum.inverse_associator_obj_inr_inl CategoryTheory.sum.inverseAssociator_obj_inr_inlₓ'. -/
@[simp]
theorem inverseAssociator_obj_inr_inl (X) :
(inverseAssociator C D E).obj (inr (inl X)) = inl (inr X) :=
rfl
#align category_theory.sum.inverse_associator_obj_inr_inl CategoryTheory.sum.inverseAssociator_obj_inr_inl
+/- warning: category_theory.sum.inverse_associator_obj_inr_inr -> CategoryTheory.sum.inverseAssociator_obj_inr_inr is a dubious translation:
+lean 3 declaration is
+ forall (C : Type.{u2}) [_inst_1 : CategoryTheory.Category.{u1, u2} C] (D : Type.{u2}) [_inst_2 : CategoryTheory.Category.{u1, u2} D] (E : Type.{u2}) [_inst_3 : CategoryTheory.Category.{u1, u2} E] (X : E), Eq.{succ u2} (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (CategoryTheory.Functor.obj.{u1, u1, u2, u2} (Sum.{u2, u2} C (Sum.{u2, u2} D E)) (CategoryTheory.sum.{u1, u2} C _inst_1 (Sum.{u2, u2} D E) (CategoryTheory.sum.{u1, u2} D _inst_2 E _inst_3)) (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (CategoryTheory.sum.{u1, u2} (Sum.{u2, u2} C D) (CategoryTheory.sum.{u1, u2} C _inst_1 D _inst_2) E _inst_3) (CategoryTheory.sum.inverseAssociator.{u1, u2} C _inst_1 D _inst_2 E _inst_3) (Sum.inr.{u2, u2} C (Sum.{u2, u2} D E) (Sum.inr.{u2, u2} D E X))) (Sum.inr.{u2, u2} (Sum.{u2, u2} C D) E X)
+but is expected to have type
+ forall (C : Type.{u2}) [_inst_1 : CategoryTheory.Category.{u1, u2} C] (D : Type.{u2}) [_inst_2 : CategoryTheory.Category.{u1, u2} D] (E : Type.{u2}) [_inst_3 : CategoryTheory.Category.{u1, u2} E] (X : E), Eq.{succ u2} (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (Prefunctor.obj.{succ u1, succ u1, u2, u2} (Sum.{u2, u2} C (Sum.{u2, u2} D E)) (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} (Sum.{u2, u2} C (Sum.{u2, u2} D E)) (CategoryTheory.Category.toCategoryStruct.{u1, u2} (Sum.{u2, u2} C (Sum.{u2, u2} D E)) (CategoryTheory.sum.{u1, u2} C _inst_1 (Sum.{u2, u2} D E) (CategoryTheory.sum.{u1, u2} D _inst_2 E _inst_3)))) (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (CategoryTheory.Category.toCategoryStruct.{u1, u2} (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (CategoryTheory.sum.{u1, u2} (Sum.{u2, u2} C D) (CategoryTheory.sum.{u1, u2} C _inst_1 D _inst_2) E _inst_3))) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u2, u2} (Sum.{u2, u2} C (Sum.{u2, u2} D E)) (CategoryTheory.sum.{u1, u2} C _inst_1 (Sum.{u2, u2} D E) (CategoryTheory.sum.{u1, u2} D _inst_2 E _inst_3)) (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (CategoryTheory.sum.{u1, u2} (Sum.{u2, u2} C D) (CategoryTheory.sum.{u1, u2} C _inst_1 D _inst_2) E _inst_3) (CategoryTheory.sum.inverseAssociator.{u1, u2} C _inst_1 D _inst_2 E _inst_3)) (Sum.inr.{u2, u2} C (Sum.{u2, u2} D E) (Sum.inr.{u2, u2} D E X))) (Sum.inr.{u2, u2} (Sum.{u2, u2} C D) E X)
+Case conversion may be inaccurate. Consider using '#align category_theory.sum.inverse_associator_obj_inr_inr CategoryTheory.sum.inverseAssociator_obj_inr_inrₓ'. -/
@[simp]
theorem inverseAssociator_obj_inr_inr (X) : (inverseAssociator C D E).obj (inr (inr X)) = inr X :=
rfl
#align category_theory.sum.inverse_associator_obj_inr_inr CategoryTheory.sum.inverseAssociator_obj_inr_inr
+/- warning: category_theory.sum.inverse_associator_map_inl -> CategoryTheory.sum.inverseAssociator_map_inl is a dubious translation:
+lean 3 declaration is
+ forall (C : Type.{u2}) [_inst_1 : CategoryTheory.Category.{u1, u2} C] (D : Type.{u2}) [_inst_2 : CategoryTheory.Category.{u1, u2} D] (E : Type.{u2}) [_inst_3 : CategoryTheory.Category.{u1, u2} E] {X : C} {Y : C} (f : Quiver.Hom.{succ u1, u2} (Sum.{u2, u2} C (Sum.{u2, u2} D E)) (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} (Sum.{u2, u2} C (Sum.{u2, u2} D E)) (CategoryTheory.Category.toCategoryStruct.{u1, u2} (Sum.{u2, u2} C (Sum.{u2, u2} D E)) (CategoryTheory.sum.{u1, u2} C _inst_1 (Sum.{u2, u2} D E) (CategoryTheory.sum.{u1, u2} D _inst_2 E _inst_3)))) (Sum.inl.{u2, u2} C (Sum.{u2, u2} D E) X) (Sum.inl.{u2, u2} C (Sum.{u2, u2} D E) Y)), Eq.{succ u1} (Quiver.Hom.{succ u1, u2} (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (CategoryTheory.Category.toCategoryStruct.{u1, u2} (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (CategoryTheory.sum.{u1, u2} (Sum.{u2, u2} C D) (CategoryTheory.sum.{u1, u2} C _inst_1 D _inst_2) E _inst_3))) (CategoryTheory.Functor.obj.{u1, u1, u2, u2} (Sum.{u2, u2} C (Sum.{u2, u2} D E)) (CategoryTheory.sum.{u1, u2} C _inst_1 (Sum.{u2, u2} D E) (CategoryTheory.sum.{u1, u2} D _inst_2 E _inst_3)) (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (CategoryTheory.sum.{u1, u2} (Sum.{u2, u2} C D) (CategoryTheory.sum.{u1, u2} C _inst_1 D _inst_2) E _inst_3) (CategoryTheory.sum.inverseAssociator.{u1, u2} C _inst_1 D _inst_2 E _inst_3) (Sum.inl.{u2, u2} C (Sum.{u2, u2} D E) X)) (CategoryTheory.Functor.obj.{u1, u1, u2, u2} (Sum.{u2, u2} C (Sum.{u2, u2} D E)) (CategoryTheory.sum.{u1, u2} C _inst_1 (Sum.{u2, u2} D E) (CategoryTheory.sum.{u1, u2} D _inst_2 E _inst_3)) (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (CategoryTheory.sum.{u1, u2} (Sum.{u2, u2} C D) (CategoryTheory.sum.{u1, u2} C _inst_1 D _inst_2) E _inst_3) (CategoryTheory.sum.inverseAssociator.{u1, u2} C _inst_1 D _inst_2 E _inst_3) (Sum.inl.{u2, u2} C (Sum.{u2, u2} D E) Y))) (CategoryTheory.Functor.map.{u1, u1, u2, u2} (Sum.{u2, u2} C (Sum.{u2, u2} D E)) (CategoryTheory.sum.{u1, u2} C _inst_1 (Sum.{u2, u2} D E) (CategoryTheory.sum.{u1, u2} D _inst_2 E _inst_3)) (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (CategoryTheory.sum.{u1, u2} (Sum.{u2, u2} C D) (CategoryTheory.sum.{u1, u2} C _inst_1 D _inst_2) E _inst_3) (CategoryTheory.sum.inverseAssociator.{u1, u2} C _inst_1 D _inst_2 E _inst_3) (Sum.inl.{u2, u2} C (Sum.{u2, u2} D E) X) (Sum.inl.{u2, u2} C (Sum.{u2, u2} D E) Y) f) f
+but is expected to have type
+ forall (C : Type.{u2}) [_inst_1 : CategoryTheory.Category.{u1, u2} C] (D : Type.{u2}) [_inst_2 : CategoryTheory.Category.{u1, u2} D] (E : Type.{u2}) [_inst_3 : CategoryTheory.Category.{u1, u2} E] {X : C} {Y : C} (f : Quiver.Hom.{succ u1, u2} (Sum.{u2, u2} C (Sum.{u2, u2} D E)) (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} (Sum.{u2, u2} C (Sum.{u2, u2} D E)) (CategoryTheory.Category.toCategoryStruct.{u1, u2} (Sum.{u2, u2} C (Sum.{u2, u2} D E)) (CategoryTheory.sum.{u1, u2} C _inst_1 (Sum.{u2, u2} D E) (CategoryTheory.sum.{u1, u2} D _inst_2 E _inst_3)))) (Sum.inl.{u2, u2} C (Sum.{u2, u2} D E) X) (Sum.inl.{u2, u2} C (Sum.{u2, u2} D E) Y)), Eq.{succ u1} (Quiver.Hom.{succ u1, u2} (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (CategoryTheory.Category.toCategoryStruct.{u1, u2} (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (CategoryTheory.sum.{u1, u2} (Sum.{u2, u2} C D) (CategoryTheory.sum.{u1, u2} C _inst_1 D _inst_2) E _inst_3))) (Prefunctor.obj.{succ u1, succ u1, u2, u2} (Sum.{u2, u2} C (Sum.{u2, u2} D E)) (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} (Sum.{u2, u2} C (Sum.{u2, u2} D E)) (CategoryTheory.Category.toCategoryStruct.{u1, u2} (Sum.{u2, u2} C (Sum.{u2, u2} D E)) (CategoryTheory.sum.{u1, u2} C _inst_1 (Sum.{u2, u2} D E) (CategoryTheory.sum.{u1, u2} D _inst_2 E _inst_3)))) (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (CategoryTheory.Category.toCategoryStruct.{u1, u2} (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (CategoryTheory.sum.{u1, u2} (Sum.{u2, u2} C D) (CategoryTheory.sum.{u1, u2} C _inst_1 D _inst_2) E _inst_3))) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u2, u2} (Sum.{u2, u2} C (Sum.{u2, u2} D E)) (CategoryTheory.sum.{u1, u2} C _inst_1 (Sum.{u2, u2} D E) (CategoryTheory.sum.{u1, u2} D _inst_2 E _inst_3)) (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (CategoryTheory.sum.{u1, u2} (Sum.{u2, u2} C D) (CategoryTheory.sum.{u1, u2} C _inst_1 D _inst_2) E _inst_3) (CategoryTheory.sum.inverseAssociator.{u1, u2} C _inst_1 D _inst_2 E _inst_3)) (Sum.inl.{u2, u2} C (Sum.{u2, u2} D E) X)) (Prefunctor.obj.{succ u1, succ u1, u2, u2} (Sum.{u2, u2} C (Sum.{u2, u2} D E)) (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} (Sum.{u2, u2} C (Sum.{u2, u2} D E)) (CategoryTheory.Category.toCategoryStruct.{u1, u2} (Sum.{u2, u2} C (Sum.{u2, u2} D E)) (CategoryTheory.sum.{u1, u2} C _inst_1 (Sum.{u2, u2} D E) (CategoryTheory.sum.{u1, u2} D _inst_2 E _inst_3)))) (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (CategoryTheory.Category.toCategoryStruct.{u1, u2} (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (CategoryTheory.sum.{u1, u2} (Sum.{u2, u2} C D) (CategoryTheory.sum.{u1, u2} C _inst_1 D _inst_2) E _inst_3))) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u2, u2} (Sum.{u2, u2} C (Sum.{u2, u2} D E)) (CategoryTheory.sum.{u1, u2} C _inst_1 (Sum.{u2, u2} D E) (CategoryTheory.sum.{u1, u2} D _inst_2 E _inst_3)) (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (CategoryTheory.sum.{u1, u2} (Sum.{u2, u2} C D) (CategoryTheory.sum.{u1, u2} C _inst_1 D _inst_2) E _inst_3) (CategoryTheory.sum.inverseAssociator.{u1, u2} C _inst_1 D _inst_2 E _inst_3)) (Sum.inl.{u2, u2} C (Sum.{u2, u2} D E) Y))) (Prefunctor.map.{succ u1, succ u1, u2, u2} (Sum.{u2, u2} C (Sum.{u2, u2} D E)) (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} (Sum.{u2, u2} C (Sum.{u2, u2} D E)) (CategoryTheory.Category.toCategoryStruct.{u1, u2} (Sum.{u2, u2} C (Sum.{u2, u2} D E)) (CategoryTheory.sum.{u1, u2} C _inst_1 (Sum.{u2, u2} D E) (CategoryTheory.sum.{u1, u2} D _inst_2 E _inst_3)))) (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (CategoryTheory.Category.toCategoryStruct.{u1, u2} (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (CategoryTheory.sum.{u1, u2} (Sum.{u2, u2} C D) (CategoryTheory.sum.{u1, u2} C _inst_1 D _inst_2) E _inst_3))) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u2, u2} (Sum.{u2, u2} C (Sum.{u2, u2} D E)) (CategoryTheory.sum.{u1, u2} C _inst_1 (Sum.{u2, u2} D E) (CategoryTheory.sum.{u1, u2} D _inst_2 E _inst_3)) (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (CategoryTheory.sum.{u1, u2} (Sum.{u2, u2} C D) (CategoryTheory.sum.{u1, u2} C _inst_1 D _inst_2) E _inst_3) (CategoryTheory.sum.inverseAssociator.{u1, u2} C _inst_1 D _inst_2 E _inst_3)) (Sum.inl.{u2, u2} C (Sum.{u2, u2} D E) X) (Sum.inl.{u2, u2} C (Sum.{u2, u2} D E) Y) f) f
+Case conversion may be inaccurate. Consider using '#align category_theory.sum.inverse_associator_map_inl CategoryTheory.sum.inverseAssociator_map_inlₓ'. -/
@[simp]
theorem inverseAssociator_map_inl {X Y : C} (f : inl X ⟶ inl Y) :
(inverseAssociator C D E).map f = f :=
rfl
#align category_theory.sum.inverse_associator_map_inl CategoryTheory.sum.inverseAssociator_map_inl
+/- warning: category_theory.sum.inverse_associator_map_inr_inl -> CategoryTheory.sum.inverseAssociator_map_inr_inl is a dubious translation:
+lean 3 declaration is
+ forall (C : Type.{u2}) [_inst_1 : CategoryTheory.Category.{u1, u2} C] (D : Type.{u2}) [_inst_2 : CategoryTheory.Category.{u1, u2} D] (E : Type.{u2}) [_inst_3 : CategoryTheory.Category.{u1, u2} E] {X : D} {Y : D} (f : Quiver.Hom.{succ u1, u2} (Sum.{u2, u2} C (Sum.{u2, u2} D E)) (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} (Sum.{u2, u2} C (Sum.{u2, u2} D E)) (CategoryTheory.Category.toCategoryStruct.{u1, u2} (Sum.{u2, u2} C (Sum.{u2, u2} D E)) (CategoryTheory.sum.{u1, u2} C _inst_1 (Sum.{u2, u2} D E) (CategoryTheory.sum.{u1, u2} D _inst_2 E _inst_3)))) (Sum.inr.{u2, u2} C (Sum.{u2, u2} D E) (Sum.inl.{u2, u2} D E X)) (Sum.inr.{u2, u2} C (Sum.{u2, u2} D E) (Sum.inl.{u2, u2} D E Y))), Eq.{succ u1} (Quiver.Hom.{succ u1, u2} (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (CategoryTheory.Category.toCategoryStruct.{u1, u2} (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (CategoryTheory.sum.{u1, u2} (Sum.{u2, u2} C D) (CategoryTheory.sum.{u1, u2} C _inst_1 D _inst_2) E _inst_3))) (CategoryTheory.Functor.obj.{u1, u1, u2, u2} (Sum.{u2, u2} C (Sum.{u2, u2} D E)) (CategoryTheory.sum.{u1, u2} C _inst_1 (Sum.{u2, u2} D E) (CategoryTheory.sum.{u1, u2} D _inst_2 E _inst_3)) (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (CategoryTheory.sum.{u1, u2} (Sum.{u2, u2} C D) (CategoryTheory.sum.{u1, u2} C _inst_1 D _inst_2) E _inst_3) (CategoryTheory.sum.inverseAssociator.{u1, u2} C _inst_1 D _inst_2 E _inst_3) (Sum.inr.{u2, u2} C (Sum.{u2, u2} D E) (Sum.inl.{u2, u2} D E X))) (CategoryTheory.Functor.obj.{u1, u1, u2, u2} (Sum.{u2, u2} C (Sum.{u2, u2} D E)) (CategoryTheory.sum.{u1, u2} C _inst_1 (Sum.{u2, u2} D E) (CategoryTheory.sum.{u1, u2} D _inst_2 E _inst_3)) (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (CategoryTheory.sum.{u1, u2} (Sum.{u2, u2} C D) (CategoryTheory.sum.{u1, u2} C _inst_1 D _inst_2) E _inst_3) (CategoryTheory.sum.inverseAssociator.{u1, u2} C _inst_1 D _inst_2 E _inst_3) (Sum.inr.{u2, u2} C (Sum.{u2, u2} D E) (Sum.inl.{u2, u2} D E Y)))) (CategoryTheory.Functor.map.{u1, u1, u2, u2} (Sum.{u2, u2} C (Sum.{u2, u2} D E)) (CategoryTheory.sum.{u1, u2} C _inst_1 (Sum.{u2, u2} D E) (CategoryTheory.sum.{u1, u2} D _inst_2 E _inst_3)) (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (CategoryTheory.sum.{u1, u2} (Sum.{u2, u2} C D) (CategoryTheory.sum.{u1, u2} C _inst_1 D _inst_2) E _inst_3) (CategoryTheory.sum.inverseAssociator.{u1, u2} C _inst_1 D _inst_2 E _inst_3) (Sum.inr.{u2, u2} C (Sum.{u2, u2} D E) (Sum.inl.{u2, u2} D E X)) (Sum.inr.{u2, u2} C (Sum.{u2, u2} D E) (Sum.inl.{u2, u2} D E Y)) f) f
+but is expected to have type
+ forall (C : Type.{u2}) [_inst_1 : CategoryTheory.Category.{u1, u2} C] (D : Type.{u2}) [_inst_2 : CategoryTheory.Category.{u1, u2} D] (E : Type.{u2}) [_inst_3 : CategoryTheory.Category.{u1, u2} E] {X : D} {Y : D} (f : Quiver.Hom.{succ u1, u2} (Sum.{u2, u2} C (Sum.{u2, u2} D E)) (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} (Sum.{u2, u2} C (Sum.{u2, u2} D E)) (CategoryTheory.Category.toCategoryStruct.{u1, u2} (Sum.{u2, u2} C (Sum.{u2, u2} D E)) (CategoryTheory.sum.{u1, u2} C _inst_1 (Sum.{u2, u2} D E) (CategoryTheory.sum.{u1, u2} D _inst_2 E _inst_3)))) (Sum.inr.{u2, u2} C (Sum.{u2, u2} D E) (Sum.inl.{u2, u2} D E X)) (Sum.inr.{u2, u2} C (Sum.{u2, u2} D E) (Sum.inl.{u2, u2} D E Y))), Eq.{succ u1} (Quiver.Hom.{succ u1, u2} (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (CategoryTheory.Category.toCategoryStruct.{u1, u2} (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (CategoryTheory.sum.{u1, u2} (Sum.{u2, u2} C D) (CategoryTheory.sum.{u1, u2} C _inst_1 D _inst_2) E _inst_3))) (Prefunctor.obj.{succ u1, succ u1, u2, u2} (Sum.{u2, u2} C (Sum.{u2, u2} D E)) (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} (Sum.{u2, u2} C (Sum.{u2, u2} D E)) (CategoryTheory.Category.toCategoryStruct.{u1, u2} (Sum.{u2, u2} C (Sum.{u2, u2} D E)) (CategoryTheory.sum.{u1, u2} C _inst_1 (Sum.{u2, u2} D E) (CategoryTheory.sum.{u1, u2} D _inst_2 E _inst_3)))) (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (CategoryTheory.Category.toCategoryStruct.{u1, u2} (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (CategoryTheory.sum.{u1, u2} (Sum.{u2, u2} C D) (CategoryTheory.sum.{u1, u2} C _inst_1 D _inst_2) E _inst_3))) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u2, u2} (Sum.{u2, u2} C (Sum.{u2, u2} D E)) (CategoryTheory.sum.{u1, u2} C _inst_1 (Sum.{u2, u2} D E) (CategoryTheory.sum.{u1, u2} D _inst_2 E _inst_3)) (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (CategoryTheory.sum.{u1, u2} (Sum.{u2, u2} C D) (CategoryTheory.sum.{u1, u2} C _inst_1 D _inst_2) E _inst_3) (CategoryTheory.sum.inverseAssociator.{u1, u2} C _inst_1 D _inst_2 E _inst_3)) (Sum.inr.{u2, u2} C (Sum.{u2, u2} D E) (Sum.inl.{u2, u2} D E X))) (Prefunctor.obj.{succ u1, succ u1, u2, u2} (Sum.{u2, u2} C (Sum.{u2, u2} D E)) (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} (Sum.{u2, u2} C (Sum.{u2, u2} D E)) (CategoryTheory.Category.toCategoryStruct.{u1, u2} (Sum.{u2, u2} C (Sum.{u2, u2} D E)) (CategoryTheory.sum.{u1, u2} C _inst_1 (Sum.{u2, u2} D E) (CategoryTheory.sum.{u1, u2} D _inst_2 E _inst_3)))) (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (CategoryTheory.Category.toCategoryStruct.{u1, u2} (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (CategoryTheory.sum.{u1, u2} (Sum.{u2, u2} C D) (CategoryTheory.sum.{u1, u2} C _inst_1 D _inst_2) E _inst_3))) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u2, u2} (Sum.{u2, u2} C (Sum.{u2, u2} D E)) (CategoryTheory.sum.{u1, u2} C _inst_1 (Sum.{u2, u2} D E) (CategoryTheory.sum.{u1, u2} D _inst_2 E _inst_3)) (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (CategoryTheory.sum.{u1, u2} (Sum.{u2, u2} C D) (CategoryTheory.sum.{u1, u2} C _inst_1 D _inst_2) E _inst_3) (CategoryTheory.sum.inverseAssociator.{u1, u2} C _inst_1 D _inst_2 E _inst_3)) (Sum.inr.{u2, u2} C (Sum.{u2, u2} D E) (Sum.inl.{u2, u2} D E Y)))) (Prefunctor.map.{succ u1, succ u1, u2, u2} (Sum.{u2, u2} C (Sum.{u2, u2} D E)) (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} (Sum.{u2, u2} C (Sum.{u2, u2} D E)) (CategoryTheory.Category.toCategoryStruct.{u1, u2} (Sum.{u2, u2} C (Sum.{u2, u2} D E)) (CategoryTheory.sum.{u1, u2} C _inst_1 (Sum.{u2, u2} D E) (CategoryTheory.sum.{u1, u2} D _inst_2 E _inst_3)))) (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (CategoryTheory.Category.toCategoryStruct.{u1, u2} (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (CategoryTheory.sum.{u1, u2} (Sum.{u2, u2} C D) (CategoryTheory.sum.{u1, u2} C _inst_1 D _inst_2) E _inst_3))) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u2, u2} (Sum.{u2, u2} C (Sum.{u2, u2} D E)) (CategoryTheory.sum.{u1, u2} C _inst_1 (Sum.{u2, u2} D E) (CategoryTheory.sum.{u1, u2} D _inst_2 E _inst_3)) (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (CategoryTheory.sum.{u1, u2} (Sum.{u2, u2} C D) (CategoryTheory.sum.{u1, u2} C _inst_1 D _inst_2) E _inst_3) (CategoryTheory.sum.inverseAssociator.{u1, u2} C _inst_1 D _inst_2 E _inst_3)) (Sum.inr.{u2, u2} C (Sum.{u2, u2} D E) (Sum.inl.{u2, u2} D E X)) (Sum.inr.{u2, u2} C (Sum.{u2, u2} D E) (Sum.inl.{u2, u2} D E Y)) f) f
+Case conversion may be inaccurate. Consider using '#align category_theory.sum.inverse_associator_map_inr_inl CategoryTheory.sum.inverseAssociator_map_inr_inlₓ'. -/
@[simp]
theorem inverseAssociator_map_inr_inl {X Y : D} (f : inr (inl X) ⟶ inr (inl Y)) :
(inverseAssociator C D E).map f = f :=
rfl
#align category_theory.sum.inverse_associator_map_inr_inl CategoryTheory.sum.inverseAssociator_map_inr_inl
+/- warning: category_theory.sum.inverse_associator_map_inr_inr -> CategoryTheory.sum.inverseAssociator_map_inr_inr is a dubious translation:
+lean 3 declaration is
+ forall (C : Type.{u2}) [_inst_1 : CategoryTheory.Category.{u1, u2} C] (D : Type.{u2}) [_inst_2 : CategoryTheory.Category.{u1, u2} D] (E : Type.{u2}) [_inst_3 : CategoryTheory.Category.{u1, u2} E] {X : E} {Y : E} (f : Quiver.Hom.{succ u1, u2} (Sum.{u2, u2} C (Sum.{u2, u2} D E)) (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} (Sum.{u2, u2} C (Sum.{u2, u2} D E)) (CategoryTheory.Category.toCategoryStruct.{u1, u2} (Sum.{u2, u2} C (Sum.{u2, u2} D E)) (CategoryTheory.sum.{u1, u2} C _inst_1 (Sum.{u2, u2} D E) (CategoryTheory.sum.{u1, u2} D _inst_2 E _inst_3)))) (Sum.inr.{u2, u2} C (Sum.{u2, u2} D E) (Sum.inr.{u2, u2} D E X)) (Sum.inr.{u2, u2} C (Sum.{u2, u2} D E) (Sum.inr.{u2, u2} D E Y))), Eq.{succ u1} (Quiver.Hom.{succ u1, u2} (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (CategoryTheory.Category.toCategoryStruct.{u1, u2} (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (CategoryTheory.sum.{u1, u2} (Sum.{u2, u2} C D) (CategoryTheory.sum.{u1, u2} C _inst_1 D _inst_2) E _inst_3))) (CategoryTheory.Functor.obj.{u1, u1, u2, u2} (Sum.{u2, u2} C (Sum.{u2, u2} D E)) (CategoryTheory.sum.{u1, u2} C _inst_1 (Sum.{u2, u2} D E) (CategoryTheory.sum.{u1, u2} D _inst_2 E _inst_3)) (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (CategoryTheory.sum.{u1, u2} (Sum.{u2, u2} C D) (CategoryTheory.sum.{u1, u2} C _inst_1 D _inst_2) E _inst_3) (CategoryTheory.sum.inverseAssociator.{u1, u2} C _inst_1 D _inst_2 E _inst_3) (Sum.inr.{u2, u2} C (Sum.{u2, u2} D E) (Sum.inr.{u2, u2} D E X))) (CategoryTheory.Functor.obj.{u1, u1, u2, u2} (Sum.{u2, u2} C (Sum.{u2, u2} D E)) (CategoryTheory.sum.{u1, u2} C _inst_1 (Sum.{u2, u2} D E) (CategoryTheory.sum.{u1, u2} D _inst_2 E _inst_3)) (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (CategoryTheory.sum.{u1, u2} (Sum.{u2, u2} C D) (CategoryTheory.sum.{u1, u2} C _inst_1 D _inst_2) E _inst_3) (CategoryTheory.sum.inverseAssociator.{u1, u2} C _inst_1 D _inst_2 E _inst_3) (Sum.inr.{u2, u2} C (Sum.{u2, u2} D E) (Sum.inr.{u2, u2} D E Y)))) (CategoryTheory.Functor.map.{u1, u1, u2, u2} (Sum.{u2, u2} C (Sum.{u2, u2} D E)) (CategoryTheory.sum.{u1, u2} C _inst_1 (Sum.{u2, u2} D E) (CategoryTheory.sum.{u1, u2} D _inst_2 E _inst_3)) (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (CategoryTheory.sum.{u1, u2} (Sum.{u2, u2} C D) (CategoryTheory.sum.{u1, u2} C _inst_1 D _inst_2) E _inst_3) (CategoryTheory.sum.inverseAssociator.{u1, u2} C _inst_1 D _inst_2 E _inst_3) (Sum.inr.{u2, u2} C (Sum.{u2, u2} D E) (Sum.inr.{u2, u2} D E X)) (Sum.inr.{u2, u2} C (Sum.{u2, u2} D E) (Sum.inr.{u2, u2} D E Y)) f) f
+but is expected to have type
+ forall (C : Type.{u2}) [_inst_1 : CategoryTheory.Category.{u1, u2} C] (D : Type.{u2}) [_inst_2 : CategoryTheory.Category.{u1, u2} D] (E : Type.{u2}) [_inst_3 : CategoryTheory.Category.{u1, u2} E] {X : E} {Y : E} (f : Quiver.Hom.{succ u1, u2} (Sum.{u2, u2} C (Sum.{u2, u2} D E)) (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} (Sum.{u2, u2} C (Sum.{u2, u2} D E)) (CategoryTheory.Category.toCategoryStruct.{u1, u2} (Sum.{u2, u2} C (Sum.{u2, u2} D E)) (CategoryTheory.sum.{u1, u2} C _inst_1 (Sum.{u2, u2} D E) (CategoryTheory.sum.{u1, u2} D _inst_2 E _inst_3)))) (Sum.inr.{u2, u2} C (Sum.{u2, u2} D E) (Sum.inr.{u2, u2} D E X)) (Sum.inr.{u2, u2} C (Sum.{u2, u2} D E) (Sum.inr.{u2, u2} D E Y))), Eq.{succ u1} (Quiver.Hom.{succ u1, u2} (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (CategoryTheory.Category.toCategoryStruct.{u1, u2} (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (CategoryTheory.sum.{u1, u2} (Sum.{u2, u2} C D) (CategoryTheory.sum.{u1, u2} C _inst_1 D _inst_2) E _inst_3))) (Prefunctor.obj.{succ u1, succ u1, u2, u2} (Sum.{u2, u2} C (Sum.{u2, u2} D E)) (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} (Sum.{u2, u2} C (Sum.{u2, u2} D E)) (CategoryTheory.Category.toCategoryStruct.{u1, u2} (Sum.{u2, u2} C (Sum.{u2, u2} D E)) (CategoryTheory.sum.{u1, u2} C _inst_1 (Sum.{u2, u2} D E) (CategoryTheory.sum.{u1, u2} D _inst_2 E _inst_3)))) (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (CategoryTheory.Category.toCategoryStruct.{u1, u2} (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (CategoryTheory.sum.{u1, u2} (Sum.{u2, u2} C D) (CategoryTheory.sum.{u1, u2} C _inst_1 D _inst_2) E _inst_3))) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u2, u2} (Sum.{u2, u2} C (Sum.{u2, u2} D E)) (CategoryTheory.sum.{u1, u2} C _inst_1 (Sum.{u2, u2} D E) (CategoryTheory.sum.{u1, u2} D _inst_2 E _inst_3)) (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (CategoryTheory.sum.{u1, u2} (Sum.{u2, u2} C D) (CategoryTheory.sum.{u1, u2} C _inst_1 D _inst_2) E _inst_3) (CategoryTheory.sum.inverseAssociator.{u1, u2} C _inst_1 D _inst_2 E _inst_3)) (Sum.inr.{u2, u2} C (Sum.{u2, u2} D E) (Sum.inr.{u2, u2} D E X))) (Prefunctor.obj.{succ u1, succ u1, u2, u2} (Sum.{u2, u2} C (Sum.{u2, u2} D E)) (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} (Sum.{u2, u2} C (Sum.{u2, u2} D E)) (CategoryTheory.Category.toCategoryStruct.{u1, u2} (Sum.{u2, u2} C (Sum.{u2, u2} D E)) (CategoryTheory.sum.{u1, u2} C _inst_1 (Sum.{u2, u2} D E) (CategoryTheory.sum.{u1, u2} D _inst_2 E _inst_3)))) (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (CategoryTheory.Category.toCategoryStruct.{u1, u2} (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (CategoryTheory.sum.{u1, u2} (Sum.{u2, u2} C D) (CategoryTheory.sum.{u1, u2} C _inst_1 D _inst_2) E _inst_3))) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u2, u2} (Sum.{u2, u2} C (Sum.{u2, u2} D E)) (CategoryTheory.sum.{u1, u2} C _inst_1 (Sum.{u2, u2} D E) (CategoryTheory.sum.{u1, u2} D _inst_2 E _inst_3)) (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (CategoryTheory.sum.{u1, u2} (Sum.{u2, u2} C D) (CategoryTheory.sum.{u1, u2} C _inst_1 D _inst_2) E _inst_3) (CategoryTheory.sum.inverseAssociator.{u1, u2} C _inst_1 D _inst_2 E _inst_3)) (Sum.inr.{u2, u2} C (Sum.{u2, u2} D E) (Sum.inr.{u2, u2} D E Y)))) (Prefunctor.map.{succ u1, succ u1, u2, u2} (Sum.{u2, u2} C (Sum.{u2, u2} D E)) (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} (Sum.{u2, u2} C (Sum.{u2, u2} D E)) (CategoryTheory.Category.toCategoryStruct.{u1, u2} (Sum.{u2, u2} C (Sum.{u2, u2} D E)) (CategoryTheory.sum.{u1, u2} C _inst_1 (Sum.{u2, u2} D E) (CategoryTheory.sum.{u1, u2} D _inst_2 E _inst_3)))) (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (CategoryTheory.Category.toCategoryStruct.{u1, u2} (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (CategoryTheory.sum.{u1, u2} (Sum.{u2, u2} C D) (CategoryTheory.sum.{u1, u2} C _inst_1 D _inst_2) E _inst_3))) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u2, u2} (Sum.{u2, u2} C (Sum.{u2, u2} D E)) (CategoryTheory.sum.{u1, u2} C _inst_1 (Sum.{u2, u2} D E) (CategoryTheory.sum.{u1, u2} D _inst_2 E _inst_3)) (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (CategoryTheory.sum.{u1, u2} (Sum.{u2, u2} C D) (CategoryTheory.sum.{u1, u2} C _inst_1 D _inst_2) E _inst_3) (CategoryTheory.sum.inverseAssociator.{u1, u2} C _inst_1 D _inst_2 E _inst_3)) (Sum.inr.{u2, u2} C (Sum.{u2, u2} D E) (Sum.inr.{u2, u2} D E X)) (Sum.inr.{u2, u2} C (Sum.{u2, u2} D E) (Sum.inr.{u2, u2} D E Y)) f) f
+Case conversion may be inaccurate. Consider using '#align category_theory.sum.inverse_associator_map_inr_inr CategoryTheory.sum.inverseAssociator_map_inr_inrₓ'. -/
@[simp]
theorem inverseAssociator_map_inr_inr {X Y : E} (f : inr (inr X) ⟶ inr (inr Y)) :
(inverseAssociator C D E).map f = f :=
rfl
#align category_theory.sum.inverse_associator_map_inr_inr CategoryTheory.sum.inverseAssociator_map_inr_inr
+/- warning: category_theory.sum.associativity -> CategoryTheory.sum.associativity is a dubious translation:
+lean 3 declaration is
+ forall (C : Type.{u2}) [_inst_1 : CategoryTheory.Category.{u1, u2} C] (D : Type.{u2}) [_inst_2 : CategoryTheory.Category.{u1, u2} D] (E : Type.{u2}) [_inst_3 : CategoryTheory.Category.{u1, u2} E], CategoryTheory.Equivalence.{u1, u1, u2, u2} (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (CategoryTheory.sum.{u1, u2} (Sum.{u2, u2} C D) (CategoryTheory.sum.{u1, u2} C _inst_1 D _inst_2) E _inst_3) (Sum.{u2, u2} C (Sum.{u2, u2} D E)) (CategoryTheory.sum.{u1, u2} C _inst_1 (Sum.{u2, u2} D E) (CategoryTheory.sum.{u1, u2} D _inst_2 E _inst_3))
+but is expected to have type
+ forall (C : Type.{u2}) [_inst_1 : CategoryTheory.Category.{u1, u2} C] (D : Type.{u2}) [_inst_2 : CategoryTheory.Category.{u1, u2} D] (E : Type.{u2}) [_inst_3 : CategoryTheory.Category.{u1, u2} E], CategoryTheory.Equivalence.{u1, u1, u2, u2} (Sum.{u2, u2} (Sum.{u2, u2} C D) E) (Sum.{u2, u2} C (Sum.{u2, u2} D E)) (CategoryTheory.sum.{u1, u2} (Sum.{u2, u2} C D) (CategoryTheory.sum.{u1, u2} C _inst_1 D _inst_2) E _inst_3) (CategoryTheory.sum.{u1, u2} C _inst_1 (Sum.{u2, u2} D E) (CategoryTheory.sum.{u1, u2} D _inst_2 E _inst_3))
+Case conversion may be inaccurate. Consider using '#align category_theory.sum.associativity CategoryTheory.sum.associativityₓ'. -/
/-- The equivalence of categories expressing associativity of sums of categories.
-/
def associativity : Sum (Sum C D) E ≌ Sum C (Sum D E) :=
@@ -133,13 +215,17 @@ def associativity : Sum (Sum C D) E ≌ Sum C (Sum D E) :=
(NatIso.ofComponents (fun X => eqToIso (by tidy)) (by tidy))
#align category_theory.sum.associativity CategoryTheory.sum.associativity
+#print CategoryTheory.sum.associatorIsEquivalence /-
instance associatorIsEquivalence : IsEquivalence (associator C D E) :=
(by infer_instance : IsEquivalence (associativity C D E).Functor)
#align category_theory.sum.associator_is_equivalence CategoryTheory.sum.associatorIsEquivalence
+-/
+#print CategoryTheory.sum.inverseAssociatorIsEquivalence /-
instance inverseAssociatorIsEquivalence : IsEquivalence (inverseAssociator C D E) :=
(by infer_instance : IsEquivalence (associativity C D E).inverse)
#align category_theory.sum.inverse_associator_is_equivalence CategoryTheory.sum.inverseAssociatorIsEquivalence
+-/
-- TODO unitors?
-- TODO pentagon natural transformation? ...satisfying?
mathlib commit https://github.com/leanprover-community/mathlib/commit/bd9851ca476957ea4549eb19b40e7b5ade9428cc
These notions on functors are now Functor.Full
, Functor.Faithful
, Functor.EssSurj
, Functor.IsEquivalence
, Functor.ReflectsIsomorphisms
. Deprecated aliases are introduced for the previous names.
@@ -140,12 +140,12 @@ def associativity : Sum (Sum C D) E ≌ Sum C (Sum D E) :=
(by rintro (_|(_|_)) (_|(_|_)) f <;> first | cases f | aesop_cat))
#align category_theory.sum.associativity CategoryTheory.sum.associativity
-instance associatorIsEquivalence : IsEquivalence (associator C D E) :=
- (by infer_instance : IsEquivalence (associativity C D E).functor)
+instance associatorIsEquivalence : (associator C D E).IsEquivalence :=
+ (by infer_instance : (associativity C D E).functor.IsEquivalence)
#align category_theory.sum.associator_is_equivalence CategoryTheory.sum.associatorIsEquivalence
-instance inverseAssociatorIsEquivalence : IsEquivalence (inverseAssociator C D E) :=
- (by infer_instance : IsEquivalence (associativity C D E).inverse)
+instance inverseAssociatorIsEquivalence : (inverseAssociator C D E).IsEquivalence :=
+ (by infer_instance : (associativity C D E).inverse.IsEquivalence)
#align category_theory.sum.inverse_associator_is_equivalence CategoryTheory.sum.inverseAssociatorIsEquivalence
-- TODO unitors?
@@ -133,10 +133,10 @@ theorem inverseAssociator_map_inr_inr {X Y : E} (f : inr (inr X) ⟶ inr (inr Y)
def associativity : Sum (Sum C D) E ≌ Sum C (Sum D E) :=
Equivalence.mk (associator C D E) (inverseAssociator C D E)
(NatIso.ofComponents (fun X => eqToIso
- (by rcases X with ((_|_)|_) <;> rfl)) -- Porting notes: aesop_cat fails
+ (by rcases X with ((_|_)|_) <;> rfl)) -- Porting note: aesop_cat fails
(by rintro ((_|_)|_) ((_|_)|_) f <;> first | cases f | aesop_cat))
(NatIso.ofComponents (fun X => eqToIso
- (by rcases X with (_|(_|_)) <;> rfl)) -- Porting notes: aesop_cat fails
+ (by rcases X with (_|(_|_)) <;> rfl)) -- Porting note: aesop_cat fails
(by rintro (_|(_|_)) (_|(_|_)) f <;> first | cases f | aesop_cat))
#align category_theory.sum.associativity CategoryTheory.sum.associativity
@@ -2,14 +2,11 @@
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-
-! This file was ported from Lean 3 source module category_theory.sums.associator
-! leanprover-community/mathlib commit 590f43db91071eb3134fef935ec9d7cd2a3bd4ce
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
-/
import Mathlib.CategoryTheory.Sums.Basic
+#align_import category_theory.sums.associator from "leanprover-community/mathlib"@"590f43db91071eb3134fef935ec9d7cd2a3bd4ce"
+
/-!
# Associator for binary disjoint union of categories.
All dependencies are ported!