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category_theory.sums.associator

Associator for binary disjoint union of categories. #

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The associator functor ((C ⊕ D) ⊕ E) ⥤ (C ⊕ (D ⊕ E)) and its inverse form an equivalence.

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def category_theory.sum.associator (C : Type u) [category_theory.category C] (D : Type u) [category_theory.category D] (E : Type u) [category_theory.category E] :
(C ⊕ D) ⊕ E ⥤ C ⊕ D ⊕ E

The associator functor (C ⊕ D) ⊕ E ⥤ C ⊕ (D ⊕ E) for sums of categories.

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Instances for category_theory.sum.associator
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@[simp]
theorem category_theory.sum.associator_obj_inl_inl (C : Type u) [category_theory.category C] (D : Type u) [category_theory.category D] (E : Type u) [category_theory.category E] (X : C) :
(category_theory.sum.associator C D E).obj (sum.inl (sum.inl X)) = sum.inl X
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@[simp]
theorem category_theory.sum.associator_obj_inl_inr (C : Type u) [category_theory.category C] (D : Type u) [category_theory.category D] (E : Type u) [category_theory.category E] (X : D) :
(category_theory.sum.associator C D E).obj (sum.inl (sum.inr X)) = sum.inr (sum.inl X)
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@[simp]
theorem category_theory.sum.associator_obj_inr (C : Type u) [category_theory.category C] (D : Type u) [category_theory.category D] (E : Type u) [category_theory.category E] (X : E) :
(category_theory.sum.associator C D E).obj (sum.inr X) = sum.inr (sum.inr X)
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@[simp]
theorem category_theory.sum.associator_map_inl_inl (C : Type u) [category_theory.category C] (D : Type u) [category_theory.category D] (E : Type u) [category_theory.category E] {X Y : C} (f : sum.inl (sum.inl X) ⟶ sum.inl (sum.inl Y)) :
(category_theory.sum.associator C D E).map f = f
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@[simp]
theorem category_theory.sum.associator_map_inl_inr (C : Type u) [category_theory.category C] (D : Type u) [category_theory.category D] (E : Type u) [category_theory.category E] {X Y : D} (f : sum.inl (sum.inr X) ⟶ sum.inl (sum.inr Y)) :
(category_theory.sum.associator C D E).map f = f
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@[simp]
theorem category_theory.sum.associator_map_inr (C : Type u) [category_theory.category C] (D : Type u) [category_theory.category D] (E : Type u) [category_theory.category E] {X Y : E} (f : sum.inr X ⟶ sum.inr Y) :
(category_theory.sum.associator C D E).map f = f
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def category_theory.sum.inverse_associator (C : Type u) [category_theory.category C] (D : Type u) [category_theory.category D] (E : Type u) [category_theory.category E] :
C ⊕ D ⊕ E ⥤ (C ⊕ D) ⊕ E

The inverse associator functor C ⊕ (D ⊕ E) ⥤ (C ⊕ D) ⊕ E for sums of categories.

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Instances for category_theory.sum.inverse_associator
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@[simp]
theorem category_theory.sum.inverse_associator_obj_inl (C : Type u) [category_theory.category C] (D : Type u) [category_theory.category D] (E : Type u) [category_theory.category E] (X : C) :
(category_theory.sum.inverse_associator C D E).obj (sum.inl X) = sum.inl (sum.inl X)
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@[simp]
theorem category_theory.sum.inverse_associator_obj_inr_inl (C : Type u) [category_theory.category C] (D : Type u) [category_theory.category D] (E : Type u) [category_theory.category E] (X : D) :
(category_theory.sum.inverse_associator C D E).obj (sum.inr (sum.inl X)) = sum.inl (sum.inr X)
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@[simp]
theorem category_theory.sum.inverse_associator_obj_inr_inr (C : Type u) [category_theory.category C] (D : Type u) [category_theory.category D] (E : Type u) [category_theory.category E] (X : E) :
(category_theory.sum.inverse_associator C D E).obj (sum.inr (sum.inr X)) = sum.inr X
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@[simp]
theorem category_theory.sum.inverse_associator_map_inl (C : Type u) [category_theory.category C] (D : Type u) [category_theory.category D] (E : Type u) [category_theory.category E] {X Y : C} (f : sum.inl X ⟶ sum.inl Y) :
(category_theory.sum.inverse_associator C D E).map f = f
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@[simp]
theorem category_theory.sum.inverse_associator_map_inr_inl (C : Type u) [category_theory.category C] (D : Type u) [category_theory.category D] (E : Type u) [category_theory.category E] {X Y : D} (f : sum.inr (sum.inl X) ⟶ sum.inr (sum.inl Y)) :
(category_theory.sum.inverse_associator C D E).map f = f
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@[simp]
theorem category_theory.sum.inverse_associator_map_inr_inr (C : Type u) [category_theory.category C] (D : Type u) [category_theory.category D] (E : Type u) [category_theory.category E] {X Y : E} (f : sum.inr (sum.inr X) ⟶ sum.inr (sum.inr Y)) :
(category_theory.sum.inverse_associator C D E).map f = f
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def category_theory.sum.associativity (C : Type u) [category_theory.category C] (D : Type u) [category_theory.category D] (E : Type u) [category_theory.category E] :
(C ⊕ D) ⊕ E ≌ C ⊕ D ⊕ E

The equivalence of categories expressing associativity of sums of categories.

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@[protected, instance]
def category_theory.sum.associator_is_equivalence (C : Type u) [category_theory.category C] (D : Type u) [category_theory.category D] (E : Type u) [category_theory.category E] :
category_theory.is_equivalence (category_theory.sum.associator C D E)
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@[protected, instance]
def category_theory.sum.inverse_associator_is_equivalence (C : Type u) [category_theory.category C] (D : Type u) [category_theory.category D] (E : Type u) [category_theory.category E] :
category_theory.is_equivalence (category_theory.sum.inverse_associator C D E)
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