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Mathlib.CategoryTheory.Sums.Associator

Associator for binary disjoint union of categories. #

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

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def CategoryTheory.sum.associator (C : Type u) [CategoryTheory.Category.{v, u} C] (D : Type u) [CategoryTheory.Category.{v, u} D] (E : Type u) [CategoryTheory.Category.{v, u} E] :
CategoryTheory.Functor ((C ⊕ D) ⊕ E) (C ⊕ D ⊕ E)

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

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    @[simp]
    theorem CategoryTheory.sum.associator_obj_inl_inl (C : Type u) [CategoryTheory.Category.{v, u} C] (D : Type u) [CategoryTheory.Category.{v, u} D] (E : Type u) [CategoryTheory.Category.{v, u} E] (X : C) :
    (CategoryTheory.sum.associator C D E).obj (Sum.inl (Sum.inl X)) = Sum.inl X
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    @[simp]
    theorem CategoryTheory.sum.associator_obj_inl_inr (C : Type u) [CategoryTheory.Category.{v, u} C] (D : Type u) [CategoryTheory.Category.{v, u} D] (E : Type u) [CategoryTheory.Category.{v, u} E] (X : D) :
    (CategoryTheory.sum.associator C D E).obj (Sum.inl (Sum.inr X)) = Sum.inr (Sum.inl X)
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    @[simp]
    theorem CategoryTheory.sum.associator_obj_inr (C : Type u) [CategoryTheory.Category.{v, u} C] (D : Type u) [CategoryTheory.Category.{v, u} D] (E : Type u) [CategoryTheory.Category.{v, u} E] (X : E) :
    (CategoryTheory.sum.associator C D E).obj (Sum.inr X) = Sum.inr (Sum.inr X)
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    @[simp]
    theorem CategoryTheory.sum.associator_map_inl_inl (C : Type u) [CategoryTheory.Category.{v, u} C] (D : Type u) [CategoryTheory.Category.{v, u} D] (E : Type u) [CategoryTheory.Category.{v, u} E] {X : C} {Y : C} (f : Sum.inl (Sum.inl X) ⟶ Sum.inl (Sum.inl Y)) :
    (CategoryTheory.sum.associator C D E).map f = f
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    @[simp]
    theorem CategoryTheory.sum.associator_map_inl_inr (C : Type u) [CategoryTheory.Category.{v, u} C] (D : Type u) [CategoryTheory.Category.{v, u} D] (E : Type u) [CategoryTheory.Category.{v, u} E] {X : D} {Y : D} (f : Sum.inl (Sum.inr X) ⟶ Sum.inl (Sum.inr Y)) :
    (CategoryTheory.sum.associator C D E).map f = f
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    @[simp]
    theorem CategoryTheory.sum.associator_map_inr (C : Type u) [CategoryTheory.Category.{v, u} C] (D : Type u) [CategoryTheory.Category.{v, u} D] (E : Type u) [CategoryTheory.Category.{v, u} E] {X : E} {Y : E} (f : Sum.inr X ⟶ Sum.inr Y) :
    (CategoryTheory.sum.associator C D E).map f = f
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    def CategoryTheory.sum.inverseAssociator (C : Type u) [CategoryTheory.Category.{v, u} C] (D : Type u) [CategoryTheory.Category.{v, u} D] (E : Type u) [CategoryTheory.Category.{v, u} E] :
    CategoryTheory.Functor (C ⊕ D ⊕ E) ((C ⊕ D) ⊕ E)

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

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      @[simp]
      theorem CategoryTheory.sum.inverseAssociator_obj_inl (C : Type u) [CategoryTheory.Category.{v, u} C] (D : Type u) [CategoryTheory.Category.{v, u} D] (E : Type u) [CategoryTheory.Category.{v, u} E] (X : C) :
      (CategoryTheory.sum.inverseAssociator C D E).obj (Sum.inl X) = Sum.inl (Sum.inl X)
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      @[simp]
      theorem CategoryTheory.sum.inverseAssociator_obj_inr_inl (C : Type u) [CategoryTheory.Category.{v, u} C] (D : Type u) [CategoryTheory.Category.{v, u} D] (E : Type u) [CategoryTheory.Category.{v, u} E] (X : D) :
      (CategoryTheory.sum.inverseAssociator C D E).obj (Sum.inr (Sum.inl X)) = Sum.inl (Sum.inr X)
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      @[simp]
      theorem CategoryTheory.sum.inverseAssociator_obj_inr_inr (C : Type u) [CategoryTheory.Category.{v, u} C] (D : Type u) [CategoryTheory.Category.{v, u} D] (E : Type u) [CategoryTheory.Category.{v, u} E] (X : E) :
      (CategoryTheory.sum.inverseAssociator C D E).obj (Sum.inr (Sum.inr X)) = Sum.inr X
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      @[simp]
      theorem CategoryTheory.sum.inverseAssociator_map_inl (C : Type u) [CategoryTheory.Category.{v, u} C] (D : Type u) [CategoryTheory.Category.{v, u} D] (E : Type u) [CategoryTheory.Category.{v, u} E] {X : C} {Y : C} (f : Sum.inl X ⟶ Sum.inl Y) :
      (CategoryTheory.sum.inverseAssociator C D E).map f = f
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      @[simp]
      theorem CategoryTheory.sum.inverseAssociator_map_inr_inl (C : Type u) [CategoryTheory.Category.{v, u} C] (D : Type u) [CategoryTheory.Category.{v, u} D] (E : Type u) [CategoryTheory.Category.{v, u} E] {X : D} {Y : D} (f : Sum.inr (Sum.inl X) ⟶ Sum.inr (Sum.inl Y)) :
      (CategoryTheory.sum.inverseAssociator C D E).map f = f
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      @[simp]
      theorem CategoryTheory.sum.inverseAssociator_map_inr_inr (C : Type u) [CategoryTheory.Category.{v, u} C] (D : Type u) [CategoryTheory.Category.{v, u} D] (E : Type u) [CategoryTheory.Category.{v, u} E] {X : E} {Y : E} (f : Sum.inr (Sum.inr X) ⟶ Sum.inr (Sum.inr Y)) :
      (CategoryTheory.sum.inverseAssociator C D E).map f = f
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      @[simp]
      theorem CategoryTheory.sum.associativity_inverse (C : Type u) [CategoryTheory.Category.{v, u} C] (D : Type u) [CategoryTheory.Category.{v, u} D] (E : Type u) [CategoryTheory.Category.{v, u} E] :
      (CategoryTheory.sum.associativity C D E).inverse = CategoryTheory.sum.inverseAssociator C D E
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      @[simp]
      theorem CategoryTheory.sum.associativity_functor (C : Type u) [CategoryTheory.Category.{v, u} C] (D : Type u) [CategoryTheory.Category.{v, u} D] (E : Type u) [CategoryTheory.Category.{v, u} E] :
      (CategoryTheory.sum.associativity C D E).functor = CategoryTheory.sum.associator C D E
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      def CategoryTheory.sum.associativity (C : Type u) [CategoryTheory.Category.{v, u} C] (D : Type u) [CategoryTheory.Category.{v, u} D] (E : Type u) [CategoryTheory.Category.{v, u} E] :
      (C ⊕ D) ⊕ E ≌ C ⊕ D ⊕ E

      The equivalence of categories expressing associativity of sums of categories.

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        instance CategoryTheory.sum.associatorIsEquivalence (C : Type u) [CategoryTheory.Category.{v, u} C] (D : Type u) [CategoryTheory.Category.{v, u} D] (E : Type u) [CategoryTheory.Category.{v, u} E] :
        (CategoryTheory.sum.associator C D E).IsEquivalence
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        • ⋯ = ⋯
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        instance CategoryTheory.sum.inverseAssociatorIsEquivalence (C : Type u) [CategoryTheory.Category.{v, u} C] (D : Type u) [CategoryTheory.Category.{v, u} D] (E : Type u) [CategoryTheory.Category.{v, u} E] :
        (CategoryTheory.sum.inverseAssociator C D E).IsEquivalence
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        • ⋯ = ⋯