File: Mathlib/CategoryTheory/Sums/Associator.lean

full_name: CategoryTheory.sum.associator
code: /-- The associator functor `(C ⊕ D) ⊕ E ⥤ C ⊕ (D ⊕ E)` for sums of categories.
-/
def associator : (C ⊕ D) ⊕ E ⥤ C ⊕ (D ⊕ E) 
goal: (C ⊕ D) ⊕ E ⥤ C ⊕ D ⊕ E

full_name: CategoryTheory.sum.associator_obj_inl_inl
code: @[simp]
theorem associator_obj_inl_inl (X) : (associator C D E).obj (inl (inl X)) = inl X 
goal: (CategoryTheory.sum.associator C D E).obj (Sum.inl (Sum.inl X)) = Sum.inl X

full_name: CategoryTheory.sum.associator_obj_inl_inr
code: @[simp]
theorem associator_obj_inl_inr (X) : (associator C D E).obj (inl (inr X)) = inr (inl X) 
goal: (CategoryTheory.sum.associator C D E).obj (Sum.inl (Sum.inr X)) = Sum.inr (Sum.inl X)

full_name: CategoryTheory.sum.associator_obj_inr
code: @[simp]
theorem associator_obj_inr (X) : (associator C D E).obj (inr X) = inr (inr X) 
goal: (CategoryTheory.sum.associator C D E).obj (Sum.inr X) = Sum.inr (Sum.inr X)

full_name: CategoryTheory.sum.associator_map_inl_inl
code: @[simp]
theorem associator_map_inl_inl {X Y : C} (f : inl (inl X) ⟶ inl (inl Y)) :
    (associator C D E).map f = f 
goal: (CategoryTheory.sum.associator C D E).map f = f

full_name: CategoryTheory.sum.associator_map_inl_inr
code: @[simp]
theorem associator_map_inl_inr {X Y : D} (f : inl (inr X) ⟶ inl (inr Y)) :
    (associator C D E).map f = f 
goal: (CategoryTheory.sum.associator C D E).map f = f

full_name: CategoryTheory.sum.associator_map_inr
code: @[simp]
theorem associator_map_inr {X Y : E} (f : inr X ⟶ inr Y) : (associator C D E).map f = f 
goal: (CategoryTheory.sum.associator C D E).map f = f

full_name: CategoryTheory.sum.inverseAssociator
code: /-- The inverse associator functor `C ⊕ (D ⊕ E) ⥤ (C ⊕ D) ⊕ E` for sums of categories.
-/
def inverseAssociator : C ⊕ (D ⊕ E) ⥤ (C ⊕ D) ⊕ E 
goal: C ⊕ D ⊕ E ⥤ (C ⊕ D) ⊕ E

full_name: CategoryTheory.sum.inverseAssociator_obj_inl
code: @[simp]
theorem inverseAssociator_obj_inl (X) : (inverseAssociator C D E).obj (inl X) = inl (inl X) 
goal: (CategoryTheory.sum.inverseAssociator C D E).obj (Sum.inl X) = Sum.inl (Sum.inl X)

full_name: CategoryTheory.sum.inverseAssociator_obj_inr_inl
code: @[simp]
theorem inverseAssociator_obj_inr_inl (X) :
    (inverseAssociator C D E).obj (inr (inl X)) = inl (inr X) 
goal: (CategoryTheory.sum.inverseAssociator C D E).obj (Sum.inr (Sum.inl X)) = Sum.inl (Sum.inr X)

full_name: CategoryTheory.sum.inverseAssociator_obj_inr_inr
code: @[simp]
theorem inverseAssociator_obj_inr_inr (X) : (inverseAssociator C D E).obj (inr (inr X)) = inr X 
goal: (CategoryTheory.sum.inverseAssociator C D E).obj (Sum.inr (Sum.inr X)) = Sum.inr X

full_name: CategoryTheory.sum.inverseAssociator_map_inl
code: @[simp]
theorem inverseAssociator_map_inl {X Y : C} (f : inl X ⟶ inl Y) :
    (inverseAssociator C D E).map f = f 
goal: (CategoryTheory.sum.inverseAssociator C D E).map f = f

full_name: CategoryTheory.sum.inverseAssociator_map_inr_inl
code: @[simp]
theorem inverseAssociator_map_inr_inl {X Y : D} (f : inr (inl X) ⟶ inr (inl Y)) :
    (inverseAssociator C D E).map f = f 
goal: (CategoryTheory.sum.inverseAssociator C D E).map f = f

full_name: CategoryTheory.sum.inverseAssociator_map_inr_inr
code: @[simp]
theorem inverseAssociator_map_inr_inr {X Y : E} (f : inr (inr X) ⟶ inr (inr Y)) :
    (inverseAssociator C D E).map f = f 
goal: (CategoryTheory.sum.inverseAssociator C D E).map f = f

full_name: CategoryTheory.sum.associativity
code: /-- The equivalence of categories expressing associativity of sums of categories.
-/
def associativity : (C ⊕ D) ⊕ E ≌ C ⊕ (D ⊕ E) 
goal: (C ⊕ D) ⊕ E ≌ C ⊕ D ⊕ E

full_name: CategoryTheory.sum.associatorIsEquivalence
code: instance associatorIsEquivalence : (associator C D E).IsEquivalence 
goal: (CategoryTheory.sum.associator C D E).IsEquivalence

full_name: CategoryTheory.sum.inverseAssociatorIsEquivalence
code: instance inverseAssociatorIsEquivalence : (inverseAssociator C D E).IsEquivalence 
goal: (CategoryTheory.sum.inverseAssociator C D E).IsEquivalence