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[Merged by Bors] - chore(category_theory/*): doc-strings #4453

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9 changes: 9 additions & 0 deletions src/category_theory/endomorphism.lean
Original file line number Diff line number Diff line change
Expand Up @@ -23,6 +23,7 @@ section struct
variables {C : Type u} [category_struct.{v} C] (X : C)

instance has_one : has_one (End X) := ⟨𝟙 X⟩
instance inhabited : inhabited (End X) := ⟨𝟙 X⟩

/-- Multiplication of endomorphisms agrees with `function.comp`, not `category_struct.comp`. -/
instance has_mul : has_mul (End X) := ⟨λ x y, y ≫ x⟩
Expand Down Expand Up @@ -50,12 +51,20 @@ end End

variables {C : Type u} [category.{v} C] (X : C)

/--
Automorphisms of an object in a category.

The order of arguments in multiplication agrees with
`function.comp`, not with `category.comp`.
-/
def Aut (X : C) := X ≅ X

attribute [ext Aut] iso.ext

namespace Aut

instance inhabited : inhabited (Aut X) := ⟨iso.refl X⟩

instance : group (Aut X) :=
by refine { one := iso.refl X,
inv := iso.symm,
Expand Down
8 changes: 8 additions & 0 deletions src/category_theory/groupoid.lean
Original file line number Diff line number Diff line change
Expand Up @@ -20,7 +20,15 @@ restate_axiom groupoid.comp_inv'

attribute [simp] groupoid.inv_comp groupoid.comp_inv

/--
A `large_groupoid` is a groupoid
where the objects live in `Type (u+1)` while the morphisms live in `Type u`.
-/
abbreviation large_groupoid (C : Type (u+1)) : Type (u+1) := groupoid.{u} C
/--
A `small_groupoid` is a groupoid
where the objects and morphisms live in the same universe.
-/
abbreviation small_groupoid (C : Type u) : Type (u+1) := groupoid.{u} C

section
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10 changes: 10 additions & 0 deletions src/category_theory/monoidal/functor.lean
Original file line number Diff line number Diff line change
Expand Up @@ -75,9 +75,15 @@ attribute [instance] monoidal_functor.ε_is_iso monoidal_functor.μ_is_iso

variables {C D}

/--
The unit morphism of a (strong) monoidal functor as an isomorphism.
-/
def monoidal_functor.ε_iso (F : monoidal_functor.{v₁ v₂} C D) :
tensor_unit D ≅ F.obj (tensor_unit C) :=
as_iso F.ε
/--
The tensorator of a (strong) monoidal functor as an isomorphism.
-/
def monoidal_functor.μ_iso (F : monoidal_functor.{v₁ v₂} C D) (X Y : C) :
(F.obj X) ⊗ (F.obj Y) ≅ F.obj (X ⊗ Y) :=
as_iso (F.μ X Y)
Expand All @@ -96,6 +102,8 @@ variables (C : Type u₁) [category.{v₁} C] [monoidal_category.{v₁} C]
μ := λ X Y, 𝟙 _,
.. 𝟭 C }

instance : inhabited (lax_monoidal_functor C C) := ⟨id C⟩

end lax_monoidal_functor

namespace monoidal_functor
Expand Down Expand Up @@ -141,6 +149,8 @@ variables (C : Type u₁) [category.{v₁} C] [monoidal_category.{v₁} C]
μ := λ X Y, 𝟙 _,
.. 𝟭 C }

instance : inhabited (monoidal_functor C C) := ⟨id C⟩

end

end monoidal_functor
Expand Down