Skip to content
New issue

Have a question about this project? Sign up for a free GitHub account to open an issue and contact its maintainers and the community.

Sign up for GitHub

By clicking “Sign up for GitHub”, you agree to our terms of service and privacy statement. We’ll occasionally send you account related emails.

Already on GitHub? Sign in to your account

Jump to bottom

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

Closed
wants to merge 2 commits into from
Closed

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

Show file tree
Hide file tree
Changes from 1 commit
Commits
Show all changes
2 commits
Select commit Hold shift + click to select a range
0160e65
chore(category_theory/*): doc-strings
semorrison Oct 5, 2020
4d7fa45
Apply suggestions from code review
semorrison Oct 6, 2020
File filter

Filter by extension

Filter by extension
Conversations
Failed to load comments.
Loading
Jump to
Jump to file
Failed to load files.
Loading
Diff view
Diff view
9 changes: 9 additions & 0 deletions src/category_theory/endomorphism.lean
View file
Open in desktop
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.

Arguments order in multiplication agrees with
semorrison marked this conversation as resolved.
Show resolved Hide resolved
`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
View file
Open in desktop
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 morphims live in the same universe.
semorrison marked this conversation as resolved.
Show resolved Hide resolved
-/
abbreviation small_groupoid (C : Type u) : Type (u+1) := groupoid.{u} C

section
Expand Down
10 changes: 10 additions & 0 deletions src/category_theory/monoidal/functor.lean
View file
Open in desktop
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