Library UniMath.CategoryTheory.Bicategories.Bicategories.Examples.Initial
Initial bicategory and proof that it's univalent.
Require Import UniMath.Foundations.All.
Require Import UniMath.MoreFoundations.All.
Require Import UniMath.CategoryTheory.Categories.
Require Import UniMath.CategoryTheory.categories.StandardCategories.
Require Import UniMath.CategoryTheory.functor_categories.
Require Import UniMath.CategoryTheory.PrecategoryBinProduct.
Require Import UniMath.CategoryTheory.Bicategories.Bicategories.Bicat. Import Bicat.Notations.
Require Import UniMath.CategoryTheory.Bicategories.Bicategories.Univalence.
Require Import UniMath.CategoryTheory.Bicategories.Bicategories.Adjunctions.
Require Import UniMath.CategoryTheory.Bicategories.Bicategories.EquivToAdjequiv.
Require Import UniMath.CategoryTheory.Bicategories.Bicategories.AdjointUnique.
Local Open Scope cat.
Section Initial_Bicategory.
Definition initial_1_id_comp_cells : prebicat_1_id_comp_cells
:= tpair (λ C : precategory_data, prebicat_2cell_struct C)
empty_category
(λ (a b : empty_category) (f g : a --> b), ∅).
Definition initial_2_id_comp_struct
: prebicat_2_id_comp_struct initial_1_id_comp_cells.
Proof.
repeat split; exact (λ u : ∅, fromempty u).
Defined.
Definition initial_prebicat_data : prebicat_data
:= initial_1_id_comp_cells,, initial_2_id_comp_struct.
Lemma initial_bicat_laws : prebicat_laws initial_prebicat_data.
Proof.
repeat split; exact (λ u : ∅, fromempty u).
Qed.
Definition initial_prebicat : prebicat
:= initial_prebicat_data,, initial_bicat_laws.
Definition cellset_initial_prebicat
: isaset_cells initial_prebicat
:= λ u : ∅, fromempty u.
Definition initial_bicat : bicat
:= initial_prebicat,, cellset_initial_prebicat.
Definition initial_bicat_invertible_2cell
{x y : initial_bicat}
{f g : x --> y}
(α : f ==> g)
: is_invertible_2cell α.
Proof.
exact (fromempty x).
Defined.
Definition initial_bicat_adjoint_equivalence
{x y : initial_bicat}
(f : x --> y)
: left_adjoint_equivalence f.
Proof.
exact (fromempty x).
Defined.
It is univalent
Definition initial_bicat_is_univalent_2_1
: is_univalent_2_1 initial_bicat.
Proof.
intros x.
exact (fromempty x).
Defined.
Definition initial_bicat_is_univalent_2_0
: is_univalent_2_0 initial_bicat.
Proof.
intros x.
exact (fromempty x).
Defined.
End Initial_Bicategory.
: is_univalent_2_1 initial_bicat.
Proof.
intros x.
exact (fromempty x).
Defined.
Definition initial_bicat_is_univalent_2_0
: is_univalent_2_0 initial_bicat.
Proof.
intros x.
exact (fromempty x).
Defined.
End Initial_Bicategory.