Library UniMath.Bicategories.PseudoFunctors.Examples.PathGroupoid
Bequivalence between 1-types and univalent groupoids
Require Import UniMath.Foundations.All.
Require Import UniMath.MoreFoundations.All.
Require Import UniMath.CategoryTheory.Core.Categories.
Require Import UniMath.CategoryTheory.Core.Functors.
Require Import UniMath.CategoryTheory.Core.NaturalTransformations.
Require Import UniMath.CategoryTheory.Core.Isos.
Require Import UniMath.CategoryTheory.Core.Univalence.
Require Import UniMath.CategoryTheory.Groupoids.
Require Import UniMath.CategoryTheory.categories.StandardCategories.
Require Import UniMath.Bicategories.Core.Bicat. Import Bicat.Notations.
Require Import UniMath.Bicategories.Core.Invertible_2cells.
Require Import UniMath.Bicategories.Core.BicategoryLaws.
Require Import UniMath.Bicategories.PseudoFunctors.Display.PseudoFunctorBicat.
Require Import UniMath.Bicategories.PseudoFunctors.Display.Base.
Require Import UniMath.Bicategories.PseudoFunctors.Display.Map1Cells.
Require Import UniMath.Bicategories.PseudoFunctors.Display.Map2Cells.
Require Import UniMath.Bicategories.PseudoFunctors.Display.StrictIdentitor.
Require Import UniMath.Bicategories.PseudoFunctors.Display.StrictCompositor.
Require Import UniMath.Bicategories.PseudoFunctors.PseudoFunctor.
Require Import UniMath.Bicategories.PseudoFunctors.Biequivalence.
Require Import UniMath.Bicategories.PseudoFunctors.Examples.Identity.
Require Import UniMath.Bicategories.PseudoFunctors.Examples.Composition.
Import PseudoFunctor.Notations.
Require Import UniMath.Bicategories.PseudoFunctors.Biadjunction.
Require Import UniMath.Bicategories.Transformations.PseudoTransformation.
Require Import UniMath.Bicategories.Transformations.Examples.Whiskering.
Require Import UniMath.Bicategories.Transformations.Examples.Unitality.
Require Import UniMath.Bicategories.Transformations.Examples.Associativity.
Require Import UniMath.Bicategories.Modifications.Modification.
Require Import UniMath.Bicategories.Core.Examples.OneTypes.
Require Import UniMath.Bicategories.Core.Examples.Groupoids.
Require Import UniMath.Bicategories.Core.Examples.BicatOfUnivCats.
Local Open Scope cat.
Definition one_type_to_groupoid
(X : one_type)
: univalent_groupoid
:= path_univalent_groupoid (pr2 X).
Require Import UniMath.MoreFoundations.All.
Require Import UniMath.CategoryTheory.Core.Categories.
Require Import UniMath.CategoryTheory.Core.Functors.
Require Import UniMath.CategoryTheory.Core.NaturalTransformations.
Require Import UniMath.CategoryTheory.Core.Isos.
Require Import UniMath.CategoryTheory.Core.Univalence.
Require Import UniMath.CategoryTheory.Groupoids.
Require Import UniMath.CategoryTheory.categories.StandardCategories.
Require Import UniMath.Bicategories.Core.Bicat. Import Bicat.Notations.
Require Import UniMath.Bicategories.Core.Invertible_2cells.
Require Import UniMath.Bicategories.Core.BicategoryLaws.
Require Import UniMath.Bicategories.PseudoFunctors.Display.PseudoFunctorBicat.
Require Import UniMath.Bicategories.PseudoFunctors.Display.Base.
Require Import UniMath.Bicategories.PseudoFunctors.Display.Map1Cells.
Require Import UniMath.Bicategories.PseudoFunctors.Display.Map2Cells.
Require Import UniMath.Bicategories.PseudoFunctors.Display.StrictIdentitor.
Require Import UniMath.Bicategories.PseudoFunctors.Display.StrictCompositor.
Require Import UniMath.Bicategories.PseudoFunctors.PseudoFunctor.
Require Import UniMath.Bicategories.PseudoFunctors.Biequivalence.
Require Import UniMath.Bicategories.PseudoFunctors.Examples.Identity.
Require Import UniMath.Bicategories.PseudoFunctors.Examples.Composition.
Import PseudoFunctor.Notations.
Require Import UniMath.Bicategories.PseudoFunctors.Biadjunction.
Require Import UniMath.Bicategories.Transformations.PseudoTransformation.
Require Import UniMath.Bicategories.Transformations.Examples.Whiskering.
Require Import UniMath.Bicategories.Transformations.Examples.Unitality.
Require Import UniMath.Bicategories.Transformations.Examples.Associativity.
Require Import UniMath.Bicategories.Modifications.Modification.
Require Import UniMath.Bicategories.Core.Examples.OneTypes.
Require Import UniMath.Bicategories.Core.Examples.Groupoids.
Require Import UniMath.Bicategories.Core.Examples.BicatOfUnivCats.
Local Open Scope cat.
Definition one_type_to_groupoid
(X : one_type)
: univalent_groupoid
:= path_univalent_groupoid (pr2 X).
Action on morphisms
Definition function_to_functor_data
{X Y : one_type}
(f : X → Y)
: functor_data (one_type_to_groupoid X) (one_type_to_groupoid Y).
Show proof.
Definition function_to_functor_laws
{X Y : one_type}
(f : X → Y)
: is_functor (function_to_functor_data f).
Show proof.
Definition function_to_functor
{X Y : one_type}
(f : X → Y)
: one_type_to_groupoid X ⟶ one_type_to_groupoid Y.
Show proof.
{X Y : one_type}
(f : X → Y)
: functor_data (one_type_to_groupoid X) (one_type_to_groupoid Y).
Show proof.
Definition function_to_functor_laws
{X Y : one_type}
(f : X → Y)
: is_functor (function_to_functor_data f).
Show proof.
Definition function_to_functor
{X Y : one_type}
(f : X → Y)
: one_type_to_groupoid X ⟶ one_type_to_groupoid Y.
Show proof.
Action on cells
Definition path_to_nattrans
{X Y : one_type}
{f g : X → Y}
(p : homotsec f g)
: function_to_functor f ⟹ function_to_functor g.
Show proof.
{X Y : one_type}
{f g : X → Y}
(p : homotsec f g)
: function_to_functor f ⟹ function_to_functor g.
Show proof.
use make_nat_trans.
- exact p.
- abstract
(intros x y q ; cbn in * ;
induction q ; simpl ;
refine (!_) ;
apply pathscomp0rid).
- exact p.
- abstract
(intros x y q ; cbn in * ;
induction q ; simpl ;
refine (!_) ;
apply pathscomp0rid).
Identitor
Definition path_groupoid_identitor
(X : one_type)
: functor_identity _ ⟹ function_to_functor (λ (x : X), x).
Show proof.
(X : one_type)
: functor_identity _ ⟹ function_to_functor (λ (x : X), x).
Show proof.
use make_nat_trans.
- exact idpath.
- abstract
(intros x y p ; cbn in * ;
induction p ; cbn ;
apply idpath).
- exact idpath.
- abstract
(intros x y p ; cbn in * ;
induction p ; cbn ;
apply idpath).
Compositor
Definition path_groupoid_compositor
{X Y Z : one_type}
(f : X → Y)
(g : Y → Z)
: (function_to_functor f ∙ function_to_functor g)
⟹
function_to_functor (λ x, g(f x)).
Show proof.
{X Y Z : one_type}
(f : X → Y)
(g : Y → Z)
: (function_to_functor f ∙ function_to_functor g)
⟹
function_to_functor (λ x, g(f x)).
Show proof.
use make_nat_trans.
- exact (λ x, idpath (g(f x))).
- abstract
(intros x y p ; cbn in * ;
induction p ; cbn ;
apply idpath).
- exact (λ x, idpath (g(f x))).
- abstract
(intros x y p ; cbn in * ;
induction p ; cbn ;
apply idpath).
Data of path groupoid pseudofunctor
Definition path_groupoid_data
: psfunctor_data one_types grpds.
Show proof.
: psfunctor_data one_types grpds.
Show proof.
use make_psfunctor_data.
- exact one_type_to_groupoid.
- exact (λ _ _ f, function_to_functor f ,, tt).
- exact (λ _ _ _ _ p, path_to_nattrans p ,, tt).
- exact (λ X, path_groupoid_identitor X ,, tt).
- exact (λ _ _ _ f g, path_groupoid_compositor f g ,, tt).
- exact one_type_to_groupoid.
- exact (λ _ _ f, function_to_functor f ,, tt).
- exact (λ _ _ _ _ p, path_to_nattrans p ,, tt).
- exact (λ X, path_groupoid_identitor X ,, tt).
- exact (λ _ _ _ f g, path_groupoid_compositor f g ,, tt).
Laws
Definition path_groupoid_laws
: psfunctor_laws path_groupoid_data.
Show proof.
: psfunctor_laws path_groupoid_data.
Show proof.
repeat split.
- intros X Y f.
use subtypePath.
{ intro ; apply isapropunit. }
apply nat_trans_eq.
{ apply homset_property. }
intro x ; cbn.
apply idpath.
- intros X Y f g h p q.
use subtypePath.
{ intro ; apply isapropunit. }
apply nat_trans_eq.
{ apply homset_property. }
intros x ; cbn.
apply idpath.
- intros X Y f.
use subtypePath.
{ intro ; apply isapropunit. }
apply nat_trans_eq.
{ apply homset_property. }
intros x ; cbn.
apply idpath.
- intros X Y f.
use subtypePath.
{ intro ; apply isapropunit. }
apply nat_trans_eq.
{ apply homset_property. }
intros x ; cbn.
apply idpath.
- intros W X Y Z f g h.
use subtypePath.
{ intro ; apply isapropunit. }
apply nat_trans_eq.
{ apply homset_property. }
intros x ; cbn.
apply idpath.
- intros X Y Z f g₁ g₂ p.
use subtypePath.
{ intro ; apply isapropunit. }
apply nat_trans_eq.
{ apply homset_property. }
intros x ; cbn.
cbn.
refine (!_).
apply pathscomp0rid.
- intros X Y Z f₁ f₂ g p.
use subtypePath.
{ intro ; apply isapropunit. }
apply nat_trans_eq.
{ apply homset_property. }
intros x ; cbn.
cbn.
refine (!_).
apply pathscomp0rid.
- intros X Y f.
use subtypePath.
{ intro ; apply isapropunit. }
apply nat_trans_eq.
{ apply homset_property. }
intro x ; cbn.
apply idpath.
- intros X Y f g h p q.
use subtypePath.
{ intro ; apply isapropunit. }
apply nat_trans_eq.
{ apply homset_property. }
intros x ; cbn.
apply idpath.
- intros X Y f.
use subtypePath.
{ intro ; apply isapropunit. }
apply nat_trans_eq.
{ apply homset_property. }
intros x ; cbn.
apply idpath.
- intros X Y f.
use subtypePath.
{ intro ; apply isapropunit. }
apply nat_trans_eq.
{ apply homset_property. }
intros x ; cbn.
apply idpath.
- intros W X Y Z f g h.
use subtypePath.
{ intro ; apply isapropunit. }
apply nat_trans_eq.
{ apply homset_property. }
intros x ; cbn.
apply idpath.
- intros X Y Z f g₁ g₂ p.
use subtypePath.
{ intro ; apply isapropunit. }
apply nat_trans_eq.
{ apply homset_property. }
intros x ; cbn.
cbn.
refine (!_).
apply pathscomp0rid.
- intros X Y Z f₁ f₂ g p.
use subtypePath.
{ intro ; apply isapropunit. }
apply nat_trans_eq.
{ apply homset_property. }
intros x ; cbn.
cbn.
refine (!_).
apply pathscomp0rid.
The identitor and compositor are invertible
The pseudofunctor
Definition path_groupoid
: psfunctor one_types grpds.
Show proof.
: psfunctor one_types grpds.
Show proof.
use make_psfunctor.
- exact path_groupoid_data.
- exact path_groupoid_laws.
- exact path_groupoid_invertible_cells.
- exact path_groupoid_data.
- exact path_groupoid_laws.
- exact path_groupoid_invertible_cells.
The adjoint
Definition objects_of_grpd_data
: psfunctor_data grpds one_types.
Show proof.
Definition objects_of_grpd_is_psfunctor
: psfunctor_laws objects_of_grpd_data.
Show proof.
Definition objects_of_grpd
: psfunctor grpds one_types.
Show proof.
: psfunctor_data grpds one_types.
Show proof.
use make_psfunctor_data.
- refine (λ G, make_one_type _ (univalent_category_has_groupoid_ob _)).
apply G.
- exact (λ _ _ F x, (pr1 F : _ ⟶ _) x).
- intros G₁ G₂ F₁ F₂ α ; simpl.
intro x.
apply isotoid.
{ apply (pr1 G₂). }
use make_z_iso'.
+ exact ((pr1 α : _ ⟹ _) x).
+ exact (pr2 G₂ _ _ ((pr1 α : _ ⟹ _) x)).
- exact (λ _ _, idpath _).
- exact (λ _ _ _ _ _ _, idpath _).
- refine (λ G, make_one_type _ (univalent_category_has_groupoid_ob _)).
apply G.
- exact (λ _ _ F x, (pr1 F : _ ⟶ _) x).
- intros G₁ G₂ F₁ F₂ α ; simpl.
intro x.
apply isotoid.
{ apply (pr1 G₂). }
use make_z_iso'.
+ exact ((pr1 α : _ ⟹ _) x).
+ exact (pr2 G₂ _ _ ((pr1 α : _ ⟹ _) x)).
- exact (λ _ _, idpath _).
- exact (λ _ _ _ _ _ _, idpath _).
Definition objects_of_grpd_is_psfunctor
: psfunctor_laws objects_of_grpd_data.
Show proof.
repeat split.
- intros G₁ G₂ F ; cbn.
use funextsec.
intro x.
refine (_ @ isotoid_identity_iso _ _ _).
apply maponpaths.
use subtypePath.
{ intro ; apply isaprop_is_z_isomorphism. }
apply idpath.
- intros G₁ G₂ F₁ F₂ F₃ α₁ α₂ ; cbn.
use funextsec.
intro x ; unfold homotcomp.
rewrite <- isotoid_comp.
apply maponpaths.
use subtypePath.
{ intro ; apply isaprop_is_z_isomorphism. }
apply idpath.
- intros G₁ G₂ F ; cbn.
use funextsec.
intro x ; unfold homotcomp, homotfun ; cbn.
refine (!_).
refine (_ @ isotoid_identity_iso _ _ _).
apply maponpaths.
use subtypePath.
{ intro ; apply isaprop_is_z_isomorphism. }
apply idpath.
- intros G₁ G₂ F ; cbn.
use funextsec.
intro x ; unfold homotcomp, homotfun ; cbn.
refine (!_).
refine (_ @ isotoid_identity_iso _ _ _).
apply maponpaths.
use subtypePath.
{ intro ; apply isaprop_is_z_isomorphism. }
apply idpath.
- intros G₁ G₂ G₃ G₄ F₁ F₂ F₃ ; cbn.
use funextsec.
intro x ; unfold homotcomp, homotfun ; cbn.
refine (_ @ isotoid_identity_iso _ _ _).
apply maponpaths.
use subtypePath.
{ intro ; apply isaprop_is_z_isomorphism. }
apply idpath.
- intros G₁ G₂ G₃ F₁ F₂ F₃ α ; cbn.
use funextsec.
intro x ; unfold homotcomp, homotfun, funhomotsec ; cbn.
rewrite pathscomp0rid.
apply maponpaths.
use subtypePath.
{ intro ; apply isaprop_is_z_isomorphism. }
apply idpath.
- intros G₁ G₂ G₃ F₁ F₂ F₃ α ; cbn.
use funextsec.
intro x ; unfold homotcomp, homotfun, funhomotsec ; cbn.
rewrite pathscomp0rid.
refine (!_).
etrans.
{
apply maponpaths_isotoid.
}
apply maponpaths.
use subtypePath.
{ intro ; apply isaprop_is_z_isomorphism. }
apply idpath.
- intros G₁ G₂ F ; cbn.
use funextsec.
intro x.
refine (_ @ isotoid_identity_iso _ _ _).
apply maponpaths.
use subtypePath.
{ intro ; apply isaprop_is_z_isomorphism. }
apply idpath.
- intros G₁ G₂ F₁ F₂ F₃ α₁ α₂ ; cbn.
use funextsec.
intro x ; unfold homotcomp.
rewrite <- isotoid_comp.
apply maponpaths.
use subtypePath.
{ intro ; apply isaprop_is_z_isomorphism. }
apply idpath.
- intros G₁ G₂ F ; cbn.
use funextsec.
intro x ; unfold homotcomp, homotfun ; cbn.
refine (!_).
refine (_ @ isotoid_identity_iso _ _ _).
apply maponpaths.
use subtypePath.
{ intro ; apply isaprop_is_z_isomorphism. }
apply idpath.
- intros G₁ G₂ F ; cbn.
use funextsec.
intro x ; unfold homotcomp, homotfun ; cbn.
refine (!_).
refine (_ @ isotoid_identity_iso _ _ _).
apply maponpaths.
use subtypePath.
{ intro ; apply isaprop_is_z_isomorphism. }
apply idpath.
- intros G₁ G₂ G₃ G₄ F₁ F₂ F₃ ; cbn.
use funextsec.
intro x ; unfold homotcomp, homotfun ; cbn.
refine (_ @ isotoid_identity_iso _ _ _).
apply maponpaths.
use subtypePath.
{ intro ; apply isaprop_is_z_isomorphism. }
apply idpath.
- intros G₁ G₂ G₃ F₁ F₂ F₃ α ; cbn.
use funextsec.
intro x ; unfold homotcomp, homotfun, funhomotsec ; cbn.
rewrite pathscomp0rid.
apply maponpaths.
use subtypePath.
{ intro ; apply isaprop_is_z_isomorphism. }
apply idpath.
- intros G₁ G₂ G₃ F₁ F₂ F₃ α ; cbn.
use funextsec.
intro x ; unfold homotcomp, homotfun, funhomotsec ; cbn.
rewrite pathscomp0rid.
refine (!_).
etrans.
{
apply maponpaths_isotoid.
}
apply maponpaths.
use subtypePath.
{ intro ; apply isaprop_is_z_isomorphism. }
apply idpath.
Definition objects_of_grpd
: psfunctor grpds one_types.
Show proof.
use make_psfunctor.
- exact objects_of_grpd_data.
- exact objects_of_grpd_is_psfunctor.
- split ; intros ; apply one_type_2cell_iso.
- exact objects_of_grpd_data.
- exact objects_of_grpd_is_psfunctor.
- split ; intros ; apply one_type_2cell_iso.
The biadjunction
Definition path_groupoid_unit_data
: pstrans_data
(id_psfunctor one_types)
(comp_psfunctor objects_of_grpd path_groupoid).
Show proof.
Lemma eq_isotoid
{C : univalent_category}
{X Y : C}
(f : z_iso X Y) (p : X = Y)
: f = idtoiso p → isotoid C (pr2 C) f = p.
Show proof.
Definition path_groupoid_inv_is_pstrans
: is_pstrans path_groupoid_unit_data.
Show proof.
Definition path_groupoid_unit
: pstrans
(id_psfunctor one_types)
(comp_psfunctor objects_of_grpd path_groupoid).
Show proof.
Definition path_groupoid_counit_data_functor_data
(G : univalent_groupoid)
: functor_data (path_groupoid (objects_of_grpd G) : univalent_groupoid) G.
Show proof.
Definition path_groupoid_counit_data_functor_is_functor
(G : univalent_groupoid)
: is_functor (path_groupoid_counit_data_functor_data G).
Show proof.
Definition path_groupoid_counit_data_functor
(G : univalent_groupoid)
: (path_groupoid (objects_of_grpd G) : univalent_groupoid) ⟶ G.
Show proof.
Definition path_groupoid_counit_data_nat_trans_data
(G₁ G₂ : grpds)
(F : G₁ --> G₂)
: nat_trans_data
(path_groupoid_counit_data_functor G₁ ∙ pr1 F)
((function_to_functor (# objects_of_grpd F))
∙ path_groupoid_counit_data_functor G₂)
:= λ _, id₁ _.
Definition path_groupoid_counit_data_nat_trans_is_nat_trans
(G₁ G₂ : grpds)
(F : G₁ --> G₂)
: is_nat_trans _ _ (path_groupoid_counit_data_nat_trans_data G₁ G₂ F).
Show proof.
Definition path_groupoid_counit_data_nat_trans
(G₁ G₂ : grpds)
(F : G₁ --> G₂)
: (path_groupoid_counit_data_functor G₁ ∙ pr1 F)
⟹
function_to_functor (#objects_of_grpd F)
∙ path_groupoid_counit_data_functor G₂.
Show proof.
Definition path_groupoid_counit_data
: pstrans_data
(comp_psfunctor path_groupoid objects_of_grpd)
(id_psfunctor grpds).
Show proof.
Definition path_groupoid_counit_is_pstrans
: is_pstrans path_groupoid_counit_data.
Show proof.
Definition path_groupoid_counit
: pstrans
(comp_psfunctor path_groupoid objects_of_grpd)
(id_psfunctor grpds).
Show proof.
Definition path_groupoid_biadj_unit_counit
: left_biadj_unit_counit path_groupoid.
Show proof.
Definition path_groupoid_biadj_triangle_l_data
: invertible_modification_data
(biadj_triangle_l_lhs path_groupoid_biadj_unit_counit)
(id_pstrans path_groupoid).
Show proof.
Definition path_groupoid_biadj_triangle_l_is_modification
: is_modification path_groupoid_biadj_triangle_l_data.
Show proof.
Definition path_groupoid_biadj_triangle_l
: biadj_triangle_l_law path_groupoid_biadj_unit_counit.
Show proof.
Definition path_groupoid_biadj_triangle_r_data
: invertible_modification_data
(biadj_triangle_r_lhs path_groupoid_biadj_unit_counit)
(id_pstrans path_groupoid_biadj_unit_counit).
Show proof.
Definition path_groupoid_biadj_triangle_r_is_modification
: is_modification path_groupoid_biadj_triangle_r_data.
Show proof.
Definition path_groupoid_biadj_triangle_r
: biadj_triangle_r_law path_groupoid_biadj_unit_counit.
Show proof.
Definition path_groupoid_biadj_data
: left_biadj_data path_groupoid.
Show proof.
: pstrans_data
(id_psfunctor one_types)
(comp_psfunctor objects_of_grpd path_groupoid).
Show proof.
Lemma eq_isotoid
{C : univalent_category}
{X Y : C}
(f : z_iso X Y) (p : X = Y)
: f = idtoiso p → isotoid C (pr2 C) f = p.
Show proof.
Definition path_groupoid_inv_is_pstrans
: is_pstrans path_groupoid_unit_data.
Show proof.
repeat split.
- intros X Y f g p.
use funextsec ; intro x.
refine (pathscomp0rid _ @ _ @ !(maponpathsidfun _)).
cbn ; unfold funhomotsec.
apply (@eq_isotoid (path_univalent_groupoid _)).
use subtypePath.
{ intro ; apply isaprop_is_z_isomorphism. }
cbn.
induction (p x).
apply idpath.
- intros X ; cbn.
use funextsec ; intro x.
unfold homotcomp, funhomotsec, homotfun, homotrefl ; cbn.
rewrite pathscomp0rid ; cbn in *.
apply (@eq_isotoid (path_univalent_groupoid _)).
use subtypePath.
{ intro ; apply isaprop_is_z_isomorphism. }
cbn.
apply idpath.
- intros X Y Z f g ; cbn.
use funextsec ; intro x.
unfold homotcomp, funhomotsec, homotfun, homotrefl ; cbn.
rewrite pathscomp0rid ; cbn in *.
apply (@eq_isotoid (path_univalent_groupoid _)).
use subtypePath.
{ intro ; apply isaprop_is_z_isomorphism. }
cbn.
apply idpath.
- intros X Y f g p.
use funextsec ; intro x.
refine (pathscomp0rid _ @ _ @ !(maponpathsidfun _)).
cbn ; unfold funhomotsec.
apply (@eq_isotoid (path_univalent_groupoid _)).
use subtypePath.
{ intro ; apply isaprop_is_z_isomorphism. }
cbn.
induction (p x).
apply idpath.
- intros X ; cbn.
use funextsec ; intro x.
unfold homotcomp, funhomotsec, homotfun, homotrefl ; cbn.
rewrite pathscomp0rid ; cbn in *.
apply (@eq_isotoid (path_univalent_groupoid _)).
use subtypePath.
{ intro ; apply isaprop_is_z_isomorphism. }
cbn.
apply idpath.
- intros X Y Z f g ; cbn.
use funextsec ; intro x.
unfold homotcomp, funhomotsec, homotfun, homotrefl ; cbn.
rewrite pathscomp0rid ; cbn in *.
apply (@eq_isotoid (path_univalent_groupoid _)).
use subtypePath.
{ intro ; apply isaprop_is_z_isomorphism. }
cbn.
apply idpath.
Definition path_groupoid_unit
: pstrans
(id_psfunctor one_types)
(comp_psfunctor objects_of_grpd path_groupoid).
Show proof.
Definition path_groupoid_counit_data_functor_data
(G : univalent_groupoid)
: functor_data (path_groupoid (objects_of_grpd G) : univalent_groupoid) G.
Show proof.
Definition path_groupoid_counit_data_functor_is_functor
(G : univalent_groupoid)
: is_functor (path_groupoid_counit_data_functor_data G).
Show proof.
split.
- intros x ; apply idpath.
- intros x y z f g ; cbn.
exact (maponpaths pr1 (idtoiso_concat _ _ _ _ f g)).
- intros x ; apply idpath.
- intros x y z f g ; cbn.
exact (maponpaths pr1 (idtoiso_concat _ _ _ _ f g)).
Definition path_groupoid_counit_data_functor
(G : univalent_groupoid)
: (path_groupoid (objects_of_grpd G) : univalent_groupoid) ⟶ G.
Show proof.
use make_functor.
- exact (path_groupoid_counit_data_functor_data G).
- exact (path_groupoid_counit_data_functor_is_functor G).
- exact (path_groupoid_counit_data_functor_data G).
- exact (path_groupoid_counit_data_functor_is_functor G).
Definition path_groupoid_counit_data_nat_trans_data
(G₁ G₂ : grpds)
(F : G₁ --> G₂)
: nat_trans_data
(path_groupoid_counit_data_functor G₁ ∙ pr1 F)
((function_to_functor (# objects_of_grpd F))
∙ path_groupoid_counit_data_functor G₂)
:= λ _, id₁ _.
Definition path_groupoid_counit_data_nat_trans_is_nat_trans
(G₁ G₂ : grpds)
(F : G₁ --> G₂)
: is_nat_trans _ _ (path_groupoid_counit_data_nat_trans_data G₁ G₂ F).
Show proof.
intros x y f ; cbn.
rewrite id_left, id_right.
refine (!_).
exact (maponpaths pr1 (maponpaths_idtoiso _ _ _ _ _ _)).
rewrite id_left, id_right.
refine (!_).
exact (maponpaths pr1 (maponpaths_idtoiso _ _ _ _ _ _)).
Definition path_groupoid_counit_data_nat_trans
(G₁ G₂ : grpds)
(F : G₁ --> G₂)
: (path_groupoid_counit_data_functor G₁ ∙ pr1 F)
⟹
function_to_functor (#objects_of_grpd F)
∙ path_groupoid_counit_data_functor G₂.
Show proof.
use make_nat_trans.
- exact (path_groupoid_counit_data_nat_trans_data G₁ G₂ F).
- exact (path_groupoid_counit_data_nat_trans_is_nat_trans G₁ G₂ F).
- exact (path_groupoid_counit_data_nat_trans_data G₁ G₂ F).
- exact (path_groupoid_counit_data_nat_trans_is_nat_trans G₁ G₂ F).
Definition path_groupoid_counit_data
: pstrans_data
(comp_psfunctor path_groupoid objects_of_grpd)
(id_psfunctor grpds).
Show proof.
use make_pstrans_data.
- exact (λ G, path_groupoid_counit_data_functor G ,, tt).
- intros G₁ G₂ F.
use make_invertible_2cell.
+ exact (path_groupoid_counit_data_nat_trans G₁ G₂ F ,, tt).
+ apply locally_groupoid_grpds.
- exact (λ G, path_groupoid_counit_data_functor G ,, tt).
- intros G₁ G₂ F.
use make_invertible_2cell.
+ exact (path_groupoid_counit_data_nat_trans G₁ G₂ F ,, tt).
+ apply locally_groupoid_grpds.
Definition path_groupoid_counit_is_pstrans
: is_pstrans path_groupoid_counit_data.
Show proof.
repeat split.
- intros G₁ G₂ F₁ F₂ α.
use subtypePath.
{ intro ; apply isapropunit. }
use nat_trans_eq.
{ apply homset_property. }
intro x ; cbn.
rewrite id_left, id_right.
rewrite idtoiso_isotoid.
apply idpath.
- intros G.
use subtypePath.
{ intro ; apply isapropunit. }
use nat_trans_eq.
{ apply homset_property. }
intro x ; cbn.
rewrite id_left, !id_right.
apply idpath.
- intros G₁ G₂ G₃ F₁ F₂.
use subtypePath.
{ intro ; apply isapropunit. }
use nat_trans_eq.
{ apply homset_property. }
intro x ; cbn.
rewrite !id_left, !id_right.
rewrite (functor_id (pr1 F₂)).
apply idpath.
- intros G₁ G₂ F₁ F₂ α.
use subtypePath.
{ intro ; apply isapropunit. }
use nat_trans_eq.
{ apply homset_property. }
intro x ; cbn.
rewrite id_left, id_right.
rewrite idtoiso_isotoid.
apply idpath.
- intros G.
use subtypePath.
{ intro ; apply isapropunit. }
use nat_trans_eq.
{ apply homset_property. }
intro x ; cbn.
rewrite id_left, !id_right.
apply idpath.
- intros G₁ G₂ G₃ F₁ F₂.
use subtypePath.
{ intro ; apply isapropunit. }
use nat_trans_eq.
{ apply homset_property. }
intro x ; cbn.
rewrite !id_left, !id_right.
rewrite (functor_id (pr1 F₂)).
apply idpath.
Definition path_groupoid_counit
: pstrans
(comp_psfunctor path_groupoid objects_of_grpd)
(id_psfunctor grpds).
Show proof.
Definition path_groupoid_biadj_unit_counit
: left_biadj_unit_counit path_groupoid.
Show proof.
use make_biadj_unit_counit.
- exact objects_of_grpd.
- exact path_groupoid_unit.
- exact path_groupoid_counit.
- exact objects_of_grpd.
- exact path_groupoid_unit.
- exact path_groupoid_counit.
Definition path_groupoid_biadj_triangle_l_data
: invertible_modification_data
(biadj_triangle_l_lhs path_groupoid_biadj_unit_counit)
(id_pstrans path_groupoid).
Show proof.
intros X.
use make_invertible_2cell.
- refine (_ ,, tt).
use make_nat_trans.
+ exact idpath.
+ abstract
(intros x y p ; cbn in * ;
induction p ; simpl ;
apply idpath).
- apply locally_groupoid_grpds.
use make_invertible_2cell.
- refine (_ ,, tt).
use make_nat_trans.
+ exact idpath.
+ abstract
(intros x y p ; cbn in * ;
induction p ; simpl ;
apply idpath).
- apply locally_groupoid_grpds.
Definition path_groupoid_biadj_triangle_l_is_modification
: is_modification path_groupoid_biadj_triangle_l_data.
Show proof.
intros X Y f.
use subtypePath.
{ intro ; apply isapropunit. }
use nat_trans_eq.
{ apply homset_property. }
intro x.
apply idpath.
use subtypePath.
{ intro ; apply isapropunit. }
use nat_trans_eq.
{ apply homset_property. }
intro x.
apply idpath.
Definition path_groupoid_biadj_triangle_l
: biadj_triangle_l_law path_groupoid_biadj_unit_counit.
Show proof.
use make_invertible_modification.
- exact path_groupoid_biadj_triangle_l_data.
- exact path_groupoid_biadj_triangle_l_is_modification.
- exact path_groupoid_biadj_triangle_l_data.
- exact path_groupoid_biadj_triangle_l_is_modification.
Definition path_groupoid_biadj_triangle_r_data
: invertible_modification_data
(biadj_triangle_r_lhs path_groupoid_biadj_unit_counit)
(id_pstrans path_groupoid_biadj_unit_counit).
Show proof.
Definition path_groupoid_biadj_triangle_r_is_modification
: is_modification path_groupoid_biadj_triangle_r_data.
Show proof.
intros G₁ G₂ F.
use funextsec.
intro x.
repeat (refine (pathscomp0rid _ @ _)).
refine (maponpathsidfun _ @ _).
refine (pathscomp0rid _ @ _).
cbn ; unfold path_groupoid_counit_data_nat_trans_data.
use eq_isotoid.
use subtypePath.
{ intro ; apply isaprop_is_z_isomorphism. }
apply idpath.
use funextsec.
intro x.
repeat (refine (pathscomp0rid _ @ _)).
refine (maponpathsidfun _ @ _).
refine (pathscomp0rid _ @ _).
cbn ; unfold path_groupoid_counit_data_nat_trans_data.
use eq_isotoid.
use subtypePath.
{ intro ; apply isaprop_is_z_isomorphism. }
apply idpath.
Definition path_groupoid_biadj_triangle_r
: biadj_triangle_r_law path_groupoid_biadj_unit_counit.
Show proof.
use make_invertible_modification.
- exact path_groupoid_biadj_triangle_r_data.
- exact path_groupoid_biadj_triangle_r_is_modification.
- exact path_groupoid_biadj_triangle_r_data.
- exact path_groupoid_biadj_triangle_r_is_modification.
Definition path_groupoid_biadj_data
: left_biadj_data path_groupoid.
Show proof.
use make_biadj_data.
- exact path_groupoid_biadj_unit_counit.
- exact path_groupoid_biadj_triangle_l.
- exact path_groupoid_biadj_triangle_r.
- exact path_groupoid_biadj_unit_counit.
- exact path_groupoid_biadj_triangle_l.
- exact path_groupoid_biadj_triangle_r.
Inverse of unit
Definition path_groupoid_unit_inv_data
: pstrans_data
(comp_psfunctor objects_of_grpd path_groupoid)
(id_psfunctor one_types).
Show proof.
Definition path_groupoid_unit_inv_is_pstrans
: is_pstrans path_groupoid_unit_inv_data.
Show proof.
Definition path_groupoid_unit_inv
: pstrans
(comp_psfunctor objects_of_grpd path_groupoid)
(id_psfunctor one_types).
Show proof.
Definition path_groupoid_unit_unit_inv
: invertible_modification
(comp_pstrans path_groupoid_unit path_groupoid_unit_inv)
(id_pstrans _).
Show proof.
Definition path_groupoid_unit_inv_unit
: invertible_modification
(comp_pstrans path_groupoid_unit_inv path_groupoid_unit)
(id_pstrans _).
Show proof.
: pstrans_data
(comp_psfunctor objects_of_grpd path_groupoid)
(id_psfunctor one_types).
Show proof.
Definition path_groupoid_unit_inv_is_pstrans
: is_pstrans path_groupoid_unit_inv_data.
Show proof.
repeat split.
- intros X Y f g p.
use funextsec ; intro x.
refine (pathscomp0rid _ @ _ @ !(maponpathsidfun _)).
cbn ; unfold funhomotsec.
refine (!_).
apply (@eq_isotoid (path_univalent_groupoid _)).
use subtypePath.
{ intro ; apply isaprop_is_z_isomorphism. }
cbn.
induction (p x).
apply idpath.
- intros X ; cbn.
use funextsec ; intro x.
unfold homotcomp, funhomotsec, homotfun, homotrefl ; cbn.
refine (!_).
rewrite maponpathsidfun.
apply (@eq_isotoid (path_univalent_groupoid _)).
use subtypePath.
{ intro ; apply isaprop_is_z_isomorphism. }
apply idpath.
- intros X Y Z f g ; cbn.
use funextsec ; intro x.
unfold homotcomp, funhomotsec, homotfun, homotrefl ; cbn.
refine (!_).
rewrite maponpathsidfun.
apply (@eq_isotoid (path_univalent_groupoid _)).
use subtypePath.
{ intro ; apply isaprop_is_z_isomorphism. }
apply idpath.
- intros X Y f g p.
use funextsec ; intro x.
refine (pathscomp0rid _ @ _ @ !(maponpathsidfun _)).
cbn ; unfold funhomotsec.
refine (!_).
apply (@eq_isotoid (path_univalent_groupoid _)).
use subtypePath.
{ intro ; apply isaprop_is_z_isomorphism. }
cbn.
induction (p x).
apply idpath.
- intros X ; cbn.
use funextsec ; intro x.
unfold homotcomp, funhomotsec, homotfun, homotrefl ; cbn.
refine (!_).
rewrite maponpathsidfun.
apply (@eq_isotoid (path_univalent_groupoid _)).
use subtypePath.
{ intro ; apply isaprop_is_z_isomorphism. }
apply idpath.
- intros X Y Z f g ; cbn.
use funextsec ; intro x.
unfold homotcomp, funhomotsec, homotfun, homotrefl ; cbn.
refine (!_).
rewrite maponpathsidfun.
apply (@eq_isotoid (path_univalent_groupoid _)).
use subtypePath.
{ intro ; apply isaprop_is_z_isomorphism. }
apply idpath.
Definition path_groupoid_unit_inv
: pstrans
(comp_psfunctor objects_of_grpd path_groupoid)
(id_psfunctor one_types).
Show proof.
Definition path_groupoid_unit_unit_inv
: invertible_modification
(comp_pstrans path_groupoid_unit path_groupoid_unit_inv)
(id_pstrans _).
Show proof.
use make_invertible_modification.
- intro.
apply id2_invertible_2cell.
- abstract
(intros X Y f ;
use funextsec ;
intro ;
apply idpath).
- intro.
apply id2_invertible_2cell.
- abstract
(intros X Y f ;
use funextsec ;
intro ;
apply idpath).
Definition path_groupoid_unit_inv_unit
: invertible_modification
(comp_pstrans path_groupoid_unit_inv path_groupoid_unit)
(id_pstrans _).
Show proof.
use make_invertible_modification.
- intro.
apply id2_invertible_2cell.
- abstract
(intros X Y f ;
use funextsec ;
intro x ;
apply idpath).
- intro.
apply id2_invertible_2cell.
- abstract
(intros X Y f ;
use funextsec ;
intro x ;
apply idpath).
Inverse of counit
Definition path_groupoid_counit_inv_data_functor_data
(G : univalent_groupoid)
: functor_data G (path_groupoid (objects_of_grpd G) : univalent_groupoid).
Show proof.
Definition path_groupoid_counit_inv_data_functor_is_functor
(G : univalent_groupoid)
: is_functor (path_groupoid_counit_inv_data_functor_data G).
Show proof.
Definition path_groupoid_counit_inv_data_functor
(G : univalent_groupoid)
: G ⟶ (path_groupoid (objects_of_grpd G) : univalent_groupoid).
Show proof.
Definition path_groupoid_counit_inv_data_nat_trans_data
(G₁ G₂ : grpds)
(F : G₁ --> G₂)
: nat_trans_data
((path_groupoid_counit_inv_data_functor G₁)
∙ function_to_functor (# objects_of_grpd F))
(pr1 F ∙ path_groupoid_counit_inv_data_functor G₂)
:= λ _, id₁ _.
Definition path_groupoid_counit_inv_data_nat_trans_is_nat_trans
(G₁ G₂ : grpds)
(F : G₁ --> G₂)
: is_nat_trans _ _ (path_groupoid_counit_inv_data_nat_trans_data G₁ G₂ F).
Show proof.
Definition path_groupoid_counit_inv_data_nat_trans
(G₁ G₂ : grpds)
(F : G₁ --> G₂)
: ((path_groupoid_counit_inv_data_functor G₁)
∙ function_to_functor (#objects_of_grpd F))
⟹
(pr1 F∙ path_groupoid_counit_inv_data_functor G₂).
Show proof.
Definition path_groupoid_counit_inv_data
: pstrans_data
(id_psfunctor grpds)
(comp_psfunctor path_groupoid objects_of_grpd).
Show proof.
Definition path_groupoid_counit_inv_is_pstrans
: is_pstrans path_groupoid_counit_inv_data.
Show proof.
Definition path_groupoid_counit_inv
: pstrans
(id_psfunctor grpds)
(comp_psfunctor path_groupoid objects_of_grpd).
Show proof.
Definition path_groupoid_counit_counit_inv
: invertible_modification
(comp_pstrans path_groupoid_counit path_groupoid_counit_inv)
(id_pstrans _).
Show proof.
Definition path_groupoid_counit_inv_counit
: invertible_modification
(comp_pstrans path_groupoid_counit_inv path_groupoid_counit)
(id_pstrans _).
Show proof.
Definition is_biequiv_path_groupoid : is_biequivalence path_groupoid.
Show proof.
Definition biequiv_path_groupoid : biequivalence one_types grpds
:= path_groupoid ,, is_biequiv_path_groupoid.
(G : univalent_groupoid)
: functor_data G (path_groupoid (objects_of_grpd G) : univalent_groupoid).
Show proof.
Definition path_groupoid_counit_inv_data_functor_is_functor
(G : univalent_groupoid)
: is_functor (path_groupoid_counit_inv_data_functor_data G).
Show proof.
split.
- intro x ; cbn.
apply eq_isotoid.
use subtypePath.
{ intro ; apply isaprop_is_z_isomorphism. }
apply idpath.
- intros x y z f g ; cbn.
apply eq_isotoid.
use subtypePath.
{ intro ; apply isaprop_is_z_isomorphism. }
rewrite idtoiso_concat.
rewrite !idtoiso_isotoid.
apply idpath.
- intro x ; cbn.
apply eq_isotoid.
use subtypePath.
{ intro ; apply isaprop_is_z_isomorphism. }
apply idpath.
- intros x y z f g ; cbn.
apply eq_isotoid.
use subtypePath.
{ intro ; apply isaprop_is_z_isomorphism. }
rewrite idtoiso_concat.
rewrite !idtoiso_isotoid.
apply idpath.
Definition path_groupoid_counit_inv_data_functor
(G : univalent_groupoid)
: G ⟶ (path_groupoid (objects_of_grpd G) : univalent_groupoid).
Show proof.
use make_functor.
- exact (path_groupoid_counit_inv_data_functor_data G).
- exact (path_groupoid_counit_inv_data_functor_is_functor G).
- exact (path_groupoid_counit_inv_data_functor_data G).
- exact (path_groupoid_counit_inv_data_functor_is_functor G).
Definition path_groupoid_counit_inv_data_nat_trans_data
(G₁ G₂ : grpds)
(F : G₁ --> G₂)
: nat_trans_data
((path_groupoid_counit_inv_data_functor G₁)
∙ function_to_functor (# objects_of_grpd F))
(pr1 F ∙ path_groupoid_counit_inv_data_functor G₂)
:= λ _, id₁ _.
Definition path_groupoid_counit_inv_data_nat_trans_is_nat_trans
(G₁ G₂ : grpds)
(F : G₁ --> G₂)
: is_nat_trans _ _ (path_groupoid_counit_inv_data_nat_trans_data G₁ G₂ F).
Show proof.
intros x y f ; cbn.
rewrite pathscomp0rid.
rewrite (maponpaths_isotoid _ _ (pr1 F) _ (pr21 G₂)).
apply maponpaths.
apply subtypePath.
{ intro ; apply isaprop_is_z_isomorphism. }
apply idpath.
rewrite pathscomp0rid.
rewrite (maponpaths_isotoid _ _ (pr1 F) _ (pr21 G₂)).
apply maponpaths.
apply subtypePath.
{ intro ; apply isaprop_is_z_isomorphism. }
apply idpath.
Definition path_groupoid_counit_inv_data_nat_trans
(G₁ G₂ : grpds)
(F : G₁ --> G₂)
: ((path_groupoid_counit_inv_data_functor G₁)
∙ function_to_functor (#objects_of_grpd F))
⟹
(pr1 F∙ path_groupoid_counit_inv_data_functor G₂).
Show proof.
use make_nat_trans.
- exact (path_groupoid_counit_inv_data_nat_trans_data G₁ G₂ F).
- exact (path_groupoid_counit_inv_data_nat_trans_is_nat_trans G₁ G₂ F).
- exact (path_groupoid_counit_inv_data_nat_trans_data G₁ G₂ F).
- exact (path_groupoid_counit_inv_data_nat_trans_is_nat_trans G₁ G₂ F).
Definition path_groupoid_counit_inv_data
: pstrans_data
(id_psfunctor grpds)
(comp_psfunctor path_groupoid objects_of_grpd).
Show proof.
use make_pstrans_data.
- exact (λ G, path_groupoid_counit_inv_data_functor G ,, tt).
- intros G₁ G₂ F.
use make_invertible_2cell.
+ exact (path_groupoid_counit_inv_data_nat_trans G₁ G₂ F ,, tt).
+ apply locally_groupoid_grpds.
- exact (λ G, path_groupoid_counit_inv_data_functor G ,, tt).
- intros G₁ G₂ F.
use make_invertible_2cell.
+ exact (path_groupoid_counit_inv_data_nat_trans G₁ G₂ F ,, tt).
+ apply locally_groupoid_grpds.
Definition path_groupoid_counit_inv_is_pstrans
: is_pstrans path_groupoid_counit_inv_data.
Show proof.
repeat split.
- intros G₁ G₂ F₁ F₂ α.
use subtypePath.
{ intro ; apply isapropunit. }
use nat_trans_eq.
{ apply homset_property. }
intro x ; cbn.
rewrite pathscomp0rid.
apply maponpaths.
use subtypePath.
{ intro ; apply isaprop_is_z_isomorphism. }
apply idpath.
- intros G.
use subtypePath.
{ intro ; apply isapropunit. }
use nat_trans_eq.
{ apply homset_property. }
intro x ; cbn.
refine (!_).
apply eq_isotoid.
use subtypePath.
{ intro ; apply isaprop_is_z_isomorphism. }
apply idpath.
- intros G₁ G₂ G₃ F₁ F₂.
use subtypePath.
{ intro ; apply isapropunit. }
use nat_trans_eq.
{ apply homset_property. }
intro x ; cbn.
refine (!_).
apply eq_isotoid.
use subtypePath.
{ intro ; apply isaprop_is_z_isomorphism. }
apply idpath.
- intros G₁ G₂ F₁ F₂ α.
use subtypePath.
{ intro ; apply isapropunit. }
use nat_trans_eq.
{ apply homset_property. }
intro x ; cbn.
rewrite pathscomp0rid.
apply maponpaths.
use subtypePath.
{ intro ; apply isaprop_is_z_isomorphism. }
apply idpath.
- intros G.
use subtypePath.
{ intro ; apply isapropunit. }
use nat_trans_eq.
{ apply homset_property. }
intro x ; cbn.
refine (!_).
apply eq_isotoid.
use subtypePath.
{ intro ; apply isaprop_is_z_isomorphism. }
apply idpath.
- intros G₁ G₂ G₃ F₁ F₂.
use subtypePath.
{ intro ; apply isapropunit. }
use nat_trans_eq.
{ apply homset_property. }
intro x ; cbn.
refine (!_).
apply eq_isotoid.
use subtypePath.
{ intro ; apply isaprop_is_z_isomorphism. }
apply idpath.
Definition path_groupoid_counit_inv
: pstrans
(id_psfunctor grpds)
(comp_psfunctor path_groupoid objects_of_grpd).
Show proof.
use make_pstrans.
- exact path_groupoid_counit_inv_data.
- exact path_groupoid_counit_inv_is_pstrans.
- exact path_groupoid_counit_inv_data.
- exact path_groupoid_counit_inv_is_pstrans.
Definition path_groupoid_counit_counit_inv
: invertible_modification
(comp_pstrans path_groupoid_counit path_groupoid_counit_inv)
(id_pstrans _).
Show proof.
use make_invertible_modification.
- intro.
use make_invertible_2cell.
+ refine (_ ,, tt).
use make_nat_trans.
* exact idpath.
* abstract
(intros x y p ; cbn ;
rewrite pathscomp0rid ;
apply eq_isotoid ;
use subtypePath ; try (intro ; apply isaprop_is_z_isomorphism) ;
apply idpath).
+ apply locally_groupoid_grpds.
- abstract
(intros G₁ G₂ F ;
use subtypePath ; try (intro ; apply isapropunit) ;
use nat_trans_eq ; try (apply homset_property) ;
intro ; cbn ;
rewrite !pathscomp0rid ;
apply eq_isotoid ;
use subtypePath ; try (intro ; apply isaprop_is_z_isomorphism) ;
apply idpath).
- intro.
use make_invertible_2cell.
+ refine (_ ,, tt).
use make_nat_trans.
* exact idpath.
* abstract
(intros x y p ; cbn ;
rewrite pathscomp0rid ;
apply eq_isotoid ;
use subtypePath ; try (intro ; apply isaprop_is_z_isomorphism) ;
apply idpath).
+ apply locally_groupoid_grpds.
- abstract
(intros G₁ G₂ F ;
use subtypePath ; try (intro ; apply isapropunit) ;
use nat_trans_eq ; try (apply homset_property) ;
intro ; cbn ;
rewrite !pathscomp0rid ;
apply eq_isotoid ;
use subtypePath ; try (intro ; apply isaprop_is_z_isomorphism) ;
apply idpath).
Definition path_groupoid_counit_inv_counit
: invertible_modification
(comp_pstrans path_groupoid_counit_inv path_groupoid_counit)
(id_pstrans _).
Show proof.
use make_invertible_modification.
- intro.
use make_invertible_2cell.
+ refine (_ ,, tt).
use make_nat_trans.
* exact identity.
* abstract
(intros x y p ; cbn ;
rewrite id_right, id_left, idtoiso_isotoid ;
apply idpath).
+ apply locally_groupoid_grpds.
- abstract
(intros G₁ G₂ F ;
use subtypePath ; try (intro ; apply isapropunit) ;
use nat_trans_eq ; try (apply homset_property) ;
intro ; cbn ;
rewrite !id_left, id_right ;
rewrite (functor_id (pr1 F)) ;
apply idpath).
- intro.
use make_invertible_2cell.
+ refine (_ ,, tt).
use make_nat_trans.
* exact identity.
* abstract
(intros x y p ; cbn ;
rewrite id_right, id_left, idtoiso_isotoid ;
apply idpath).
+ apply locally_groupoid_grpds.
- abstract
(intros G₁ G₂ F ;
use subtypePath ; try (intro ; apply isapropunit) ;
use nat_trans_eq ; try (apply homset_property) ;
intro ; cbn ;
rewrite !id_left, id_right ;
rewrite (functor_id (pr1 F)) ;
apply idpath).
Definition is_biequiv_path_groupoid : is_biequivalence path_groupoid.
Show proof.
use make_is_biequivalence.
- exact objects_of_grpd.
- exact path_groupoid_unit.
- exact path_groupoid_unit_inv.
- exact path_groupoid_counit.
- exact path_groupoid_counit_inv.
- exact (inv_of_invertible_2cell path_groupoid_unit_inv_unit).
- exact path_groupoid_unit_unit_inv.
- exact (inv_of_invertible_2cell path_groupoid_counit_counit_inv).
- exact path_groupoid_counit_inv_counit.
- exact objects_of_grpd.
- exact path_groupoid_unit.
- exact path_groupoid_unit_inv.
- exact path_groupoid_counit.
- exact path_groupoid_counit_inv.
- exact (inv_of_invertible_2cell path_groupoid_unit_inv_unit).
- exact path_groupoid_unit_unit_inv.
- exact (inv_of_invertible_2cell path_groupoid_counit_counit_inv).
- exact path_groupoid_counit_inv_counit.
Definition biequiv_path_groupoid : biequivalence one_types grpds
:= path_groupoid ,, is_biequiv_path_groupoid.