Definition Monad_disp_biequiv_Ktriple : disp_is_biequivalence_data (monad bicat_of_univ_cats) kleisli_triple_disp_bicat (id_is_biequivalence _) Monad_biequiv_Ktriple_unit_counit. Proof. simple refine (_ ,, _ ,, ((_ ,, _) ,, (_ ,, _))). - exact Monad_biequiv_Ktriple_unit_inv. - exact Monad_biequiv_Ktriple_counit_inv. - use make_disp_invmodification. + exact (disp_2cells_isaprop_monad bicat_of_univ_cats univalent_cat_is_univalent_2). + exact (disp_locally_groupoid_monad bicat_of_univ_cats univalent_cat_is_univalent_2). + abstract (intros x xx ; use make_cat_monad_cell ; intros z ; simpl ; rewrite !id_left ; rewrite (functor_id (pr11 xx)), (functor_id (monad_endo xx)) ; exact (!(id_left _))). - use make_disp_invmodification. + exact (disp_2cells_isaprop_monad bicat_of_univ_cats univalent_cat_is_univalent_2). + exact (disp_locally_groupoid_monad bicat_of_univ_cats univalent_cat_is_univalent_2). + abstract (intros x xx ; use make_cat_monad_cell ; intros z ; simpl ; rewrite !id_left ; refine (!_) ; refine (bind_bind (Monad_to_Ktriple_data xx) (unit_kt (Monad_to_Ktriple_data xx) z) (unit_kt (Monad_to_Ktriple_data xx) z) @ _) ; apply maponpaths ; apply unit_bind). - use make_disp_invmodification. + exact disp_2cells_isaprop_kleisli. + exact disp_locally_groupoid_kleisli. + abstract (intros x xx z ; simpl ; rewrite !id_left, id_right ; rewrite bind_unit ; apply idpath). - use make_disp_invmodification. + exact disp_2cells_isaprop_kleisli. + exact disp_locally_groupoid_kleisli. + abstract (intros x xx z ; simpl ; rewrite !id_left, id_right ; rewrite bind_unit ; apply idpath). Defined.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Prelude. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Univalence. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.whiskering. Require Import UniMath.CategoryTheory.opp_precat. Require Import UniMath.CategoryTheory.DisplayedCats.Core. Require Import UniMath.Bicategories.Core.Examples.BicatOfUnivCats. Require Import UniMath.Bicategories.Core.Bicat. Import Bicat.Notations. Require Import UniMath.Bicategories.Morphisms.Adjunctions. Require Import UniMath.Bicategories.Core.AdjointUnique. Require Import UniMath.Bicategories.Core.Invertible_2cells. Require Import UniMath.Bicategories.Core.Univalence. Require Import UniMath.Bicategories.PseudoFunctors.Display.PseudoFunctorBicat. Require Import UniMath.Bicategories.PseudoFunctors.PseudoFunctor. Import PseudoFunctor.Notations. Require Import UniMath.Bicategories.DisplayedBicats.DispBicat. Import DispBicat.Notations. Require Import UniMath.Bicategories.DisplayedBicats.DispAdjunctions. Require Import UniMath.Bicategories.DisplayedBicats.DispInvertibles. Require Import UniMath.Bicategories.DisplayedBicats.DispUnivalence. Require Import UniMath.Bicategories.PseudoFunctors.Examples.Identity. Require Import UniMath.Bicategories.PseudoFunctors.Examples.Composition. Require Import UniMath.Bicategories.PseudoFunctors.Examples.Projection. Require Import UniMath.Bicategories.Transformations.PseudoTransformation. Require Import UniMath.Bicategories.Transformations.Examples.AlgebraMap. Require Import UniMath.Bicategories.Modifications.Modification. Require Import UniMath.Bicategories.PseudoFunctors.Biequivalence. Require Import UniMath.Bicategories.Core.Unitors. Require Import UniMath.Bicategories.DisplayedBicats.DispPseudofunctor. Require Import UniMath.Bicategories.DisplayedBicats.Examples.Algebras. Require Import UniMath.Bicategories.DisplayedBicats.Examples.Add2Cell. Require Import UniMath.Bicategories.DisplayedBicats.Examples.Prod. Require Import UniMath.Bicategories.DisplayedBicats.Examples.FullSub. Require Import UniMath.Bicategories.DisplayedBicats.Examples.Monads. Require Import UniMath.Bicategories.DisplayedBicats.Examples.KleisliTriple. Require Import UniMath.Bicategories.DisplayedBicats.DispBuilders. Require Import UniMath.Bicategories.DisplayedBicats.DispTransformation. Require Import UniMath.Bicategories.DisplayedBicats.DispModification. Require Import UniMath.Bicategories.DisplayedBicats.DispBiequivalence. Require Import UniMath.Bicategories.Transformations.Examples.Unitality. Require Import UniMath.Bicategories.DisplayedBicats.Examples.KleisliTriple. Require Import UniMath.CategoryTheory.Equivalences.CompositesAndInverses.