Definition cat_with_chosen_pullbacks_has_RezkCompletion : disp_left_universal_arrow LUR (disp_psfunctor_on_cat_to_univ_cat disp_bicat_chosen_pullbacks (disp_2cells_isaprop_from_disp_2cells_iscontr _ disp_2cells_is_contr_chosen_pullbacks)). Proof. use make_disp_left_universal_arrow_if_contr_CAT_from_weak_equiv. - exact η_weak_equiv. - intros C1 C2 C2_univ F Fw P1. exact (weak_equiv_into_univ_creates_pullbacks C2_univ Fw (pr1 P1) ,, tt). - intros C P1. refine (tt ,, _). use weak_equiv_preserves_pullbacks_eq. + apply η_weak_equiv. + exact (pr2 (pr1 LUR C)). - intros C1 C2 C3 F G H α P1 P2 P3 Gw. intros [t Fprod]. exists tt. exact (weak_equiv_lifts_preserves_chosen_pullbacks_eq C2 C3 α (pr1 P1) (pr1 P2) (pr1 P3) Gw Fprod). Defined.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Prelude. Require Import UniMath.CategoryTheory.Limits.Pullbacks. Require Import UniMath.CategoryTheory.Limits.Preservation. Require Import UniMath.CategoryTheory.WeakEquivalences.Core. Require Import UniMath.CategoryTheory.WeakEquivalences.Preservation.Pullbacks. Require Import UniMath.CategoryTheory.WeakEquivalences.Reflection.Pullbacks. Require Import UniMath.CategoryTheory.WeakEquivalences.Creation.Pullbacks. Require Import UniMath.CategoryTheory.WeakEquivalences.LiftPreservation.Pullbacks. Require Import UniMath.Bicategories.Core.Bicat. Import Bicat.Notations. Require Import UniMath.Bicategories.Core.Invertible_2cells. Require Import UniMath.Bicategories.Core.Examples.BicatOfUnivCats. Require Import UniMath.Bicategories.Core.Examples.BicatOfCats. Require Import UniMath.Bicategories.Core.Univalence. Require Import UniMath.Bicategories.DisplayedBicats.DispBicat. Require Import UniMath.Bicategories.PseudoFunctors.UniversalArrow. Import PseudoFunctor.Notations. Import DispBicat.Notations. Require Import UniMath.Bicategories.PseudoFunctors.Examples.BicatOfCatToUnivCat. Require Import UniMath.Bicategories.DisplayedBicats.Examples.DispBicatOnCatToUniv. Require Import UniMath.Bicategories.DisplayedBicats.Examples.CategoriesWithStructure. Require Import UniMath.Bicategories.DisplayedBicats.DisplayedUniversalArrow. Require Import UniMath.Bicategories.DisplayedBicats.DisplayedUniversalArrowOnCat.