Library UniMath.Bicategories.Core.Examples.Image
Image of a pseudofunctor
Require Import UniMath.Foundations.All.
Require Import UniMath.MoreFoundations.All.
Require Import UniMath.CategoryTheory.Core.Categories.
Require Import UniMath.CategoryTheory.Core.Univalence.
Require Import UniMath.CategoryTheory.categories.StandardCategories.
Require Import UniMath.CategoryTheory.Core.Functors.
Require Import UniMath.CategoryTheory.PrecategoryBinProduct.
Require Import UniMath.Bicategories.Core.Bicat. Import Bicat.Notations.
Require Import UniMath.Bicategories.Core.Univalence.
Require Import UniMath.Bicategories.Core.Adjunctions.
Require Import UniMath.Bicategories.Core.EquivToAdjequiv.
Require Import UniMath.Bicategories.Core.AdjointUnique.
Require Import UniMath.Bicategories.DisplayedBicats.Examples.FullSub.
Require Import UniMath.Bicategories.PseudoFunctors.Display.PseudoFunctorBicat.
Require Import UniMath.Bicategories.PseudoFunctors.PseudoFunctor.
Local Open Scope cat.
Definition full_image
{B₁ B₂ : bicat}
(F : psfunctor B₁ B₂)
: bicat
:= fullsubbicat B₂ (λ y, ∃ (x : B₁), F x = y).
Definition is_univalent_2_1_full_image
{B₁ B₂ : bicat}
(F : psfunctor B₁ B₂)
(B₂_is_univalent_2_1 : is_univalent_2_1 B₂)
: is_univalent_2_1 (full_image F).
Proof.
apply is_univalent_2_1_fullsubbicat.
exact B₂_is_univalent_2_1.
Defined.
Definition is_univalent_2_0_full_image
{B₁ B₂ : bicat}
(F : psfunctor B₁ B₂)
(B₂_is_univalent_2 : is_univalent_2 B₂)
: is_univalent_2_0 (full_image F).
Proof.
apply is_univalent_2_0_fullsubbicat.
- exact B₂_is_univalent_2.
- intro.
apply isapropishinh.
Defined.
Definition is_univalent_2_full_image
{B₁ B₂ : bicat}
(F : psfunctor B₁ B₂)
(B₂_is_univalent_2 : is_univalent_2 B₂)
: is_univalent_2 (full_image F).
Proof.
apply is_univalent_2_fullsubbicat.
- exact B₂_is_univalent_2.
- intro.
apply isapropishinh.
Defined.