Library UniMath.CategoryTheory.EnrichedCats.Examples.Precomposition
Require Import UniMath.Foundations.All.
Require Import UniMath.MoreFoundations.All.
Require Import UniMath.CategoryTheory.Core.Prelude.
Require Import UniMath.CategoryTheory.whiskering.
Require Import UniMath.CategoryTheory.EnrichedCats.Enrichment.
Require Import UniMath.CategoryTheory.EnrichedCats.EnrichmentFunctor.
Require Import UniMath.CategoryTheory.EnrichedCats.EnrichmentTransformation.
Require Import UniMath.CategoryTheory.EnrichedCats.Examples.FunctorCategory.
Require Import UniMath.CategoryTheory.Monoidal.Categories.
Local Open Scope cat.
Section Precomp.
Context {V : monoidal_cat}
{C₁ C₂ C₃ : category}
{F : C₁ ⟶ C₂}
{E₁ : enrichment C₁ V}
{E₂ : enrichment C₂ V}
(E₃ : enrichment C₃ V)
(EF : functor_enrichment F E₁ E₂).
Definition enriched_precomp_data
: functor_data
(enriched_functor_category E₂ E₃)
(enriched_functor_category E₁ E₃).
Show proof.
Proposition is_functor_enriched_precomp
: is_functor enriched_precomp_data.
Show proof.
Definition enriched_precomp
: enriched_functor_category E₂ E₃
⟶
enriched_functor_category E₁ E₃.
Show proof.
End Precomp.
Require Import UniMath.MoreFoundations.All.
Require Import UniMath.CategoryTheory.Core.Prelude.
Require Import UniMath.CategoryTheory.whiskering.
Require Import UniMath.CategoryTheory.EnrichedCats.Enrichment.
Require Import UniMath.CategoryTheory.EnrichedCats.EnrichmentFunctor.
Require Import UniMath.CategoryTheory.EnrichedCats.EnrichmentTransformation.
Require Import UniMath.CategoryTheory.EnrichedCats.Examples.FunctorCategory.
Require Import UniMath.CategoryTheory.Monoidal.Categories.
Local Open Scope cat.
Section Precomp.
Context {V : monoidal_cat}
{C₁ C₂ C₃ : category}
{F : C₁ ⟶ C₂}
{E₁ : enrichment C₁ V}
{E₂ : enrichment C₂ V}
(E₃ : enrichment C₃ V)
(EF : functor_enrichment F E₁ E₂).
Definition enriched_precomp_data
: functor_data
(enriched_functor_category E₂ E₃)
(enriched_functor_category E₁ E₃).
Show proof.
use make_functor_data.
- exact (λ G, F ∙ _ ,, functor_comp_enrichment EF (pr2 G)).
- exact (λ _ _ τ, pre_whisker F (pr1 τ) ,, pre_whisker_enrichment EF (pr2 τ)).
- exact (λ G, F ∙ _ ,, functor_comp_enrichment EF (pr2 G)).
- exact (λ _ _ τ, pre_whisker F (pr1 τ) ,, pre_whisker_enrichment EF (pr2 τ)).
Proposition is_functor_enriched_precomp
: is_functor enriched_precomp_data.
Show proof.
split.
- intros G.
use subtypePath ; [ intro ; apply isaprop_nat_trans_enrichment | ].
use nat_trans_eq ; [ apply homset_property | ].
intro x ; cbn.
apply idpath.
- intros G₁ G₂ G₃ τ₁ τ₂.
use subtypePath ; [ intro ; apply isaprop_nat_trans_enrichment | ].
use nat_trans_eq ; [ apply homset_property | ].
intro x ; cbn.
apply idpath.
- intros G.
use subtypePath ; [ intro ; apply isaprop_nat_trans_enrichment | ].
use nat_trans_eq ; [ apply homset_property | ].
intro x ; cbn.
apply idpath.
- intros G₁ G₂ G₃ τ₁ τ₂.
use subtypePath ; [ intro ; apply isaprop_nat_trans_enrichment | ].
use nat_trans_eq ; [ apply homset_property | ].
intro x ; cbn.
apply idpath.
Definition enriched_precomp
: enriched_functor_category E₂ E₃
⟶
enriched_functor_category E₁ E₃.
Show proof.
End Precomp.