Library UniMath.CategoryTheory.Adjunctions.Examples
Require Import UniMath.Foundations.All.
Require Import UniMath.MoreFoundations.All.
Require Import UniMath.CategoryTheory.Core.Prelude.
Require Import UniMath.CategoryTheory.PrecategoryBinProduct.
Require Import UniMath.CategoryTheory.ProductCategory.
Require Import UniMath.CategoryTheory.FunctorCategory.
Require Import UniMath.CategoryTheory.Limits.Terminal.
Require Import UniMath.CategoryTheory.Limits.BinProducts.
Require Import UniMath.CategoryTheory.Limits.Products.
Require Import UniMath.CategoryTheory.Limits.Initial.
Require Import UniMath.CategoryTheory.Limits.BinCoproducts.
Require Import UniMath.CategoryTheory.Limits.Coproducts.
Require Import UniMath.CategoryTheory.Adjunctions.Core.
Require Import UniMath.CategoryTheory.whiskering.
Local Open Scope cat.
Require Import UniMath.MoreFoundations.All.
Require Import UniMath.CategoryTheory.Core.Prelude.
Require Import UniMath.CategoryTheory.PrecategoryBinProduct.
Require Import UniMath.CategoryTheory.ProductCategory.
Require Import UniMath.CategoryTheory.FunctorCategory.
Require Import UniMath.CategoryTheory.Limits.Terminal.
Require Import UniMath.CategoryTheory.Limits.BinProducts.
Require Import UniMath.CategoryTheory.Limits.Products.
Require Import UniMath.CategoryTheory.Limits.Initial.
Require Import UniMath.CategoryTheory.Limits.BinCoproducts.
Require Import UniMath.CategoryTheory.Limits.Coproducts.
Require Import UniMath.CategoryTheory.Adjunctions.Core.
Require Import UniMath.CategoryTheory.whiskering.
Local Open Scope cat.
Lemma is_left_adjoint_bindelta_functor
(PC : BinProducts C)
: is_left_adjoint (bindelta_functor C).
Proof.
apply (tpair _ (binproduct_functor PC)).
use tpair.
- split.
+ use tpair.
* simpl; intro x.
apply (BinProductArrow _ _ (identity x) (identity x)).
* abstract (intros p q f; simpl;
now rewrite precompWithBinProductArrow, id_right, postcompWithBinProductArrow, id_left).
+ use tpair.
* simpl; intro x; split; [ apply BinProductPr1 | apply BinProductPr2 ].
* abstract (intros p q f; unfold catbinprodmor, compose; simpl;
now rewrite BinProductOfArrowsPr1, BinProductOfArrowsPr2).
- abstract (split; simpl; intro x;
[ unfold catbinprodmor, compose; simpl;
now rewrite BinProductPr1Commutes, BinProductPr2Commutes
| cbn; rewrite postcompWithBinProductArrow, !id_left;
apply pathsinv0, BinProduct_endo_is_identity;
[ apply BinProductPr1Commutes | apply BinProductPr2Commutes ]]).
Defined.
(PC : BinProducts C)
: is_left_adjoint (bindelta_functor C).
Proof.
apply (tpair _ (binproduct_functor PC)).
use tpair.
- split.
+ use tpair.
* simpl; intro x.
apply (BinProductArrow _ _ (identity x) (identity x)).
* abstract (intros p q f; simpl;
now rewrite precompWithBinProductArrow, id_right, postcompWithBinProductArrow, id_left).
+ use tpair.
* simpl; intro x; split; [ apply BinProductPr1 | apply BinProductPr2 ].
* abstract (intros p q f; unfold catbinprodmor, compose; simpl;
now rewrite BinProductOfArrowsPr1, BinProductOfArrowsPr2).
- abstract (split; simpl; intro x;
[ unfold catbinprodmor, compose; simpl;
now rewrite BinProductPr1Commutes, BinProductPr2Commutes
| cbn; rewrite postcompWithBinProductArrow, !id_left;
apply pathsinv0, BinProduct_endo_is_identity;
[ apply BinProductPr1Commutes | apply BinProductPr2Commutes ]]).
Defined.
Lemma is_left_adjoint_delta_functor
(I : UU)
(PC : Products I C)
: is_left_adjoint (delta_functor I C).
Proof.
apply (tpair _ (product_functor _ PC)).
use tpair.
- split.
+ use tpair.
* simpl; intro x.
apply (ProductArrow _ _ _ (λ _, identity x)).
* abstract (intros p q f; simpl;
now rewrite precompWithProductArrow, id_right,
postcompWithProductArrow, id_left).
+ use tpair.
* intros x i; apply ProductPr.
* abstract (intros p q f; apply funextsec; intro i; unfold compose; simpl;
now rewrite ProductOfArrowsPr).
- abstract (split; simpl; intro x;
[ apply funextsec; intro i; apply (ProductPrCommutes _ _ (λ _, x))
| cbn; rewrite postcompWithProductArrow;
apply pathsinv0, Product_endo_is_identity; intro i;
eapply pathscomp0; [|apply (ProductPrCommutes I C _ (PC x))];
apply cancel_postcomposition, maponpaths, funextsec; intro j; apply id_left]).
Defined.
(I : UU)
(PC : Products I C)
: is_left_adjoint (delta_functor I C).
Proof.
apply (tpair _ (product_functor _ PC)).
use tpair.
- split.
+ use tpair.
* simpl; intro x.
apply (ProductArrow _ _ _ (λ _, identity x)).
* abstract (intros p q f; simpl;
now rewrite precompWithProductArrow, id_right,
postcompWithProductArrow, id_left).
+ use tpair.
* intros x i; apply ProductPr.
* abstract (intros p q f; apply funextsec; intro i; unfold compose; simpl;
now rewrite ProductOfArrowsPr).
- abstract (split; simpl; intro x;
[ apply funextsec; intro i; apply (ProductPrCommutes _ _ (λ _, x))
| cbn; rewrite postcompWithProductArrow;
apply pathsinv0, Product_endo_is_identity; intro i;
eapply pathscomp0; [|apply (ProductPrCommutes I C _ (PC x))];
apply cancel_postcomposition, maponpaths, funextsec; intro j; apply id_left]).
Defined.
Lemma is_left_adjoint_bincoproduct_functor
(PC : BinCoproducts C)
: is_left_adjoint (bincoproduct_functor PC).
Proof.
apply (tpair _ (bindelta_functor _)).
use tpair.
- split.
+ use tpair.
* simpl; intro p; set (x := pr1 p); set (y := pr2 p).
split; [ apply (BinCoproductIn1 (PC x y)) | apply (BinCoproductIn2 (PC x y)) ].
* abstract (intros p q f; unfold catbinprodmor, compose; simpl;
now rewrite BinCoproductOfArrowsIn1, BinCoproductOfArrowsIn2).
+ use tpair.
* intro x; apply (BinCoproductArrow _ (identity x) (identity x)).
* abstract (intros p q f; simpl;
now rewrite precompWithBinCoproductArrow, postcompWithBinCoproductArrow,
id_right, id_left).
- abstract (split; simpl; intro x;
[ cbn; rewrite precompWithBinCoproductArrow, !id_right;
apply pathsinv0, BinCoproduct_endo_is_identity;
[ apply BinCoproductIn1Commutes | apply BinCoproductIn2Commutes ]
| unfold catbinprodmor, compose; simpl;
now rewrite BinCoproductIn1Commutes, BinCoproductIn2Commutes ]).
Defined.
(PC : BinCoproducts C)
: is_left_adjoint (bincoproduct_functor PC).
Proof.
apply (tpair _ (bindelta_functor _)).
use tpair.
- split.
+ use tpair.
* simpl; intro p; set (x := pr1 p); set (y := pr2 p).
split; [ apply (BinCoproductIn1 (PC x y)) | apply (BinCoproductIn2 (PC x y)) ].
* abstract (intros p q f; unfold catbinprodmor, compose; simpl;
now rewrite BinCoproductOfArrowsIn1, BinCoproductOfArrowsIn2).
+ use tpair.
* intro x; apply (BinCoproductArrow _ (identity x) (identity x)).
* abstract (intros p q f; simpl;
now rewrite precompWithBinCoproductArrow, postcompWithBinCoproductArrow,
id_right, id_left).
- abstract (split; simpl; intro x;
[ cbn; rewrite precompWithBinCoproductArrow, !id_right;
apply pathsinv0, BinCoproduct_endo_is_identity;
[ apply BinCoproductIn1Commutes | apply BinCoproductIn2Commutes ]
| unfold catbinprodmor, compose; simpl;
now rewrite BinCoproductIn1Commutes, BinCoproductIn2Commutes ]).
Defined.
Lemma is_left_adjoint_coproduct_functor
(I : UU)
(PC : Coproducts I C)
: is_left_adjoint (coproduct_functor I PC).
Proof.
apply (tpair _ (delta_functor _ _)).
use tpair.
- split.
+ use tpair.
* intros p i; apply CoproductIn.
* abstract (intros p q f; apply funextsec; intro i; unfold compose; simpl;
now rewrite CoproductOfArrowsIn).
+ use tpair.
* intro x; apply (CoproductArrow _ _ _ (λ _, identity x)).
* abstract (intros p q f; simpl;
now rewrite precompWithCoproductArrow,
postcompWithCoproductArrow,
id_right, id_left).
- abstract (split; simpl; intro x;
[ cbn; rewrite precompWithCoproductArrow;
apply pathsinv0, Coproduct_endo_is_identity; intro i;
eapply pathscomp0; [|apply CoproductInCommutes];
apply maponpaths, maponpaths, funextsec; intro j; apply id_right
| apply funextsec; intro i; apply (CoproductInCommutes _ _ (λ _, x))]).
Defined.
End DeltaProductAdjunctions.
(I : UU)
(PC : Coproducts I C)
: is_left_adjoint (coproduct_functor I PC).
Proof.
apply (tpair _ (delta_functor _ _)).
use tpair.
- split.
+ use tpair.
* intros p i; apply CoproductIn.
* abstract (intros p q f; apply funextsec; intro i; unfold compose; simpl;
now rewrite CoproductOfArrowsIn).
+ use tpair.
* intro x; apply (CoproductArrow _ _ _ (λ _, identity x)).
* abstract (intros p q f; simpl;
now rewrite precompWithCoproductArrow,
postcompWithCoproductArrow,
id_right, id_left).
- abstract (split; simpl; intro x;
[ cbn; rewrite precompWithCoproductArrow;
apply pathsinv0, Coproduct_endo_is_identity; intro i;
eapply pathscomp0; [|apply CoproductInCommutes];
apply maponpaths, maponpaths, funextsec; intro j; apply id_right
| apply funextsec; intro i; apply (CoproductInCommutes _ _ (λ _, x))]).
Defined.
End DeltaProductAdjunctions.
Section functor_swap.
Local Notation "[ C , D ]" := (functor_category C D).
Lemma functor_swap {C D : precategory} {E : category} : functor C [D,E] → functor D [C,E].
Proof.
intros F.
use tpair.
- use tpair.
+ intro d; simpl.
{ use tpair.
- use tpair.
+ intro c.
apply (pr1 (F c) d).
+ intros a b f; apply (# F f).
- abstract (split;
[ now intro x; simpl; rewrite (functor_id F)
| now intros a b c f g; simpl; rewrite (functor_comp F)]).
}
+ intros a b f; simpl.
{ use tpair.
- intros x; apply (# (pr1 (F x)) f).
- abstract (intros c d g; simpl; apply pathsinv0, nat_trans_ax).
}
- abstract (split;
[ intros d; apply (nat_trans_eq (homset_property E)); intro c; simpl; apply functor_id
| intros a b c f g; apply (nat_trans_eq (homset_property E)); intro x; simpl; apply functor_comp]).
Defined.
Lemma functor_cat_swap_nat_trans {C D : precategory} {E : category}
(F G : functor C [D, E]) (α : nat_trans F G) :
nat_trans (functor_swap F) (functor_swap G).
Proof.
use tpair.
+ intros d; simpl.
use tpair.
* intro c; apply (α c).
* abstract (intros a b f; apply (nat_trans_eq_pointwise (nat_trans_ax α _ _ f) d)).
+ abstract (intros a b f; apply (nat_trans_eq (homset_property E)); intro c; simpl; apply nat_trans_ax).
Defined.
Lemma functor_cat_swap (C D : precategory) (E : category) : functor [C, [D, E]] [D, [C, E]].
Proof.
use tpair.
- use tpair.
+ apply functor_swap.
+ cbn. apply functor_cat_swap_nat_trans.
- abstract (split;
[ intro F; apply (nat_trans_eq (functor_category_has_homsets _ _ (homset_property E))); simpl; intro d;
now apply (nat_trans_eq (homset_property E))
| intros F G H α β; cbn; apply (nat_trans_eq (functor_category_has_homsets _ _ (homset_property E))); intro d;
now apply (nat_trans_eq (homset_property E))]).
Defined.
Definition id_functor_cat_swap (C D : precategory) (E : category) :
nat_trans (functor_identity [C,[D,E]])
(functor_composite (functor_cat_swap C D E) (functor_cat_swap D C E)).
Proof.
set (hsE := homset_property E).
use tpair.
+ intros F.
use tpair.
- intro c.
use tpair.
* now intro f; apply identity.
* abstract (now intros a b f; rewrite id_left, id_right).
- abstract (now intros a b f; apply (nat_trans_eq hsE); intro d; simpl; rewrite id_left, id_right).
+ abstract (now intros a b f; apply nat_trans_eq; [apply functor_category_has_homsets|]; intro c;
apply (nat_trans_eq hsE); intro d; simpl; rewrite id_left, id_right).
Defined.
Definition functor_cat_swap_id (C D : precategory) (E : category) :
nat_trans (functor_composite (functor_cat_swap D C E) (functor_cat_swap C D E))
(functor_identity [D,[C,E]]).
Proof.
set (hsE := homset_property E).
use tpair.
+ intros F.
use tpair.
- intro c.
use tpair.
* now intro f; apply identity.
* abstract (now intros a b f; rewrite id_left, id_right).
- abstract (now intros a b f; apply (nat_trans_eq hsE); intro d; simpl; rewrite id_left, id_right).
+ abstract (now intros a b f; apply nat_trans_eq; [apply functor_category_has_homsets|]; intro c;
apply (nat_trans_eq hsE); intro d; simpl; rewrite id_left, id_right).
Defined.
Lemma form_adjunction_functor_cat_swap (C D : precategory) (E : category) :
form_adjunction _ _ (id_functor_cat_swap C D E) (functor_cat_swap_id C D E).
Proof.
set (hsE := homset_property E).
split; intro F.
+ apply (nat_trans_eq (functor_category_has_homsets _ _ hsE)
(pr1 (pr1 (functor_cat_swap C D E) F))
(pr1 (pr1 (functor_cat_swap C D E) F))).
now intro d; apply (nat_trans_eq hsE); intro c; apply id_right.
+ apply (nat_trans_eq (functor_category_has_homsets _ _ hsE)
(pr1 (pr1 (functor_cat_swap D C E) F))
(pr1 (pr1 (functor_cat_swap D C E) F))).
now intro d; apply (nat_trans_eq hsE); intro c; apply id_left.
Qed.
Lemma are_adjoint_functor_cat_swap (C D : precategory) (E : category) :
are_adjoints (@functor_cat_swap C D E) (@functor_cat_swap D C E).
Proof.
use tpair.
- split; [ apply id_functor_cat_swap | apply functor_cat_swap_id ].
- apply form_adjunction_functor_cat_swap.
Defined.
Lemma is_left_adjoint_functor_cat_swap (C D : precategory) (E : category) :
is_left_adjoint (functor_cat_swap C D E).
Proof.
use tpair.
+ apply functor_cat_swap.
+ apply are_adjoint_functor_cat_swap.
Defined.
End functor_swap.
Local Notation "[ C , D ]" := (functor_category C D).
Lemma functor_swap {C D : precategory} {E : category} : functor C [D,E] → functor D [C,E].
Proof.
intros F.
use tpair.
- use tpair.
+ intro d; simpl.
{ use tpair.
- use tpair.
+ intro c.
apply (pr1 (F c) d).
+ intros a b f; apply (# F f).
- abstract (split;
[ now intro x; simpl; rewrite (functor_id F)
| now intros a b c f g; simpl; rewrite (functor_comp F)]).
}
+ intros a b f; simpl.
{ use tpair.
- intros x; apply (# (pr1 (F x)) f).
- abstract (intros c d g; simpl; apply pathsinv0, nat_trans_ax).
}
- abstract (split;
[ intros d; apply (nat_trans_eq (homset_property E)); intro c; simpl; apply functor_id
| intros a b c f g; apply (nat_trans_eq (homset_property E)); intro x; simpl; apply functor_comp]).
Defined.
Lemma functor_cat_swap_nat_trans {C D : precategory} {E : category}
(F G : functor C [D, E]) (α : nat_trans F G) :
nat_trans (functor_swap F) (functor_swap G).
Proof.
use tpair.
+ intros d; simpl.
use tpair.
* intro c; apply (α c).
* abstract (intros a b f; apply (nat_trans_eq_pointwise (nat_trans_ax α _ _ f) d)).
+ abstract (intros a b f; apply (nat_trans_eq (homset_property E)); intro c; simpl; apply nat_trans_ax).
Defined.
Lemma functor_cat_swap (C D : precategory) (E : category) : functor [C, [D, E]] [D, [C, E]].
Proof.
use tpair.
- use tpair.
+ apply functor_swap.
+ cbn. apply functor_cat_swap_nat_trans.
- abstract (split;
[ intro F; apply (nat_trans_eq (functor_category_has_homsets _ _ (homset_property E))); simpl; intro d;
now apply (nat_trans_eq (homset_property E))
| intros F G H α β; cbn; apply (nat_trans_eq (functor_category_has_homsets _ _ (homset_property E))); intro d;
now apply (nat_trans_eq (homset_property E))]).
Defined.
Definition id_functor_cat_swap (C D : precategory) (E : category) :
nat_trans (functor_identity [C,[D,E]])
(functor_composite (functor_cat_swap C D E) (functor_cat_swap D C E)).
Proof.
set (hsE := homset_property E).
use tpair.
+ intros F.
use tpair.
- intro c.
use tpair.
* now intro f; apply identity.
* abstract (now intros a b f; rewrite id_left, id_right).
- abstract (now intros a b f; apply (nat_trans_eq hsE); intro d; simpl; rewrite id_left, id_right).
+ abstract (now intros a b f; apply nat_trans_eq; [apply functor_category_has_homsets|]; intro c;
apply (nat_trans_eq hsE); intro d; simpl; rewrite id_left, id_right).
Defined.
Definition functor_cat_swap_id (C D : precategory) (E : category) :
nat_trans (functor_composite (functor_cat_swap D C E) (functor_cat_swap C D E))
(functor_identity [D,[C,E]]).
Proof.
set (hsE := homset_property E).
use tpair.
+ intros F.
use tpair.
- intro c.
use tpair.
* now intro f; apply identity.
* abstract (now intros a b f; rewrite id_left, id_right).
- abstract (now intros a b f; apply (nat_trans_eq hsE); intro d; simpl; rewrite id_left, id_right).
+ abstract (now intros a b f; apply nat_trans_eq; [apply functor_category_has_homsets|]; intro c;
apply (nat_trans_eq hsE); intro d; simpl; rewrite id_left, id_right).
Defined.
Lemma form_adjunction_functor_cat_swap (C D : precategory) (E : category) :
form_adjunction _ _ (id_functor_cat_swap C D E) (functor_cat_swap_id C D E).
Proof.
set (hsE := homset_property E).
split; intro F.
+ apply (nat_trans_eq (functor_category_has_homsets _ _ hsE)
(pr1 (pr1 (functor_cat_swap C D E) F))
(pr1 (pr1 (functor_cat_swap C D E) F))).
now intro d; apply (nat_trans_eq hsE); intro c; apply id_right.
+ apply (nat_trans_eq (functor_category_has_homsets _ _ hsE)
(pr1 (pr1 (functor_cat_swap D C E) F))
(pr1 (pr1 (functor_cat_swap D C E) F))).
now intro d; apply (nat_trans_eq hsE); intro c; apply id_left.
Qed.
Lemma are_adjoint_functor_cat_swap (C D : precategory) (E : category) :
are_adjoints (@functor_cat_swap C D E) (@functor_cat_swap D C E).
Proof.
use tpair.
- split; [ apply id_functor_cat_swap | apply functor_cat_swap_id ].
- apply form_adjunction_functor_cat_swap.
Defined.
Lemma is_left_adjoint_functor_cat_swap (C D : precategory) (E : category) :
is_left_adjoint (functor_cat_swap C D E).
Proof.
use tpair.
+ apply functor_cat_swap.
+ apply are_adjoint_functor_cat_swap.
Defined.
End functor_swap.
Section LeftAdjointIso.
Context {C D : category}
{L₁ L₂ : C ⟶ D}
(HL₁ : is_left_adjoint L₁)
(τ : nat_z_iso L₁ L₂).
Let R : D ⟶ C := right_adjoint HL₁.
Let η : functor_identity C ⟹ L₁ ∙ R := unit_from_left_adjoint HL₁.
Let ε : R ∙ L₁ ⟹ functor_identity _ := counit_from_left_adjoint HL₁.
Let η' : functor_identity C ⟹ L₂ ∙ R
:= nat_trans_comp
_ _ _
η
(post_whisker τ R).
Let ε' : R ∙ L₂ ⟹ functor_identity D
:= nat_trans_comp
_ _ _
(pre_whisker R (nat_z_iso_inv τ))
ε.
Proposition is_left_adjoint_nat_z_iso_triangle_1
: triangle_1_statement (L₂ ,, R ,, η' ,, ε').
Proof.
intros x ; cbn -[η ε].
etrans.
{
rewrite !assoc.
apply maponpaths_2.
exact (nat_trans_ax (nat_z_iso_inv τ) _ _ (η x · #R(τ x))).
}
rewrite functor_comp.
rewrite !assoc'.
etrans.
{
do 2 apply maponpaths.
apply (nat_trans_ax ε).
}
etrans.
{
apply maponpaths.
rewrite !assoc.
apply maponpaths_2.
apply (pr122 HL₁).
}
rewrite id_left.
exact (z_iso_after_z_iso_inv (nat_z_iso_pointwise_z_iso τ x)).
Qed.
Proposition is_left_adjoint_nat_z_iso_triangle_2
: triangle_2_statement (L₂ ,, R ,, η' ,, ε').
Proof.
intros x ; cbn.
rewrite !assoc'.
rewrite <- functor_comp.
rewrite assoc.
etrans.
{
do 2 apply maponpaths.
apply maponpaths_2.
exact (z_iso_inv_after_z_iso (nat_z_iso_pointwise_z_iso τ (R x))).
}
rewrite id_left.
apply (pr222 HL₁).
Qed.
Definition is_left_adjoint_nat_z_iso
: is_left_adjoint L₂.
Proof.
simple refine (R ,, (η' ,, ε') ,, (_ ,, _)).
- exact is_left_adjoint_nat_z_iso_triangle_1.
- exact is_left_adjoint_nat_z_iso_triangle_2.
Defined.
End LeftAdjointIso.
Context {C D : category}
{L₁ L₂ : C ⟶ D}
(HL₁ : is_left_adjoint L₁)
(τ : nat_z_iso L₁ L₂).
Let R : D ⟶ C := right_adjoint HL₁.
Let η : functor_identity C ⟹ L₁ ∙ R := unit_from_left_adjoint HL₁.
Let ε : R ∙ L₁ ⟹ functor_identity _ := counit_from_left_adjoint HL₁.
Let η' : functor_identity C ⟹ L₂ ∙ R
:= nat_trans_comp
_ _ _
η
(post_whisker τ R).
Let ε' : R ∙ L₂ ⟹ functor_identity D
:= nat_trans_comp
_ _ _
(pre_whisker R (nat_z_iso_inv τ))
ε.
Proposition is_left_adjoint_nat_z_iso_triangle_1
: triangle_1_statement (L₂ ,, R ,, η' ,, ε').
Proof.
intros x ; cbn -[η ε].
etrans.
{
rewrite !assoc.
apply maponpaths_2.
exact (nat_trans_ax (nat_z_iso_inv τ) _ _ (η x · #R(τ x))).
}
rewrite functor_comp.
rewrite !assoc'.
etrans.
{
do 2 apply maponpaths.
apply (nat_trans_ax ε).
}
etrans.
{
apply maponpaths.
rewrite !assoc.
apply maponpaths_2.
apply (pr122 HL₁).
}
rewrite id_left.
exact (z_iso_after_z_iso_inv (nat_z_iso_pointwise_z_iso τ x)).
Qed.
Proposition is_left_adjoint_nat_z_iso_triangle_2
: triangle_2_statement (L₂ ,, R ,, η' ,, ε').
Proof.
intros x ; cbn.
rewrite !assoc'.
rewrite <- functor_comp.
rewrite assoc.
etrans.
{
do 2 apply maponpaths.
apply maponpaths_2.
exact (z_iso_inv_after_z_iso (nat_z_iso_pointwise_z_iso τ (R x))).
}
rewrite id_left.
apply (pr222 HL₁).
Qed.
Definition is_left_adjoint_nat_z_iso
: is_left_adjoint L₂.
Proof.
simple refine (R ,, (η' ,, ε') ,, (_ ,, _)).
- exact is_left_adjoint_nat_z_iso_triangle_1.
- exact is_left_adjoint_nat_z_iso_triangle_2.
Defined.
End LeftAdjointIso.
Section InitialTerminal.
Context {C C' : category}.
Context (I : Initial C').
Context (T : Terminal C).
Definition initial_terminal_are_adjoints
: are_adjoints (Initial_functor_precat C C' I : [_, _]) (Terminal_functor_precat C' C T : [_, _]).
Proof.
apply adjunction_homsetiso_weq.
use tpair.
- intros A B.
use weq_iso.
+ intro.
apply TerminalArrow.
+ intro.
apply InitialArrow.
+ abstract (
intro;
apply InitialArrowEq
).
+ abstract (
intro;
apply TerminalArrowEq
).
- abstract (
split;
intros;
apply TerminalArrowEq
).
Defined.
Definition initial_terminal_adjunction
: adjunction C C'
:= left_adjoint_to_adjunction initial_terminal_are_adjoints.
End InitialTerminal.
Context {C C' : category}.
Context (I : Initial C').
Context (T : Terminal C).
Definition initial_terminal_are_adjoints
: are_adjoints (Initial_functor_precat C C' I : [_, _]) (Terminal_functor_precat C' C T : [_, _]).
Proof.
apply adjunction_homsetiso_weq.
use tpair.
- intros A B.
use weq_iso.
+ intro.
apply TerminalArrow.
+ intro.
apply InitialArrow.
+ abstract (
intro;
apply InitialArrowEq
).
+ abstract (
intro;
apply TerminalArrowEq
).
- abstract (
split;
intros;
apply TerminalArrowEq
).
Defined.
Definition initial_terminal_adjunction
: adjunction C C'
:= left_adjoint_to_adjunction initial_terminal_are_adjoints.
End InitialTerminal.