Library UniMath.CategoryTheory.UnderCategories

Undercategories
Require Import UniMath.Foundations.PartD.
Require Import UniMath.Foundations.Propositions.
Require Import UniMath.Foundations.Sets.

Require Import UniMath.CategoryTheory.Core.Categories.
Require Import UniMath.CategoryTheory.Core.Functors.
Local Open Scope cat.

Section def_underprecategories.

Variable C : precategory.
Hypothesis hs : has_homsets C.
Variable c : ob C.

Definition Under_ob : UU := d, Cc, d.

Definition make_Under_ob {d : ob C} (f : Cc, d) : Under_ob := tpair _ d f.

Definition Under_ob_cod (X : Under_ob) : ob C := pr1 X.

Definition Under_ob_mor (X : Under_ob) : Cc, Under_ob_cod X := pr2 X.

Definition Under_mor (X Y : Under_ob) : UU :=
f : CUnder_ob_cod X, Under_ob_cod Y, Under_ob_mor X · f = Under_ob_mor Y.

Definition make_Under_mor (X Y : Under_ob) (f : CUnder_ob_cod X, Under_ob_cod Y)
(H : Under_ob_mor X · f = Under_ob_mor Y) : Under_mor X Y := tpair _ f H.

Definition Under_mor_mor {X Y : Under_ob} (M : Under_mor X Y) :
CUnder_ob_cod X, Under_ob_cod Y := pr1 M.

Definition Under_mor_eq {X Y : Under_ob} (M : Under_mor X Y) :
Under_ob_mor X · Under_mor_mor M = Under_ob_mor Y := pr2 M.

Definition isaset_Under_mor (X Y : Under_ob) : isaset (Under_mor X Y).
Proof.
apply (isofhleveltotal2 2).
- apply hs.
- intros x.
apply hlevelntosn.
apply hs.
Qed.

Definition Under_mor_equality (X Y : Under_ob) (f f' : Under_mor X Y) : pr1 f = pr1 f' f = f'.
Proof.
intro H.
apply subtypePath.
intro x.
apply hs.
exact H.
Qed.

Definition Under_id (X : Under_ob) : Under_mor X X := make_Under_mor X X (identity _) (id_right _ ).

Local Lemma Under_comp_eq {X Y Z : Under_ob} (f : Under_mor X Y) (g : Under_mor Y Z) :
Under_ob_mor X · (Under_mor_mor f · Under_mor_mor g) = Under_ob_mor Z.
Proof.
rewrite assoc.
rewrite (Under_mor_eq f).
exact (Under_mor_eq g).
Qed.

Definition Under_comp (X Y Z : Under_ob) : Under_mor X Y Under_mor Y Z Under_mor X Z.
Proof.
intros f g.
exact (make_Under_mor X Z (Under_mor_mor f · Under_mor_mor g) (Under_comp_eq f g)).
Defined.

Definition Undercategory_ob_mor : precategory_ob_mor.
Proof.
Under_ob.
exact Under_mor.
Defined.

Definition Undercategory_data : precategory_data.
Proof.
Undercategory_ob_mor.
split.
- exact Under_id.
- exact Under_comp.
Defined.

Definition is_precategory_Undercategory_data : is_precategory Undercategory_data.
Proof.
repeat split.
- intros. apply Under_mor_equality. apply id_left.
- intros. apply Under_mor_equality. apply id_right.
- intros. apply Under_mor_equality. apply assoc.
- intros. apply Under_mor_equality. apply assoc'.
Defined.

Definition Undercategory : precategory.
Proof.
Undercategory_data.
exact is_precategory_Undercategory_data.
Defined.

Lemma has_homsets_Under : has_homsets Undercategory.
Proof.
intros X Y.
apply (isaset_Under_mor X Y).
Qed.

End def_underprecategories.

Morphism of tips induces a functor

Section undercategories_morphisms.

Variable C : precategory.
Hypothesis hs : has_homsets C.

Local Notation "c / C" := (@Undercategory C hs c).

Definition Under_precategories_mor_ob {c c' : C} (h : Cc, c') : c' / C c / C.
Proof.
intro af.
(pr1 af).
exact (h · pr2 af).
Defined.

Local Lemma Under_precategories_mor_mor_eq {c c' : C} (h : Cc, c')
(af af' : c' / C) (g : (c' / C)⟦af, af') :
(Under_ob_mor C c (Under_precategories_mor_ob h af)) · (Under_mor_mor C c' g) =
(Under_ob_mor C c (Under_precategories_mor_ob h af')).
Proof.
cbn.
rewrite <- assoc.
apply cancel_precomposition.
set (tmp := Under_mor_eq C c' g).
unfold Under_mor_mor.
unfold Under_mor_mor, Under_ob_mor in tmp.
exact tmp.
Qed.

Definition Under_precategories_mor_mor {c c' : C} (h : Cc, c') (af af' : c' / C)
(g : (c' / C)⟦af, af') : (c / C) Under_precategories_mor_ob h af,
Under_precategories_mor_ob h af'.
Proof.
(Under_mor_mor C c' g).
exact (Under_precategories_mor_mor_eq h af af' g).
Defined.

Definition Under_precategories_mor_functor_data {c c' : C} (h : Cc, c') :
functor_data (c' / C) (c / C).
Proof.
(Under_precategories_mor_ob h).
exact (Under_precategories_mor_mor h).
Defined.

Lemma is_functor_Under_mor_functor_data {c c' : C} (h : Cc, c') :
is_functor (Under_precategories_mor_functor_data h).
Proof.
split.
- intro. apply (Under_mor_equality _ hs). apply idpath.
- intros ? ? ? ? ?. apply (Under_mor_equality _ hs). apply idpath.
Defined.

Definition functor_Under_precategories_mor {c c' : C} (h : Cc, c') :
functor _ _ := tpair _ _ (is_functor_Under_mor_functor_data h).

End undercategories_morphisms.