Library UniMath.Folds.folds_pre_2_cat
Univalent FOLDS
Benedikt Ahrens, following notes by Michael Shulman
Contents of this file:
- Definition: FOLDS pre-3-category
- Definition: FOLDS pre-2-categoy
- Isomorphism between morphisms in a FOLDS pre-2-category
- Lemma: In a FOLDS pre-2-category, folds_2_iso f g is equivalent to E f g
- Definition: univalent FOLDS pre-2-category as special FOLDS pre-2-category
- idtoiso2 is an equivalence
- is_univalent_folds_2_precat is an hProp
- Definition: FOLDS precategory as special FOLDS pre-2-category
- predicate is_folds_precategory defined as
- hom-types are sets
- axioms of category modul = rather than E
- predicate is_folds_precategory defined as
- Lemma: Logical equivalence between being a FOLDS precategory and being univalent
- since both are hProps, this entails equivalence between the types of
- univalent FOLDS pre-2-cats
- FOLDS precats
- Implications are called
- since both are hProps, this entails equivalence between the types of
Require Import UniMath.Folds.UnicodeNotations.
Require Import UniMath.Foundations.PartD.
Require Import UniMath.Foundations.Propositions.
Require Import UniMath.Foundations.Sets.
Require Import UniMath.CategoryTheory.total2_paths.
Local Notation "p ## a" := (transportf _ p a) (at level 3, only parsing).
Definition folds_3_ob_mor := ∑ a : UU, a → a → UU.
Definition folds_3_ob_mor_pair (ob : UU)(mor : ob → ob → UU) : folds_3_ob_mor
:= tpair _ ob mor.
Definition ob (C : folds_3_ob_mor) : UU := @pr1 _ _ C.
Coercion ob : folds_3_ob_mor >-> UU.
Definition folds_3_morphisms {C : folds_3_ob_mor} : C → C → UU := pr2 C.
Local Notation "a ⇒ b" := (folds_3_morphisms a b).
Definition double_transport {C : folds_3_ob_mor} {a a' b b' : ob C}
(p : a = a') (q : b = b') (f : a ⇒ b) : a' ⇒ b' :=
transportf (λ c, a' ⇒ c) q (transportf (λ c, c ⇒ b) p f).
Identity, composition, and equality, given through predicates
We do not assume those to be propositions.Definition folds_3_id_comp_eq := ∑ C : folds_3_ob_mor,
( (∏ a : C, a ⇒ a → UU)
× (∏ (a b c : C), (a ⇒ b) → (b ⇒ c) → (a ⇒ c) → UU))
× ∏ a b : C, a ⇒ b → a ⇒ b → UU.
Definition folds_ob_mor_from_folds_id_comp (C : folds_3_id_comp_eq) : folds_3_ob_mor := pr1 C.
Coercion folds_ob_mor_from_folds_id_comp : folds_3_id_comp_eq >-> folds_3_ob_mor.
Definition I {C : folds_3_id_comp_eq} : ∏ {a : C}, a ⇒ a → UU := pr1 (pr1 (pr2 C)).
Definition T {C : folds_3_id_comp_eq} :
∏ {a b c : C}, (a ⇒ b) → (b ⇒ c) → (a ⇒ c) → UU := pr2 (pr1 (pr2 C)).
Definition E {C : folds_3_id_comp_eq} :
∏ {a b : C}, a ⇒ b → a ⇒ b → UU := pr2 (pr2 C).
Definition E_is_good_to_I_and_T (C : folds_3_id_comp_eq) : UU :=
(((∏ (a b : C) (f : a ⇒ b), E f f)
× (∏ (a b : C) (f g : a ⇒ b), E f g → E g f))
× (∏ (a b : C) (f g h : a ⇒ b), E f g → E g h → E f h))
× ((∏ (a : C) (f g : a ⇒ a), E f g → I f → I g)
× (∏ (a b c : C) (f f' : a ⇒ b) (g g' : b ⇒ c) (h h' : a ⇒ c),
E f f' → E g g' → E h h' → T f g h → T f' g' h')).
Definition folds_ax_id (C : folds_3_id_comp_eq) :=
(∏ a : C, ∥ ∑ f : a ⇒ a, I f ∥ )
× ((∏ (a b : C) (f : a ⇒ b)(i : b ⇒ b), I i → T f i f)
× (∏ (a b : C) (f : a ⇒ b)(i : a ⇒ a), I i → T i f f)).
Definition folds_ax_comp (C : folds_3_id_comp_eq) :=
(∏ {a b c : C} (f : a ⇒ b) (g : b ⇒ c), ∥ ∑ h : a ⇒ c, T f g h ∥ )
× ( (∏ {a b c : C} {f : a ⇒ b} {g : b ⇒ c} {h k : a ⇒ c}, T f g h → T f g k → E h k )
× (∏ {a b c d : C} (f : a ⇒ b) (g : b ⇒ c) (h : c ⇒ d) (fg : a ⇒ c)
(gh : b ⇒ d) (fg_h : a ⇒ d) (f_gh : a ⇒ d),
T f g fg → T g h gh → T fg h fg_h → T f gh f_gh → E f_gh fg_h)).
A FOLDS pre-3-category is a package of
satisfying the categorical axioms modulo E and E is an "equality"Definition folds_pre_3_cat := ∑ C : folds_3_id_comp_eq,
(folds_ax_id C
× folds_ax_comp C)
× E_is_good_to_I_and_T C.
Definition folds_id_comp_from_folds_precat (C : folds_pre_3_cat) : folds_3_id_comp_eq := pr1 C.
Coercion folds_id_comp_from_folds_precat : folds_pre_3_cat >-> folds_3_id_comp_eq.
Definition is_folds_pre_2_cat (C : folds_pre_3_cat) :=
( (∏ (a : C) (i : a ⇒ a), isaprop (I i))
× (∏ (a b c : C) (f : a ⇒ b) (g : b ⇒ c) (h : a ⇒ c), isaprop (T f g h)))
× (∏ (a b : C) (f g : a ⇒ b), isaprop (E f g)).
Definition folds_pre_2_cat : UU := ∑ C, is_folds_pre_2_cat C.
Ltac folds_pre_2_cat_props C :=
try apply (pr2 (pr1 (pr2 C)));
try apply (pr1 (pr1 (pr2 C)));
try apply (pr2 (pr2 C)).
Definition folds_3_from_folds_2 (C : folds_pre_2_cat) : folds_pre_3_cat := pr1 C.
Coercion folds_3_from_folds_2 : folds_pre_2_cat >-> folds_pre_3_cat.
Definition folds_iso {C: folds_pre_3_cat} {a b : C} (f g : a ⇒ b) : UU :=
(((∏ (x : C) (u : x ⇒ a) (v : x ⇒ b), T u f v ≃ T u g v)
× (∏ (x : C) (u : a ⇒ x) (v : x ⇒ b), T u v f ≃ T u v g))
× (∏ (x : C) (u : b ⇒ x) (v : a ⇒ x), T f u v ≃ T g u v))
× ((((∏ (u : a ⇒ b) (p : b = a), T p ## f f u ≃ T p ## g g u)
× (∏ (u : b ⇒ b) (p : a = a), T (transportf (λ a, a ⇒ b) p f) u f ≃
T (transportf (λ a, a ⇒ b) p g) u g))
× ((∏ (u : a ⇒ a) (p : b = b), T u p ## f f ≃ T u p ## g g)
× (∏ (p : a = a) (q : b = a) (r : b = b),
T (double_transport p q f) r ## f f
≃ T (double_transport p q g) r ## g g)))
× (((∏ p : b = a, I p ## f ≃ I p ## g) × (∏ u : a ⇒ b, E f u ≃ E g u))
× ((∏ u : a ⇒ b, E u f ≃ E u g)
× (∏ (p : a = a) (q : b = b),
E (double_transport p q f) f ≃ E (double_transport p q g) g)))).
Lemma isaprop_folds_2_iso (C : folds_pre_2_cat) (a b : C) (f g : a ⇒ b) :
isaprop (folds_iso f g).
Proof.
repeat (apply isofhleveldirprod);
repeat (apply impred; intro);
apply isofhlevelsnweqtohlevelsn;
folds_pre_2_cat_props C.
Qed.
Definition folds_2_iso_id (C : folds_pre_3_cat) (a b : C) (f : a ⇒ b) : folds_iso f f.
Proof.
repeat split;
intros; apply idweq.
Defined.
Lemma E_transport_source : ∏ (C : folds_pre_2_cat) (a a' b : C) (f g : a ⇒ b) (p : a = a'),
E f g → E (transportf (λ c, c ⇒ b) p f) (transportf (λ c, c ⇒ b) p g).
Proof.
intros. destruct p.
assumption.
Defined.
Lemma E_transport_target : ∏ (C : folds_pre_2_cat) (a b b' : C) (f g : a ⇒ b) (p : b = b'),
E f g → E (transportf (λ c, a ⇒ c) p f) (transportf (λ c, a ⇒ c) p g).
Proof.
intros. destruct p.
assumption.
Defined.
Lemma E_implies_folds_iso (C : folds_pre_2_cat) (a b : C) (f g : a ⇒ b) : E f g → folds_iso f g.
Proof.
set (H' := pr2 (pr2 (pr1 C))). simpl in H'.
destruct H' as [[[Erefl Esym] Etrans] [EI ET]].
intro Efg.
repeat split; intros; apply weqimplimpl; intro; folds_pre_2_cat_props C.
- apply (ET _ _ _ u u f g v v); auto.
- apply (ET _ _ _ u u g f v v) ; auto.
- apply (ET _ _ _ u u v v f g); auto.
- apply (ET _ _ _ u u v v g f); auto.
- apply (ET _ _ _ f g u u v v); auto.
- apply (ET _ _ _ g f u u v v); auto.
- destruct p;
apply (ET _ _ _ f g f g u u); auto.
- destruct p;
apply (ET _ _ _ g f g f u u); auto.
- apply (ET _ _ _ (transportf (λ c, c ⇒ b) p f) (p ## g) u u f g);
try apply E_transport_source; auto.
- apply (ET _ _ _ (transportf (λ c, c ⇒ b) p g) (p ## f) u u g f);
try apply E_transport_source; auto.
- apply (ET _ _ _ u u (transportf (λ c, a ⇒ c) p f) (p ## g) f g);
try apply E_transport_target; auto.
- apply (ET _ _ _ u u (transportf (λ c, a ⇒ c) p g) (p ## f) g f);
try apply E_transport_target; auto.
- apply (ET _ _ _ (double_transport p q f) (double_transport p q g)
(transportf (λ c, a ⇒ c) r f) (r ## g) f g);
try apply E_transport_target; try apply E_transport_source; auto.
- apply (ET _ _ _ (double_transport p q g) (double_transport p q f)
(transportf (λ c, a ⇒ c) r g) (r ## f) g f);
try apply E_transport_target; try apply E_transport_source; auto.
- destruct p. apply (EI _ f); auto.
- destruct p; apply (EI _ g); auto.
- apply (Etrans _ _ g f u).
+ apply Esym; auto.
+ auto.
- apply (Etrans _ _ f g u); auto.
- apply (Etrans _ _ u f g); auto.
- apply (Etrans _ _ u g f).
+ auto.
+ apply Esym; auto.
- apply (Etrans _ _ (double_transport p q g) (double_transport p q f) g).
+ apply E_transport_target. apply E_transport_source. apply Esym; auto.
+ apply (Etrans _ _ (double_transport p q f) f g); auto.
- apply (Etrans _ _ (double_transport p q f) (double_transport p q g) f).
+ apply E_transport_target. apply E_transport_source. auto.
+ apply (Etrans _ _ (double_transport p q g) g f); auto.
Qed.
Lemma folds_iso_implies_E (C : folds_pre_2_cat) (a b : C) (f g : a ⇒ b) : folds_iso f g → E f g.
Proof.
intro Isofg.
set (keytojoy := pr1 (pr2 (pr2 (pr2 Isofg)))).
apply (keytojoy f).
set (H' := pr2 (pr2 (pr1 C))). simpl in H'.
destruct H' as [[[Erefl Esym] Etrans] [EI ET]].
apply Erefl.
Qed.
Definition idtoiso2 {C : folds_pre_2_cat} {a b : C} {f g : a ⇒ b} : f = g → folds_iso f g.
Proof.
destruct 1.
exact (folds_2_iso_id _ _ _ f).
Defined.
Definition is_univalent_folds_pre_2_cat (C : folds_pre_2_cat) : UU :=
∏ (a b : C) (f g : a ⇒ b), isweq (@idtoiso2 _ _ _ f g).
Lemma isaprop_is_univalent_folds_2_precat (C : folds_pre_2_cat) :
isaprop (is_univalent_folds_pre_2_cat C).
Proof.
do 4 (apply impred; intro);
apply isapropisweq.
Qed.
Definition isotoid2 (C : folds_pre_2_cat) (H : is_univalent_folds_pre_2_cat C)
(a b : C) (f g : a ⇒ b) : folds_iso f g → f = g :=
invmap (weqpair _ (H a b f g)).
FOLDS precategories
We define them as special FOLDS pre-2-categories, namely such that- hom-types are sets
- axioms of precategory modulo identity (rather than E)
Definition is_folds_precategory (C : folds_pre_2_cat) : UU :=
(∏ a b : C, isaset (a ⇒ b))
× ((∏ {a b c : C} {f : a ⇒ b} {g : b ⇒ c} {h k : a ⇒ c},
T f g h → T f g k → h = k )
× (∏ {a b c d : C} (f : a ⇒ b) (g : b ⇒ c) (h : c ⇒ d)
(fg : a ⇒ c) (gh : b ⇒ d) (fg_h : a ⇒ d) (f_gh : a ⇒ d),
T f g fg → T g h gh →
T fg h fg_h → T f gh f_gh → f_gh = fg_h)).
Lemma isaprop_is_folds_precategory (C : folds_pre_2_cat) : isaprop (is_folds_precategory C).
Proof.
apply isofhlevelsn. intro H.
repeat (apply isofhleveldirprod).
- do 2 (apply impred; intro).
apply isapropisaset.
- do 9 (apply impred; intro).
apply (pr1 H).
- do 15 (apply impred; intro).
apply H.
Qed.
Section is_univalent_implies_folds_precat.
Variable C : folds_pre_2_cat.
Hypothesis H : is_univalent_folds_pre_2_cat C.
Lemma is_univalent_implies_is_folds_precat : is_folds_precategory C.
Proof.
apply dirprodpair.
- intros a b f g.
apply (isofhlevelweqb _ (weqpair _ (H a b f g))).
apply isaprop_folds_2_iso.
- apply dirprodpair.
+ intros. apply (isotoid2 _ H).
apply E_implies_folds_iso.
set (T_unique := pr1 (pr2 (pr2 (pr1 (pr2 (pr1 C)))))).
apply (T_unique _ _ _ f g); auto.
+ intros. apply (isotoid2 _ H).
apply E_implies_folds_iso.
set (T_assoc := pr2 (pr2 (pr2 (pr1 (pr2 (pr1 C)))))). simpl in T_assoc.
apply (T_assoc _ _ _ _ f g h fg gh fg_h f_gh); auto.
Qed.
End is_univalent_implies_folds_precat.
Section folds_precat_implies_univalent.
Variable C : folds_pre_2_cat.
Hypothesis H : is_folds_precategory C.
Hypothesis standardness : ∏ (a b : C) (f g : a ⇒ b), E f g → f = g.
Lemma folds_2_iso_implies_identity (a b : C) (f g : a ⇒ b) : folds_iso f g → f = g.
Proof.
intro Isofg.
apply standardness.
apply folds_iso_implies_E.
apply Isofg.
Qed.
Lemma is_folds_precat_implies_is_univalent : is_univalent_folds_pre_2_cat C.
Proof.
intros a b f g.
apply isweqimplimpl.
- apply folds_2_iso_implies_identity.
- apply (pr1 H).
- apply isaprop_folds_2_iso.
Qed.
End folds_precat_implies_univalent.