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feat(library/hott): finish porting of natural_transformation.lean

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Jakob von Raumer 2014-12-01 23:12:57 -05:00 committed by Leonardo de Moura
parent ae618c20fb
commit 67f71ee376
2 changed files with 30 additions and 23 deletions
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49
library/hott/algebra/category/natural_transformation.lean
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@ -35,34 +35,41 @@ namespace natural_transformation
infixr `∘n`:60 := compose
/-definition foo (C : Precategory) (a b : C) (f g : hom a b) (p q : f ≈ g) : p ≈ q :=
@truncation.is_hset.elim _ !homH f g p q
definition nt_is_hset {F G : functor C D} : is_hset (F ⟹ G) := sorry
definition eta_nat_path {η₁ η₂ : F ⟹ G} (H1 : natural_map η₁ ≈ natural_map η₂)
(H2 : (H1 ▹ naturality η₁) ≈ naturality η₂) : η₁ ≈ η₂ :=
rec_on η₁ (λ eta1 nat1, rec_on η₂ (λ eta2 nat2, sorry))-/
protected definition assoc (η₃ : H ⟹ I) (η₂ : G ⟹ H) (η₁ : F ⟹ G) [fext : funext] :
η₃ ∘n (η₂ ∘n η₁) ≈ (η₃ ∘n η₂) ∘n η₁ :=
have aux4 [visible] : is_hprop (Π (a b : C) (f : hom a b), I f ∘ (η₃ ∘n η₂) a ∘ η₁ a ≈ ((η₃ ∘n η₂) b ∘ η₁ b) ∘ F f),
begin
apply trunc_pi, intro a,
apply trunc_pi, intro b,
apply trunc_pi, intro f,
apply (@succ_is_trunc _ -1 _ (I f ∘ (η₃ ∘n η₂) a ∘ η₁ a) (((η₃ ∘n η₂) b ∘ η₁ b) ∘ F f)),
end,
path.dcongr_arg2 mk (funext.path_forall _ _ (λ x, !assoc)) !is_hprop.elim
have aux [visible] : is_hprop (Π (a b : C) (f : hom a b), I f ∘ (η₃ ∘n η₂) a ∘ η₁ a ≈ ((η₃ ∘n η₂) b ∘ η₁ b) ∘ F f),
begin
repeat (apply trunc_pi; intros),
apply (succ_is_trunc -1 (I a_2 ∘ (η₃ ∘n η₂) a ∘ η₁ a)),
end,
dcongr_arg2 mk (funext.path_forall _ _ (λ x, !assoc)) !is_hprop.elim
exit
protected definition id {C D : Precategory} {F : functor C D} : natural_transformation F F :=
mk (λa, id) (λa b f, !id_right ⬝ (!id_left⁻¹))
protected definition ID {C D : Precategory} (F : functor C D) : natural_transformation F F := id
protected theorem id_left (η : F ⟹ G) [fext : funext] : natural_transformation.compose id η ≈ η :=
rec_on η (λf H, path.dcongr_arg2 mk (funext.path_forall _ _ (λ x, !id_left)) sorry)
protected theorem id_left (η : F ⟹ G) [fext : funext.{l_1 l_4}] :
id ∘n η ≈ η :=
begin
apply (rec_on η), intros (f, H),
fapply (path.dcongr_arg2 mk),
apply (funext.path_forall _ f (λa, !id_left)),
assert (H1 : is_hprop (Π {a b : C} (g : hom a b), G g ∘ f a ≈ f b ∘ F g)),
repeat (apply trunc_pi; intros),
apply (succ_is_trunc -1 (G a_2 ∘ f a) (f a_1 ∘ F a_2)),
apply (!is_hprop.elim),
end
protected theorem id_right (η : F ⟹ G) [fext : funext] : natural_transformation.compose η id ≈ η :=
rec (λf H, path.dcongr_arg2 mk (funext.path_forall _ _ (λ x, !id_right)) sorry) η
protected theorem id_right (η : F ⟹ G) [fext : funext.{l_1 l_4}] :
η ∘n id ≈ η :=
begin
apply (rec_on η), intros (f, H),
fapply (path.dcongr_arg2 mk),
apply (funext.path_forall _ f (λa, !id_right)),
assert (H1 : is_hprop (Π {a b : C} (g : hom a b), G g ∘ f a ≈ f b ∘ F g)),
repeat (apply trunc_pi; intros),
apply (succ_is_trunc -1 (G a_2 ∘ f a) (f a_1 ∘ F a_2)),
apply (!is_hprop.elim),
end
end natural_transformation

4
library/hott/trunc.lean
View file

@ -79,12 +79,12 @@ namespace truncation
variables {A B : Type}
-- maybe rename to is_trunc_succ.mk
definition is_trunc_succ (A : Type) {n : trunc_index} [H : ∀x y : A, is_trunc n (x ≈ y)]
definition is_trunc_succ (A : Type) (n : trunc_index) [H : ∀x y : A, is_trunc n (x ≈ y)]
: is_trunc n.+1 A :=
is_trunc.mk (λ x y, !is_trunc.to_internal)
-- maybe rename to is_trunc_succ.elim
definition succ_is_trunc {n : trunc_index} [H : is_trunc (n.+1) A] (x y : A) : is_trunc n (x ≈ y) :=
definition succ_is_trunc (n : trunc_index) [H : is_trunc (n.+1) A] (x y : A) : is_trunc n (x ≈ y) :=
is_trunc.mk (!is_trunc.to_internal x y)
/- contractibility -/