lean2/hott/algebra/category/constructions/functor.hlean

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/-
Copyright (c) 2015 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Module: algebra.category.constructions.functor
Authors: Floris van Doorn, Jakob von Raumer
Functor precategory and category
-/
import ..nat_trans ..category
open eq functor is_trunc nat_trans iso is_equiv
namespace category
definition precategory_functor [instance] [reducible] (D C : Precategory)
: precategory (functor C D) :=
precategory.mk (λa b, nat_trans a b)
(λ a b c g f, nat_trans.compose g f)
(λ a, nat_trans.id)
(λ a b c d h g f, !nat_trans.assoc)
(λ a b f, !nat_trans.id_left)
(λ a b f, !nat_trans.id_right)
definition Precategory_functor [reducible] (D C : Precategory) : Precategory :=
precategory.Mk (precategory_functor D C)
infixr `^c`:35 := Precategory_functor
section
/- we prove that if a natural transformation is pointwise an iso, then it is an iso -/
variables {C D : Precategory} {F G : C ⇒ D} (η : F ⟹ G) [iso : Π(a : C), is_iso (η a)]
include iso
definition nat_trans_inverse : G ⟹ F :=
nat_trans.mk
(λc, (η c)⁻¹)
(λc d f,
begin
apply comp_inverse_eq_of_eq_comp,
transitivity (natural_map η d)⁻¹ ∘ to_fun_hom G f ∘ natural_map η c,
{apply eq_inverse_comp_of_comp_eq, symmetry, apply naturality},
{apply assoc}
end)
definition nat_trans_left_inverse : nat_trans_inverse η ∘n η = nat_trans.id :=
begin
fapply (apd011 nat_trans.mk),
apply eq_of_homotopy, intro c, apply left_inverse,
apply eq_of_homotopy, intros, apply eq_of_homotopy, intros, apply eq_of_homotopy, intros,
apply is_hset.elim
end
definition nat_trans_right_inverse : η ∘n nat_trans_inverse η = nat_trans.id :=
begin
fapply (apd011 nat_trans.mk),
apply eq_of_homotopy, intro c, apply right_inverse,
apply eq_of_homotopy, intros, apply eq_of_homotopy, intros, apply eq_of_homotopy, intros,
apply is_hset.elim
end
definition is_iso_nat_trans : is_iso η :=
is_iso.mk (nat_trans_left_inverse η) (nat_trans_right_inverse η)
end
section
/- and conversely, if a natural transformation is an iso, it is componentwise an iso -/
variables {C D : Precategory} {F G : D ^c C} (η : hom F G) [isoη : is_iso η] (c : C)
include isoη
definition componentwise_is_iso : is_iso (η c) :=
@is_iso.mk _ _ _ _ _ (natural_map η⁻¹ c) (ap010 natural_map ( left_inverse η) c)
(ap010 natural_map (right_inverse η) c)
local attribute componentwise_is_iso [instance]
definition natural_map_inverse : natural_map η⁻¹ c = (η c)⁻¹ := idp
definition naturality_iso {c c' : C} (f : c ⟶ c') : G f = η c' ∘ F f ∘ (η c)⁻¹ :=
calc
G f = (G f ∘ η c) ∘ (η c)⁻¹ : by rewrite comp_inverse_cancel_right
... = (η c' ∘ F f) ∘ (η c)⁻¹ : by rewrite naturality
... = η c' ∘ F f ∘ (η c)⁻¹ : by rewrite assoc
definition naturality_iso' {c c' : C} (f : c ⟶ c') : (η c')⁻¹ ∘ G f ∘ η c = F f :=
calc
(η c')⁻¹ ∘ G f ∘ η c = (η c')⁻¹ ∘ η c' ∘ F f : by rewrite naturality
... = F f : by rewrite inverse_comp_cancel_left
omit isoη
definition componentwise_iso (η : F ≅ G) (c : C) : F c ≅ G c :=
@iso.mk _ _ _ _ (natural_map (to_hom η) c)
(@componentwise_is_iso _ _ _ _ (to_hom η) (struct η) c)
definition componentwise_iso_id (c : C) : componentwise_iso (iso.refl F) c = iso.refl (F c) :=
iso_eq (idpath (ID (F c)))
definition componentwise_iso_iso_of_eq (p : F = G) (c : C)
: componentwise_iso (iso_of_eq p) c = iso_of_eq (ap010 to_fun_ob p c) :=
eq.rec_on p !componentwise_iso_id
definition natural_map_hom_of_eq (p : F = G) (c : C)
: natural_map (hom_of_eq p) c = hom_of_eq (ap010 to_fun_ob p c) :=
eq.rec_on p idp
definition natural_map_inv_of_eq (p : F = G) (c : C)
: natural_map (inv_of_eq p) c = hom_of_eq (ap010 to_fun_ob p c)⁻¹ :=
eq.rec_on p idp
end
namespace functor
variables {C : Precategory} {D : Category} {F G : D ^c C}
definition eq_of_iso_ob (η : F ≅ G) (c : C) : F c = G c :=
by apply eq_of_iso; apply componentwise_iso; exact η
local attribute functor.to_fun_hom [quasireducible]
definition eq_of_iso (η : F ≅ G) : F = G :=
begin
fapply functor_eq,
{exact (eq_of_iso_ob η)},
{intro c c' f,
esimp [eq_of_iso_ob, inv_of_eq, hom_of_eq, eq_of_iso],
rewrite [*right_inv iso_of_eq],
esimp [function.id],
symmetry, apply naturality_iso
}
end
definition iso_of_eq_eq_of_iso (η : F ≅ G) : iso_of_eq (eq_of_iso η) = η :=
begin
apply iso_eq,
apply nat_trans_eq,
intro c,
rewrite natural_map_hom_of_eq, esimp [eq_of_iso],
rewrite ap010_functor_eq, esimp [hom_of_eq,eq_of_iso_ob],
rewrite (right_inv iso_of_eq),
end
definition eq_of_iso_iso_of_eq (p : F = G) : eq_of_iso (iso_of_eq p) = p :=
begin
apply functor_eq2,
intro c,
esimp [eq_of_iso],
rewrite ap010_functor_eq,
esimp [eq_of_iso_ob],
rewrite componentwise_iso_iso_of_eq,
rewrite (left_inv iso_of_eq)
end
definition is_univalent (D : Category) (C : Precategory) : is_univalent (D ^c C) :=
λF G, adjointify _ eq_of_iso
iso_of_eq_eq_of_iso
eq_of_iso_iso_of_eq
end functor
definition category_functor [instance] (D : Category) (C : Precategory)
: category (D ^c C) :=
category.mk (D ^c C) (functor.is_univalent D C)
definition Category_functor (D : Category) (C : Precategory) : Category :=
category.Mk (D ^c C) !category_functor
--this definition is only useful if the exponent is a category,
-- and the elaborator has trouble with inserting the coercion
definition Category_functor' (D C : Category) : Category :=
Category_functor D C
namespace ops
infixr `^c2`:35 := Category_functor
end ops
end category