lean2/hott/algebra/category/adjoint.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.precategory.adjoint
Authors: Floris van Doorn
-/
import algebra.category.constructions .constructions types.function arity
open category functor nat_trans eq is_trunc iso equiv prod trunc function
namespace category
variables {C D : Precategory} {F : C ⇒ D}
-- do we want to have a structure "is_adjoint" and define
-- structure is_left_adjoint (F : C ⇒ D) :=
-- (right_adjoint : D ⇒ C) -- G
-- (is_adjoint : adjoint F right_adjoint)
structure is_left_adjoint [class] (F : C ⇒ D) :=
(G : D ⇒ C)
(η : functor.id ⟹ G ∘f F)
(ε : F ∘f G ⟹ functor.id)
(H : Π(c : C), (ε (F c)) ∘ (F (η c)) = ID (F c))
(K : Π(d : D), (G (ε d)) ∘ (η (G d)) = ID (G d))
abbreviation right_adjoint := @is_left_adjoint.G
abbreviation unit := @is_left_adjoint.η
abbreviation counit := @is_left_adjoint.ε
-- structure is_left_adjoint [class] (F : C ⇒ D) :=
-- (right_adjoint : D ⇒ C) -- G
-- (unit : functor.id ⟹ right_adjoint ∘f F) -- η
-- (counit : F ∘f right_adjoint ⟹ functor.id) -- ε
-- (H : Π(c : C), (counit (F c)) ∘ (F (unit c)) = ID (F c))
-- (K : Π(d : D), (right_adjoint (counit d)) ∘ (unit (right_adjoint d)) = ID (right_adjoint d))
structure is_equivalence [class] (F : C ⇒ D) extends is_left_adjoint F :=
mk' ::
(is_iso_unit : is_iso η)
(is_iso_counit : is_iso ε)
structure equivalence (C D : Precategory) :=
(to_functor : C ⇒ D)
(struct : is_equivalence to_functor)
--TODO: review and change
--TODO: make some or all of these structures?
definition faithful (F : C ⇒ D) :=
Π⦃c c' : C⦄ (f f' : c ⟶ c'), F f = F f' → f = f'
definition full (F : C ⇒ D) := Π⦃c c' : C⦄ (g : F c ⟶ F c'), ∃(f : c ⟶ c'), F f = g
definition fully_faithful [reducible] (F : C ⇒ D) := Π⦃c c' : C⦄, is_equiv (@to_fun_hom _ _ F c c')
definition split_essentially_surjective (F : C ⇒ D) :=
Π⦃d : D⦄, Σ(c : C), F c ≅ d
definition essentially_surjective (F : C ⇒ D) :=
Π⦃d : D⦄, ∃(c : C), F c ≅ d
definition is_weak_equivalence (F : C ⇒ D) :=
fully_faithful F × essentially_surjective F
definition is_isomorphism (F : C ⇒ D) :=
fully_faithful F × is_equiv (to_fun_ob F)
structure isomorphism (C D : Precategory) :=
(to_functor : C ⇒ D)
(struct : is_isomorphism to_functor)
-- infix `⊣`:55 := adjoint
infix `⋍`:25 := equivalence -- \backsimeq or \equiv
infix `≌`:25 := isomorphism -- \backcong or \iso
definition is_hprop_is_left_adjoint {C : Category} {D : Precategory} (F : C ⇒ D)
: is_hprop (is_left_adjoint F) :=
begin
apply is_hprop.mk,
intro G G', cases G with G η ε H K, cases G' with G' η' ε' H' K',
fapply (apd011111 is_left_adjoint.mk),
{ fapply functor_eq,
{ intro d, apply eq_of_iso, fapply iso.MK,
{ exact (G' (ε d) ∘ η' (G d))},
{ exact (G (ε' d) ∘ η (G' d))},
{ apply sorry /-rewrite [assoc, -{((G (ε' d)) ∘ (η (G' d))) ∘ (G' (ε d))}(assoc)],-/
-- apply concat, apply (ap (λc, c ∘ η' _)), rewrite -assoc, apply idp
},
--/-rewrite [-nat_trans.assoc]-/apply sorry
---assoc (G (ε' d)) (η (G' d)) (G' (ε d))
{ apply sorry}},
{ apply sorry},
},
{ apply sorry},
{ apply sorry},
{ apply is_hprop.elim},
{ apply is_hprop.elim},
end
definition is_equivalence.mk (F : C ⇒ D) (G : D ⇒ C) (η : G ∘f F ≅ functor.id)
(ε : F ∘f G ≅ functor.id) : is_equivalence F :=
sorry
definition full_of_fully_faithful (H : fully_faithful F) : full F :=
sorry --λc c' g, trunc.elim _ _
definition faithful_of_fully_faithful (H : fully_faithful F) : faithful F :=
λc c' f f' p, is_injective_of_is_embedding p
definition fully_faithful_of_full_of_faithful (H : faithful F) (K : full F) : fully_faithful F :=
sorry
definition fully_faithful_equiv (F : C ⇒ D) : fully_faithful F ≃ (faithful F × full F) :=
sorry
definition is_equivalence_equiv (F : C ⇒ D)
: is_equivalence F ≃ (fully_faithful F × split_essentially_surjective F) :=
sorry
definition is_hprop_is_weak_equivalence (F : C ⇒ D) : is_hprop (is_weak_equivalence F) :=
sorry
definition is_hprop_is_equivalence {C D : Category} (F : C ⇒ D) : is_hprop (is_equivalence F) :=
sorry
definition is_equivalence_equiv_is_weak_equivalence {C D : Category} (F : C ⇒ D)
: is_equivalence F ≃ is_weak_equivalence F :=
sorry
definition is_hprop_is_isomorphism (F : C ⇒ D) : is_hprop (is_isomorphism F) :=
sorry
definition is_isomorphism_equiv1 (F : C ⇒ D) : is_equivalence F
≃ Σ(G : D ⇒ C) (η : functor.id = G ∘f F) (ε : F ∘f G = functor.id),
sorry ▹ ap (λ(H : C ⇒ C), F ∘f H) η = ap (λ(H : D ⇒ D), H ∘f F) ε⁻¹ :=
sorry
definition is_isomorphism_equiv2 (F : C ⇒ D) : is_equivalence F
≃ ∃(G : D ⇒ C), functor.id = G ∘f F × F ∘f G = functor.id :=
sorry
definition is_equivalence_of_isomorphism (H : is_isomorphism F) : is_equivalence F :=
sorry
definition is_isomorphism_of_is_equivalence {C D : Category} {F : C ⇒ D} (H : is_equivalence F)
: is_isomorphism F :=
sorry
definition isomorphism_of_eq {C D : Precategory} (p : C = D) : C ≌ D :=
sorry
definition is_equiv_isomorphism_of_eq (C D : Precategory) : is_equiv (@isomorphism_of_eq C D) :=
sorry
definition equivalence_of_eq {C D : Precategory} (p : C = D) : C ⋍ D :=
sorry
definition is_equiv_equivalence_of_eq (C D : Category) : is_equiv (@equivalence_of_eq C D) :=
sorry
end category