lean2/hott/algebra/category/functor/attributes.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.
Authors: Floris van Doorn
Attributes of functors (full, faithful, split essentially surjective, ...)
Adjoint functors, isomorphisms and equivalences have their own file
-/
import ..constructions.functor function arity
open eq functor trunc prod is_equiv iso equiv function is_trunc
namespace category
variables {C D E : Precategory} {F : C ⇒ D} {G : D ⇒ C}
definition faithful [class] (F : C ⇒ D) := Π⦃c c' : C⦄ ⦃f f' : c ⟶ c'⦄, F f = F f' → f = f'
definition full [class] (F : C ⇒ D) := Π⦃c c' : C⦄, is_surjective (@(to_fun_hom F) c c')
definition fully_faithful [class] (F : C ⇒ D) := Π(c c' : C), is_equiv (@(to_fun_hom F) c c')
definition split_essentially_surjective [class] (F : C ⇒ D) := Π(d : D), Σ(c : C), F c ≅ d
definition essentially_surjective [class] (F : C ⇒ D) := Π(d : D), ∃(c : C), F c ≅ d
definition is_weak_equivalence [class] (F : C ⇒ D) :=
fully_faithful F × essentially_surjective F
definition is_equiv_of_fully_faithful [instance] [reducible] (F : C ⇒ D)
[H : fully_faithful F] (c c' : C) : is_equiv (@(to_fun_hom F) c c') :=
!H
definition hom_inv [reducible] (F : C ⇒ D) [H : fully_faithful F] (c c' : C) (f : F c ⟶ F c')
: c ⟶ c' :=
(to_fun_hom F)⁻¹ᶠ f
definition reflect_is_iso [constructor] (F : C ⇒ D) [H : fully_faithful F] {c c' : C}
(f : c ⟶ c') [H : is_iso (F f)] : is_iso f :=
begin
fconstructor,
{ exact (to_fun_hom F)⁻¹ᶠ (F f)⁻¹},
{ apply eq_of_fn_eq_fn' (to_fun_hom F),
rewrite [respect_comp,right_inv (to_fun_hom F),respect_id,left_inverse]},
{ apply eq_of_fn_eq_fn' (to_fun_hom F),
rewrite [respect_comp,right_inv (to_fun_hom F),respect_id,right_inverse]},
end
definition reflect_iso [constructor] (F : C ⇒ D) [H : fully_faithful F] {c c' : C}
(f : F c ≅ F c') : c ≅ c' :=
begin
fconstructor,
{ exact (to_fun_hom F)⁻¹ᶠ f},
{ assert H : is_iso (F ((to_fun_hom F)⁻¹ᶠ f)),
{ have H' : is_iso (to_hom f), from _, exact (right_inv (to_fun_hom F) (to_hom f))⁻¹ ▸ H'},
exact reflect_is_iso F _},
end
theorem reflect_inverse (F : C ⇒ D) [H : fully_faithful F] {c c' : C} (f : c ⟶ c')
[H : is_iso f] : (to_fun_hom F)⁻¹ᶠ (F f)⁻¹ = f⁻¹ :=
inverse_eq_inverse (idp : to_hom (@(iso.mk f) (reflect_is_iso F f)) = f)
definition hom_equiv_F_hom_F [constructor] (F : C ⇒ D)
[H : fully_faithful F] (c c' : C) : (c ⟶ c') ≃ (F c ⟶ F c') :=
equiv.mk _ !H
definition iso_of_F_iso_F (F : C ⇒ D)
[H : fully_faithful F] (c c' : C) (g : F c ≅ F c') : c ≅ c' :=
begin
induction g with g G, induction G with h p q, fapply iso.MK,
{ rexact (to_fun_hom F)⁻¹ᶠ g},
{ rexact (to_fun_hom F)⁻¹ᶠ h},
{ exact abstract begin
apply eq_of_fn_eq_fn' (to_fun_hom F),
rewrite [respect_comp, respect_id,
right_inv (to_fun_hom F), right_inv (to_fun_hom F), p],
end end},
{ exact abstract begin
apply eq_of_fn_eq_fn' (to_fun_hom F),
rewrite [respect_comp, respect_id,
right_inv (to_fun_hom F), right_inv (@(to_fun_hom F) c' c), q],
end end}
end
definition iso_equiv_F_iso_F [constructor] (F : C ⇒ D)
[H : fully_faithful F] (c c' : C) : (c ≅ c') ≃ (F c ≅ F c') :=
begin
fapply equiv.MK,
{ exact to_fun_iso F},
{ apply iso_of_F_iso_F},
{ exact abstract begin
intro f, induction f with f F', induction F' with g p q, apply iso_eq,
esimp [iso_of_F_iso_F], apply right_inv end end},
{ exact abstract begin
intro f, induction f with f F', induction F' with g p q, apply iso_eq,
esimp [iso_of_F_iso_F], apply right_inv end end},
end
definition full_of_fully_faithful [instance] (F : C ⇒ D) [H : fully_faithful F] : full F :=
λc c' g, tr (fiber.mk ((@(to_fun_hom F) c c')⁻¹ᶠ g) !right_inv)
definition faithful_of_fully_faithful [instance] (F : C ⇒ D) [H : fully_faithful F]
: faithful F :=
λc c' f f' p, is_injective_of_is_embedding p
definition is_embedding_of_faithful [instance] (F : C ⇒ D) [H : faithful F] (c c' : C)
: is_embedding (to_fun_hom F : c ⟶ c' → F c ⟶ F c') :=
begin
apply is_embedding_of_is_injective,
apply H
end
definition is_surjective_of_full [instance] (F : C ⇒ D) [H : full F] (c c' : C)
: is_surjective (to_fun_hom F : c ⟶ c' → F c ⟶ F c') :=
@H c c'
definition fully_faithful_of_full_of_faithful (H : faithful F) (K : full F)
: fully_faithful F :=
begin
intro c c',
apply is_equiv_of_is_surjective_of_is_embedding,
end
theorem is_hprop_fully_faithful [instance] (F : C ⇒ D) : is_hprop (fully_faithful F) :=
by unfold fully_faithful; exact _
theorem is_hprop_full [instance] (F : C ⇒ D) : is_hprop (full F) :=
by unfold full; exact _
theorem is_hprop_faithful [instance] (F : C ⇒ D) : is_hprop (faithful F) :=
by unfold faithful; exact _
theorem is_hprop_essentially_surjective [instance] (F : C ⇒ D)
: is_hprop (essentially_surjective F) :=
by unfold essentially_surjective; exact _
theorem is_hprop_is_weak_equivalence [instance] (F : C ⇒ D) : is_hprop (is_weak_equivalence F) :=
by unfold is_weak_equivalence; exact _
definition fully_faithful_equiv (F : C ⇒ D) : fully_faithful F ≃ (faithful F × full F) :=
equiv_of_is_hprop (λH, (faithful_of_fully_faithful F, full_of_fully_faithful F))
(λH, fully_faithful_of_full_of_faithful (pr1 H) (pr2 H))
/- alternative proof using direct calculation with equivalences
definition fully_faithful_equiv (F : C ⇒ D) : fully_faithful F ≃ (faithful F × full F) :=
calc
fully_faithful F
≃ (Π(c c' : C), is_embedding (to_fun_hom F) × is_surjective (to_fun_hom F))
: pi_equiv_pi_id (λc, pi_equiv_pi_id
(λc', !is_equiv_equiv_is_embedding_times_is_surjective))
... ≃ (Π(c : C), (Π(c' : C), is_embedding (to_fun_hom F)) ×
(Π(c' : C), is_surjective (to_fun_hom F)))
: pi_equiv_pi_id (λc, !equiv_prod_corec)
... ≃ (Π(c c' : C), is_embedding (to_fun_hom F)) × full F
: equiv_prod_corec
... ≃ faithful F × full F
: prod_equiv_prod_right (pi_equiv_pi_id (λc, pi_equiv_pi_id
(λc', !is_embedding_equiv_is_injective)))
-/
end category