106 lines
4.2 KiB
Text
106 lines
4.2 KiB
Text
/-
|
|
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, Jakob von Raumer
|
|
|
|
The category of sets is complete and cocomplete
|
|
-/
|
|
|
|
import .colimits ..constructions.set hit.set_quotient
|
|
|
|
open eq functor is_trunc sigma pi sigma.ops trunc set_quotient
|
|
|
|
namespace category
|
|
local attribute Category.to.precategory [unfold 1]
|
|
local attribute category.to_precategory [unfold 2]
|
|
|
|
definition is_complete_set_cone.{u v w} [constructor]
|
|
(I : Precategory.{v w}) (F : I ⇒ set.{max u v w}) : cone_obj F :=
|
|
begin
|
|
fapply cone_obj.mk,
|
|
{ fapply trunctype.mk,
|
|
{ exact Σ(s : Π(i : I), trunctype.carrier (F i)),
|
|
Π{i j : I} (f : i ⟶ j), F f (s i) = (s j)},
|
|
{ with_options [elaborator.ignore_instances true] -- TODO: fix
|
|
( refine is_trunc_sigma _ _;
|
|
( apply is_trunc_pi);
|
|
( intro s;
|
|
refine is_trunc_pi _ _; intro i;
|
|
refine is_trunc_pi _ _; intro j;
|
|
refine is_trunc_pi _ _; intro f;
|
|
apply is_trunc_eq))}},
|
|
{ fapply nat_trans.mk,
|
|
{ intro i x, esimp at x, exact x.1 i},
|
|
{ intro i j f, esimp, apply eq_of_homotopy, intro x, esimp at x, induction x with s p,
|
|
esimp, apply p}}
|
|
end
|
|
|
|
definition is_complete_set.{u v w} [instance] : is_complete.{(max u v w)+1 (max u v w) v w} set :=
|
|
begin
|
|
intro I F, fapply has_terminal_object.mk,
|
|
{ exact is_complete_set_cone.{u v w} I F},
|
|
{ intro c, esimp at *, induction c with X η, induction η with η p, esimp at *,
|
|
fapply is_contr.mk,
|
|
{ fapply cone_hom.mk,
|
|
{ intro x, esimp at *, fapply sigma.mk,
|
|
{ intro i, exact η i x},
|
|
{ intro i j f, exact ap10 (p f) x}},
|
|
{ intro i, reflexivity}},
|
|
{ esimp, intro h, induction h with f q, apply cone_hom_eq, esimp at *,
|
|
apply eq_of_homotopy, intro x, fapply sigma_eq: esimp,
|
|
{ apply eq_of_homotopy, intro i, exact (ap10 (q i) x)⁻¹},
|
|
{ with_options [elaborator.ignore_instances true] -- TODO: fix
|
|
( refine is_hprop.elimo _ _ _;
|
|
refine is_trunc_pi _ _; intro i;
|
|
refine is_trunc_pi _ _; intro j;
|
|
refine is_trunc_pi _ _; intro f;
|
|
apply is_trunc_eq)}}}
|
|
end
|
|
|
|
definition is_cocomplete_set_cone_rel.{u v w} [unfold 3 4]
|
|
(I : Precategory.{v w}) (F : I ⇒ set.{max u v w}ᵒᵖ) : (Σ(i : I), trunctype.carrier (F i)) →
|
|
(Σ(i : I), trunctype.carrier (F i)) → hprop.{max u v w} :=
|
|
begin
|
|
intro v w, induction v with i x, induction w with j y,
|
|
fapply trunctype.mk,
|
|
{ exact ∃(f : i ⟶ j), to_fun_hom F f y = x},
|
|
{ exact _}
|
|
end
|
|
|
|
|
|
definition is_cocomplete_set_cone.{u v w} [constructor]
|
|
(I : Precategory.{v w}) (F : I ⇒ set.{max u v w}ᵒᵖ) : cone_obj F :=
|
|
begin
|
|
fapply cone_obj.mk,
|
|
{ fapply trunctype.mk,
|
|
{ apply set_quotient (is_cocomplete_set_cone_rel.{u v w} I F)},
|
|
{ apply is_hset_set_quotient}},
|
|
{ fapply nat_trans.mk,
|
|
{ intro i x, esimp, apply class_of, exact ⟨i, x⟩},
|
|
{ intro i j f, esimp, apply eq_of_homotopy, intro y, apply eq_of_rel, esimp,
|
|
exact exists.intro f idp}}
|
|
end
|
|
|
|
-- TODO: change this after induction tactic for trunc/set_quotient is implemented
|
|
definition is_cocomplete_set.{u v w} [instance]
|
|
: is_cocomplete.{(max u v w)+1 (max u v w) v w} set :=
|
|
begin
|
|
intro I F, fapply has_terminal_object.mk,
|
|
{ exact is_cocomplete_set_cone.{u v w} I F},
|
|
{ intro c, esimp at *, induction c with X η, induction η with η p, esimp at *,
|
|
fapply is_contr.mk,
|
|
{ fapply cone_hom.mk,
|
|
{ refine set_quotient.elim _ _,
|
|
{ intro v, induction v with i x, exact η i x},
|
|
{ intro v w r, induction v with i x, induction w with j y, esimp at *,
|
|
refine trunc.elim_on r _, clear r,
|
|
intro u, induction u with f q,
|
|
exact ap (η i) q⁻¹ ⬝ ap10 (p f) y}},
|
|
{ intro i, reflexivity}},
|
|
{ esimp, intro h, induction h with f q, apply cone_hom_eq, esimp at *,
|
|
apply eq_of_homotopy, refine set_quotient.rec _ _,
|
|
{ intro v, induction v with i x, esimp, exact (ap10 (q i) x)⁻¹},
|
|
{ intro v w r, apply is_hprop.elimo}}},
|
|
end
|
|
|
|
end category
|