lean2/hott/algebra/category/set.hlean

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-- Copyright (c) 2015 Jakob von Raumer. All rights reserved.
-- Released under Apache 2.0 license as described in the file LICENSE.
-- Authors: Jakob von Raumer
-- Category of sets
import .basic types.pi trunc
open truncation sigma sigma.ops pi function eq morphism precategory
open equiv
namespace precategory
universe variable l
definition set_precategory : precategory.{l+1 l} (Σ (A : Type.{l}), is_hset A) :=
begin
fapply precategory.mk.{l+1 l},
intros, apply (a.1 → a_1.1),
intros, apply trunc_pi, intros, apply b.2,
intros, intro x, exact (a_1 (a_2 x)),
intros, exact (λ (x : a.1), x),
intros, apply funext.path_pi, intro x, apply idp,
intros, apply funext.path_pi, intro x, apply idp,
intros, apply funext.path_pi, intro x, apply idp,
end
end precategory
namespace category
universe variable l
attribute precategory.set_precategory.{l+1 l} [instance]
definition set_category_equiv_iso (a b : (Σ (A : Type.{l}), is_hset A))
: (a ≅ b) = (a.1 ≃ b.1) :=
/-begin
apply ua, fapply equiv.mk,
intro H,
apply (isomorphic.rec_on H), intros (H1, H2),
apply (is_iso.rec_on H2), intros (H3, H4, H5),
fapply equiv.mk,
apply (isomorphic.rec_on H), intros (H1, H2),
exact H1,
fapply is_equiv.adjointify, exact H3,
exact sorry,
exact sorry,
end-/ sorry
definition set_category : category.{l+1 l} (Σ (A : Type.{l}), is_hset A) :=
/-begin
assert (C : precategory.{l+1 l} (Σ (A : Type.{l}), is_hset A)),
apply precategory.set_precategory,
apply category.mk,
assert (p : (λ A B p, (set_category_equiv_iso A B) ▹ iso_of_path p) = (λ A B p, @equiv_path A.1 B.1 p)),
apply is_equiv.adjointify,
intros,
apply (isomorphic.rec_on a_1), intros (iso', is_iso'),
apply (is_iso.rec_on is_iso'), intros (f', f'sect, f'retr),
fapply sigma.path,
apply ua, fapply equiv.mk, exact iso',
fapply is_equiv.adjointify,
exact f',
intros, apply (f'retr ▹ _),
intros, apply (f'sect ▹ _),
apply (@is_hprop.elim),
apply is_trunc_is_hprop,
intros,
end -/ sorry
end category