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71 lines
2.2 KiB
Lean4
71 lines
2.2 KiB
Lean4
-- Copyright (c) 2015 Jakob von Raumer. All rights reserved.
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-- Released under Apache 2.0 license as described in the file LICENSE.
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-- Authors: Jakob von Raumer
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-- Category of sets
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import .basic types.pi trunc
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open truncation sigma sigma.ops pi function eq morphism precategory
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open equiv
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namespace precategory
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universe variable l
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definition set_precategory : precategory.{l+1 l} (Σ (A : Type.{l}), is_hset A) :=
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begin
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fapply precategory.mk.{l+1 l},
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intros, apply (a.1 → a_1.1),
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intros, apply trunc_pi, intros, apply b.2,
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intros, intro x, exact (a_1 (a_2 x)),
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intros, exact (λ (x : a.1), x),
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intros, apply funext.path_pi, intro x, apply idp,
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intros, apply funext.path_pi, intro x, apply idp,
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intros, apply funext.path_pi, intro x, apply idp,
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end
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end precategory
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namespace category
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universe variable l
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local attribute precategory.set_precategory.{l+1 l} [instance]
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definition set_category_equiv_iso (a b : (Σ (A : Type.{l}), is_hset A))
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: (a ≅ b) = (a.1 ≃ b.1) :=
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/-begin
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apply ua, fapply equiv.mk,
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intro H,
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apply (isomorphic.rec_on H), intros (H1, H2),
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apply (is_iso.rec_on H2), intros (H3, H4, H5),
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fapply equiv.mk,
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apply (isomorphic.rec_on H), intros (H1, H2),
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exact H1,
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fapply is_equiv.adjointify, exact H3,
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exact sorry,
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exact sorry,
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end-/ sorry
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definition set_category : category.{l+1 l} (Σ (A : Type.{l}), is_hset A) :=
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/-begin
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assert (C : precategory.{l+1 l} (Σ (A : Type.{l}), is_hset A)),
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apply precategory.set_precategory,
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apply category.mk,
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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)),
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apply is_equiv.adjointify,
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intros,
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apply (isomorphic.rec_on a_1), intros (iso', is_iso'),
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apply (is_iso.rec_on is_iso'), intros (f', f'sect, f'retr),
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fapply sigma.path,
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apply ua, fapply equiv.mk, exact iso',
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fapply is_equiv.adjointify,
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exact f',
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intros, apply (f'retr ▹ _),
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intros, apply (f'sect ▹ _),
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apply (@is_hprop.elim),
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apply is_trunc_is_hprop,
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intros,
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end -/ sorry
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end category
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