69 lines
2.3 KiB
Text
69 lines
2.3 KiB
Text
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-- Copyright (c) 2014 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: Floris van Doorn, Jakob von Raumer
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import .basic .morphism hott.types.prod
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open path precategory sigma sigma.ops equiv is_equiv function truncation
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open prod
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namespace morphism
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variables {ob : Type} [C : precategory ob] include C
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variables {a b c : ob} {g : b ⟶ c} {f : a ⟶ b} {h : b ⟶ a}
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-- "is_iso f" is equivalent to a certain sigma type
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definition sigma_char (f : hom a b) :
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(Σ (g : hom b a), (g ∘ f ≈ id) × (f ∘ g ≈ id)) ≃ is_iso f :=
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begin
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fapply (equiv.mk),
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intro S, apply is_iso.mk,
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exact (pr₁ S.2),
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exact (pr₂ S.2),
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fapply adjointify,
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intro H, apply (is_iso.rec_on H), intros (g, η, ε),
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exact (dpair g (pair η ε)),
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intro H, apply (is_iso.rec_on H), intros (g, η, ε), apply idp,
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intro S, apply (sigma.rec_on S), intros (g, ηε),
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apply (prod.rec_on ηε), intros (η, ε), apply idp,
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end
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-- The structure for isomorphism can be characterized up to equivalence
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-- by a sigma type.
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definition sigma_is_iso_equiv ⦃a b : ob⦄ : (Σ (f : hom a b), is_iso f) ≃ (a ≅ b) :=
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begin
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fapply (equiv.mk),
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intro S, apply isomorphic.mk, apply (S.2),
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fapply adjointify,
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intro p, apply (isomorphic.rec_on p), intros (f, H),
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exact (dpair f H),
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intro p, apply (isomorphic.rec_on p), intros (f, H), apply idp,
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intro S, apply (sigma.rec_on S), intros (f, H), apply idp,
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end
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-- The statement "f is an isomorphism" is a mere proposition
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definition is_hprop_of_is_iso : is_hset (is_iso f) :=
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begin
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apply trunc_equiv,
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apply (equiv.to_is_equiv (!sigma_char)),
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apply trunc_sigma,
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apply (!homH),
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intro g, apply trunc_prod,
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repeat (apply succ_is_trunc; apply trunc_succ; apply (!homH)),
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end
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-- The type of isomorphisms between two objects is a set
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definition is_hset_iso : is_hset (a ≅ b) :=
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begin
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apply trunc_equiv,
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apply (equiv.to_is_equiv (!sigma_is_iso_equiv)),
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apply trunc_sigma,
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apply homH,
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intro f, apply is_hprop_of_is_iso,
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end
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-- In a precategory, equal objects are isomorphic
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definition iso_of_path (p : a ≈ b) : isomorphic a b :=
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path.rec_on p (isomorphic.mk id)
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end morphism
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