68 lines
2.3 KiB
Text
68 lines
2.3 KiB
Text
-- 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 types.sigma
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open eq precategory sigma sigma.ops equiv is_equiv function is_trunc
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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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protected 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, cases H with (g, η, ε),
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exact (sigma.mk g (pair η ε))},
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{intro H, cases H, apply idp},
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{intro S, cases S with (g, ηε), cases ηε, 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, cases p with (f, H), exact (sigma.mk f H)},
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{intro p, cases p, apply idp},
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{intro S, cases S, apply idp}},
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end
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set_option apply.class_instance false -- disable class instance resolution in the apply tactic
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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 is_trunc_is_equiv_closed,
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apply (equiv.to_is_equiv (!sigma_char)),
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apply is_trunc_sigma,
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apply (!homH),
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{intro g, apply is_trunc_prod,
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repeat (apply is_trunc_eq; apply is_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 is_trunc_is_equiv_closed,
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apply (equiv.to_is_equiv (!sigma_is_iso_equiv)),
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apply is_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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eq.rec_on p (isomorphic.mk id)
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end morphism
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