47 lines
1.4 KiB
Text
47 lines
1.4 KiB
Text
/-
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Copyright (c) 2015 Ulrik Buchholtz. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Ulrik Buchholtz
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-/
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import types.trunc types.arrow_2 types.fiber
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open eq is_trunc is_equiv nat equiv trunc function
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namespace homotopy
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definition is_conn (n : trunc_index) (A : Type) : Type :=
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is_contr (trunc n A)
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definition is_conn_map (n : trunc_index) {A B : Type} (f : A → B) : Type :=
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Πb : B, is_conn n (fiber f b)
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definition is_conn_of_map_to_unit (n : trunc_index) (A : Type)
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: is_conn_map n (λx : A, unit.star) → is_conn n A :=
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begin
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intro H, unfold is_conn_map at H,
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rewrite [-(ua (fiber.fiber_star_equiv A))],
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exact (H unit.star)
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end
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-- Lemma 7.5.2
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definition minus_one_conn_of_surjective {A B : Type} (f : A → B)
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: is_surjective f → is_conn_map -1 f :=
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begin
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intro H, intro b,
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exact @is_contr_of_inhabited_hprop (∥fiber f b∥) (is_trunc_trunc -1 (fiber f b)) (H b),
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end
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definition is_surjection_of_minus_one_conn {A B : Type} (f : A → B)
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: is_conn_map -1 f → is_surjective f :=
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begin
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intro H, intro b,
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exact @center (∥fiber f b∥) (H b),
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end
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definition merely_of_minus_one_conn {A : Type} : is_conn -1 A → ∥ A ∥ :=
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λH, @center (∥A∥) H
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definition minus_one_conn_of_merely {A : Type} : ∥A∥ → is_conn -1 A :=
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@is_contr_of_inhabited_hprop (∥A∥) (is_trunc_trunc -1 A)
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end homotopy
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