68 lines
2.1 KiB
Text
68 lines
2.1 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 [reducible] (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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section
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open arrow
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variables {f g : arrow}
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-- Lemma 7.5.4
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definition retract_of_conn_is_conn [instance] (r : arrow_hom f g) [H : is_retraction r]
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(n : trunc_index) [K : is_conn_map n f] : is_conn_map n g :=
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begin
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intro b, unfold is_conn,
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apply is_contr_retract (trunc_functor n (retraction_on_fiber r b)),
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exact K (on_cod (arrow.is_retraction.sect r) b)
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end
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end
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-- Corollary 7.5.5
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definition is_conn_homotopy (n : trunc_index) {A B : Type} {f g : A → B}
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(p : f ~ g) (H : is_conn_map n f) : is_conn_map n g :=
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@retract_of_conn_is_conn _ _ (arrow.arrow_hom_of_homotopy p) (arrow.is_retraction_arrow_hom_of_homotopy p) n H
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end homotopy
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