61 lines
1.8 KiB
Text
61 lines
1.8 KiB
Text
-- 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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-- Truncation properties of truncatedness
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import types.pi
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open truncation sigma sigma.ops pi function eq equiv
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namespace truncation
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definition is_contr.sigma_char (A : Type) :
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(Σ (center : A), Π (a : A), center = a) ≃ (is_contr A) :=
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begin
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fapply equiv.mk,
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intro S, apply is_contr.mk, exact S.2,
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fapply is_equiv.adjointify,
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intro H, apply sigma.mk, exact (@contr A H),
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intro H, apply (is_trunc.rec_on H), intro Hint,
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apply (contr_internal.rec_on Hint), intros (H1, H2),
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apply idp,
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intro S, apply (sigma.rec_on S), intros (H1, H2),
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apply idp,
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end
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set_option pp.implicit true
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definition is_trunc.pi_char (n : trunc_index) (A : Type) :
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(Π (x y : A), is_trunc n (x = y)) ≃ (is_trunc (n .+1) A) :=
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begin
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fapply equiv.mk,
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intro H, apply is_trunc_succ,
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fapply is_equiv.adjointify,
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intros (H, x, y), apply succ_is_trunc,
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intro H, apply (is_trunc.rec_on H), intro Hint, apply idp,
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intro P,
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exact sorry,
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end
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definition is_trunc_is_hprop {n : trunc_index} :
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Π (A : Type), is_hprop (is_trunc n A) :=
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begin
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apply (trunc_index.rec_on n),
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intro A,
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apply trunc_equiv, apply equiv.to_is_equiv,
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apply is_contr.sigma_char,
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apply (@is_hprop.mk), intros,
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fapply sigma.path, apply x.2,
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apply (@is_hprop.elim),
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apply trunc_pi, intro a,
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apply is_hprop.mk, intros (w, z),
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assert (H : is_hset A),
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apply trunc_succ, apply trunc_succ,
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apply is_contr.mk, exact y.2,
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fapply (@is_hset.elim A _ _ _ w z),
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intros (n', IH, A),
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apply trunc_equiv,
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apply equiv.to_is_equiv,
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apply is_trunc.pi_char,
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end
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end truncation
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