96 lines
2.9 KiB
Text
96 lines
2.9 KiB
Text
/-
|
|
Copyright (c) 2015 Floris van Doorn. All rights reserved.
|
|
Released under Apache 2.0 license as described in the file LICENSE.
|
|
Author: Floris van Doorn
|
|
|
|
Ported from Coq HoTT
|
|
Theorems about embeddings and surjections
|
|
-/
|
|
|
|
import hit.trunc .pi .fiber .equiv
|
|
|
|
open equiv sigma sigma.ops eq trunc is_trunc pi is_equiv fiber prod
|
|
|
|
variables {A B : Type} {f : A → B} {b : B}
|
|
|
|
structure is_embedding [class] (f : A → B) :=
|
|
(elim : Π(a a' : A), is_equiv (ap f : a = a' → f a = f a'))
|
|
|
|
structure is_surjective [class] (f : A → B) :=
|
|
(elim : Π(b : B), ∥ fiber f b ∥)
|
|
|
|
structure is_split_surjective [class] (f : A → B) :=
|
|
(elim : Π(b : B), fiber f b)
|
|
|
|
structure is_retraction [class] (f : A → B) :=
|
|
(sect : B → A)
|
|
(right_inverse : Π(b : B), f (sect b) = b)
|
|
|
|
namespace function
|
|
|
|
attribute is_embedding.elim [instance]
|
|
|
|
definition is_surjective_rec_on {P : Type} (H : is_surjective f) (b : B) [Pt : is_hprop P]
|
|
(IH : fiber f b → P) : P :=
|
|
trunc.rec_on (is_surjective.elim f b) IH
|
|
|
|
definition is_surjective_of_is_split_surjective [instance] [H : is_split_surjective f]
|
|
: is_surjective f :=
|
|
is_surjective.mk (λb, tr (is_split_surjective.elim f b))
|
|
|
|
definition is_injective_of_is_embedding [reducible] [H : is_embedding f] {a a' : A}
|
|
: f a = f a' → a = a' :=
|
|
(ap f)⁻¹
|
|
|
|
definition is_embedding_of_is_injective [HA : is_hset A] [HB : is_hset B]
|
|
(H : Π(a a' : A), f a = f a' → a = a') : is_embedding f :=
|
|
begin
|
|
fapply is_embedding.mk,
|
|
intro a a',
|
|
fapply adjointify,
|
|
{exact (H a a')},
|
|
{intro p, apply is_hset.elim},
|
|
{intro p, apply is_hset.elim}
|
|
end
|
|
|
|
definition is_hprop_is_embedding [instance] (f : A → B) : is_hprop (is_embedding f) :=
|
|
begin
|
|
have H : (Π(a a' : A), is_equiv (@ap A B f a a')) ≃ is_embedding f,
|
|
begin
|
|
fapply equiv.MK,
|
|
{exact is_embedding.mk},
|
|
{intro h, cases h, exact elim},
|
|
{intro h, cases h, apply idp},
|
|
{intro p, apply idp},
|
|
end,
|
|
apply is_trunc_equiv_closed,
|
|
exact H,
|
|
end
|
|
|
|
definition is_hprop_is_surjective [instance] (f : A → B) : is_hprop (is_surjective f) :=
|
|
begin
|
|
have H : (Π(b : B), merely (fiber f b)) ≃ is_surjective f,
|
|
begin
|
|
fapply equiv.MK,
|
|
{exact is_surjective.mk},
|
|
{intro h, cases h, exact elim},
|
|
{intro h, cases h, apply idp},
|
|
{intro p, apply idp},
|
|
end,
|
|
apply is_trunc_equiv_closed,
|
|
exact H,
|
|
end
|
|
|
|
definition is_embedding_of_is_equiv [instance] (f : A → B) [H : is_equiv f] : is_embedding f :=
|
|
is_embedding.mk _
|
|
|
|
definition is_equiv_of_is_surjective_of_is_embedding (f : A → B)
|
|
[H : is_embedding f] [H' : is_surjective f] : is_equiv f :=
|
|
@is_equiv_of_is_contr_fun _ _ _
|
|
(λb, is_surjective_rec_on H' b
|
|
(λa, is_contr.mk a
|
|
(λa',
|
|
fiber_eq ((ap f)⁻¹ ((point_eq a) ⬝ (point_eq a')⁻¹))
|
|
(by rewrite (right_inv (ap f)); rewrite inv_con_cancel_right))))
|
|
|
|
end function
|