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