81 lines
3.1 KiB
Text
81 lines
3.1 KiB
Text
-- Copyright (c) 2014 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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-- Author: Jakob von Raumer
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-- Ported from Coq HoTT
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open eq function funext
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namespace is_equiv
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context
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--Precomposition of arbitrary functions with f
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definition precomp {A B : Type} (f : A → B) (C : Type) (h : B → C) : A → C := h ∘ f
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--Postcomposition of arbitrary functions with f
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definition postcomp {A B : Type} (f : A → B) (C : Type) (l : C → A) : C → B := f ∘ l
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--Precomposing with an equivalence is an equivalence
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definition precomp_closed [instance] {A B : Type} (f : A → B) [F : funext] [Hf : is_equiv f] (C : Type)
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: is_equiv (precomp f C) :=
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adjointify (precomp f C) (λh, h ∘ f⁻¹)
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(λh, path_pi (λx, ap h (sect f x)))
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(λg, path_pi (λy, ap g (retr f y)))
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--Postcomposing with an equivalence is an equivalence
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definition postcomp_closed [instance] {A B : Type} (f : A → B) [F : funext] [Hf : is_equiv f] (C : Type)
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: is_equiv (postcomp f C) :=
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adjointify (postcomp f C) (λl, f⁻¹ ∘ l)
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(λh, path_pi (λx, retr f (h x)))
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(λg, path_pi (λy, sect f (g y)))
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--Conversely, if pre- or post-composing with a function is always an equivalence,
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--then that function is also an equivalence. It's convenient to know
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--that we only need to assume the equivalence when the other type is
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--the domain or the codomain.
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protected definition isequiv_precompose_eq {A B : Type} (f : A → B) (C D : Type)
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(Ceq : is_equiv (precomp f C)) (Deq : is_equiv (precomp f D)) (k : C → D) (h : A → C) :
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k ∘ (inv (precomp f C)) h = (inv (precomp f D)) (k ∘ h) :=
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let invD := inv (precomp f D) in
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let invC := inv (precomp f C) in
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have eq1 : invD (k ∘ h) = k ∘ (invC h),
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from calc invD (k ∘ h) = invD (k ∘ (precomp f C (invC h))) : retr (precomp f C) h
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... = k ∘ (invC h) : !sect,
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eq1⁻¹
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definition from_isequiv_precomp {A B : Type} (f : A → B) (Aeq : is_equiv (precomp f A))
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(Beq : is_equiv (precomp f B)) : (is_equiv f) :=
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let invA := inv (precomp f A) in
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let invB := inv (precomp f B) in
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let sect' : f ∘ (invA id) ∼ id := (λx,
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calc f (invA id x) = (f ∘ invA id) x : idp
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... = invB (f ∘ id) x : apD10 (!isequiv_precompose_eq)
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... = invB (precomp f B id) x : idp
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... = x : apD10 (sect (precomp f B) id))
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in
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let retr' : (invA id) ∘ f ∼ id := (λx,
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calc invA id (f x) = precomp f A (invA id) x : idp
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... = x : apD10 (retr (precomp f A) id)) in
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adjointify f (invA id) sect' retr'
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end
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end is_equiv
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--Bundled versions of the previous theorems
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namespace equiv
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definition precomp_closed [F : funext] {A B C : Type} {eqf : A ≃ B}
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: (B → C) ≃ (A → C) :=
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let f := to_fun eqf in
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let Hf := to_is_equiv eqf in
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equiv.mk (is_equiv.precomp f C)
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(@is_equiv.precomp_closed A B f F Hf C)
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definition postcomp_closed [F : funext] {A B C : Type} {eqf : A ≃ B}
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: (C → A) ≃ (C → B) :=
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let f := to_fun eqf in
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let Hf := to_is_equiv eqf in
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equiv.mk (is_equiv.postcomp f C)
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(@is_equiv.postcomp_closed A B f F Hf C)
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end equiv
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