84 lines
2.9 KiB
Text
84 lines
2.9 KiB
Text
-- Copyright (c) 2014 Microsoft Corporation. All rights reserved.
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-- Released under Apache 2.0 license as described in the file LICENSE.
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-- Author: Jeremy Avigad, Jakob von Raumer
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-- Ported from Coq HoTT
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import .equiv .funext
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open path function
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namespace IsEquiv
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context
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parameters {A B : Type} (f : A → B)
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--Precomposition of arbitrary functions with f
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definition precomp (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 (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 precompose [instance] [Hf : IsEquiv f] (C : Type):
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IsEquiv (precomp C) :=
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adjointify (precomp C) (λh, h ∘ f⁻¹)
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(λh, path_forall _ _ (λx, ap h (sect f x)))
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(λg, path_forall _ _ (λy, ap g (retr f y)))
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--Postcomposing with an equivalence is an equivalence
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definition postcompose [instance] [Hf : IsEquiv f] (C : Type):
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IsEquiv (postcomp C) :=
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adjointify (postcomp C) (λl, f⁻¹ ∘ l)
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(λh, path_forall _ _ (λx, retr f (h x)))
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(λg, path_forall _ _ (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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private definition isequiv_precompose_eq (C D : Type) (Ceq : IsEquiv (precomp C))
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(Deq : IsEquiv (precomp D)) (k : C → D) (h : A → C) :
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k ∘ (inv (precomp C)) h ≈ (inv (precomp D)) (k ∘ h) :=
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let invD := inv (precomp D) in
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let invC := inv (precomp 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 C (invC h))) : retr (precomp C) h
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... ≈ k ∘ (invC h) : !sect,
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eq1⁻¹
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definition isequiv_precompose (Aeq : IsEquiv (precomp A))
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(Beq : IsEquiv (precomp B)) : (IsEquiv f) :=
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let invA := inv (precomp A) in
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let invB := inv (precomp B) in
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let sect' : Sect (invA id) f := (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 B id) x : idp
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... ≈ x : apD10 (sect (precomp B) id))
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in
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let retr' : Sect f (invA id) := (λx,
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calc invA id (f x) ≈ precomp A (invA id) x : idp
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... ≈ x : apD10 (retr (precomp A) id)) in
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adjointify f (invA id) sect' retr'
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end
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end IsEquiv
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--Bundled versions of the previous theorems
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namespace Equiv
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context
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parameters {A B C : Type} {eqf : A ≃ B}
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private definition f := equiv_fun eqf
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private definition Hf := equiv_isequiv eqf
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definition precompose : (B → C) ≃ (A → C) :=
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Equiv_mk (IsEquiv.precomp f C)
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(@IsEquiv.precompose A B f Hf C)
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definition postcompose : (C → A) ≃ (C → B) :=
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Equiv_mk (IsEquiv.postcomp f C)
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(@IsEquiv.postcompose A B f Hf C)
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end
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end Equiv
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