144 lines
4.8 KiB
Text
144 lines
4.8 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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Author: Floris van Doorn
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Theorems about lift
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-/
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import ..function
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open eq equiv equiv.ops is_equiv is_trunc
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namespace lift
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universe variables u v
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variables {A : Type.{u}} (z z' : lift.{u v} A)
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protected definition eta : up (down z) = z :=
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by induction z; reflexivity
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protected definition code [unfold 2 3] : lift A → lift A → Type
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| code (up a) (up a') := a = a'
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protected definition decode [unfold 2 3] : Π(z z' : lift A), lift.code z z' → z = z'
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| decode (up a) (up a') := λc, ap up c
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variables {z z'}
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protected definition encode [unfold 3 4 5] (p : z = z') : lift.code z z' :=
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by induction p; induction z; esimp
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variables (z z')
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definition lift_eq_equiv : (z = z') ≃ lift.code z z' :=
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equiv.MK lift.encode
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!lift.decode
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abstract begin
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intro c, induction z with a, induction z' with a', esimp at *, induction c,
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reflexivity
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end end
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abstract begin
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intro p, induction p, induction z, reflexivity
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end end
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section
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variables {a a' : A}
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definition eq_of_up_eq_up [unfold 4] (p : up a = up a') : a = a' :=
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lift.encode p
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definition lift_transport {P : A → Type} (p : a = a') (z : lift (P a))
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: p ▸ z = up (p ▸ down z) :=
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by induction p; induction z; reflexivity
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end
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variables {A' : Type} (f : A → A') (g : lift A → lift A')
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definition lift_functor [unfold 4] : lift A → lift A'
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| lift_functor (up a) := up (f a)
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definition is_equiv_lift_functor [constructor] [Hf : is_equiv f] : is_equiv (lift_functor f) :=
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adjointify (lift_functor f)
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(lift_functor f⁻¹)
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abstract begin
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intro z', induction z' with a',
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esimp, exact ap up !right_inv
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end end
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abstract begin
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intro z, induction z with a,
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esimp, exact ap up !left_inv
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end end
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definition lift_equiv_lift_of_is_equiv [constructor] [Hf : is_equiv f] : lift A ≃ lift A' :=
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equiv.mk _ (is_equiv_lift_functor f)
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definition lift_equiv_lift [constructor] (f : A ≃ A') : lift A ≃ lift A' :=
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equiv.mk _ (is_equiv_lift_functor f)
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definition lift_equiv_lift_refl (A : Type) : lift_equiv_lift (erfl : A ≃ A) = erfl :=
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by apply equiv_eq'; intro z; induction z with a; reflexivity
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definition lift_inv_functor [unfold-full] (a : A) : A' :=
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down (g (up a))
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definition is_equiv_lift_inv_functor [constructor] [Hf : is_equiv g]
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: is_equiv (lift_inv_functor g) :=
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adjointify (lift_inv_functor g)
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(lift_inv_functor g⁻¹)
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abstract begin
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intro z', rewrite [▸*,lift.eta,right_inv g],
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end end
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abstract begin
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intro z', rewrite [▸*,lift.eta,left_inv g],
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end end
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definition equiv_of_lift_equiv_lift [constructor] (g : lift A ≃ lift A') : A ≃ A' :=
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equiv.mk _ (is_equiv_lift_inv_functor g)
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definition lift_functor_left_inv : lift_inv_functor (lift_functor f) = f :=
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eq_of_homotopy (λa, idp)
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definition lift_functor_right_inv : lift_functor (lift_inv_functor g) = g :=
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begin
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apply eq_of_homotopy, intro z, induction z with a, esimp, apply lift.eta
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end
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variables (A A')
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definition is_equiv_lift_functor_fn [constructor]
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: is_equiv (lift_functor : (A → A') → (lift A → lift A')) :=
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adjointify lift_functor
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lift_inv_functor
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lift_functor_right_inv
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lift_functor_left_inv
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definition lift_imp_lift_equiv [constructor] : (lift A → lift A') ≃ (A → A') :=
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(equiv.mk _ (is_equiv_lift_functor_fn A A'))⁻¹ᵉ
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-- can we deduce this from lift_imp_lift_equiv?
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definition lift_equiv_lift_equiv [constructor] : (lift A ≃ lift A') ≃ (A ≃ A') :=
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equiv.MK equiv_of_lift_equiv_lift
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lift_equiv_lift
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abstract begin
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intro f, apply equiv_eq, reflexivity
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end end
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abstract begin
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intro g, apply equiv_eq, esimp, apply eq_of_homotopy, intro z,
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induction z with a, esimp, apply lift.eta
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end end
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definition lift_eq_lift_equiv.{u1 u2} (A A' : Type.{u1})
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: (lift.{u1 u2} A = lift.{u1 u2} A') ≃ (A = A') :=
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!eq_equiv_equiv ⬝e !lift_equiv_lift_equiv ⬝e !eq_equiv_equiv⁻¹ᵉ
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definition is_embedding_lift [instance] : is_embedding lift :=
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begin
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intro A A', fapply is_equiv.homotopy_closed,
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exact to_inv !lift_eq_lift_equiv,
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exact _,
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{ intro p, induction p,
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esimp [lift_eq_lift_equiv,equiv.trans,equiv.symm,eq_equiv_equiv],
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rewrite [equiv_of_eq_refl,lift_equiv_lift_refl],
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apply ua_refl}
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end
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-- is_trunc_lift is defined in init.trunc
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end lift
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