229 lines
7.4 KiB
Text
229 lines
7.4 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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Authors: Floris van Doorn
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Declaration of suspension
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-/
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import hit.pushout types.pointed cubical.square
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open pushout unit eq equiv equiv.ops
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definition susp (A : Type) : Type := pushout (λ(a : A), star) (λ(a : A), star)
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namespace susp
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variable {A : Type}
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definition north {A : Type} : susp A :=
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inl star
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definition south {A : Type} : susp A :=
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inr star
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definition merid (a : A) : @north A = @south A :=
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glue a
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protected definition rec {P : susp A → Type} (PN : P north) (PS : P south)
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(Pm : Π(a : A), PN =[merid a] PS) (x : susp A) : P x :=
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begin
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induction x with u u,
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{ cases u, exact PN},
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{ cases u, exact PS},
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{ apply Pm},
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end
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protected definition rec_on [reducible] {P : susp A → Type} (y : susp A)
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(PN : P north) (PS : P south) (Pm : Π(a : A), PN =[merid a] PS) : P y :=
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susp.rec PN PS Pm y
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theorem rec_merid {P : susp A → Type} (PN : P north) (PS : P south)
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(Pm : Π(a : A), PN =[merid a] PS) (a : A)
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: apdo (susp.rec PN PS Pm) (merid a) = Pm a :=
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!rec_glue
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protected definition elim {P : Type} (PN : P) (PS : P) (Pm : A → PN = PS)
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(x : susp A) : P :=
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susp.rec PN PS (λa, pathover_of_eq (Pm a)) x
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protected definition elim_on [reducible] {P : Type} (x : susp A)
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(PN : P) (PS : P) (Pm : A → PN = PS) : P :=
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susp.elim PN PS Pm x
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theorem elim_merid {P : Type} {PN PS : P} (Pm : A → PN = PS) (a : A)
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: ap (susp.elim PN PS Pm) (merid a) = Pm a :=
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begin
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apply eq_of_fn_eq_fn_inv !(pathover_constant (merid a)),
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rewrite [▸*,-apdo_eq_pathover_of_eq_ap,↑susp.elim,rec_merid],
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end
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protected definition elim_type (PN : Type) (PS : Type) (Pm : A → PN ≃ PS)
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(x : susp A) : Type :=
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susp.elim PN PS (λa, ua (Pm a)) x
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protected definition elim_type_on [reducible] (x : susp A)
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(PN : Type) (PS : Type) (Pm : A → PN ≃ PS) : Type :=
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susp.elim_type PN PS Pm x
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theorem elim_type_merid (PN : Type) (PS : Type) (Pm : A → PN ≃ PS)
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(a : A) : transport (susp.elim_type PN PS Pm) (merid a) = Pm a :=
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by rewrite [tr_eq_cast_ap_fn,↑susp.elim_type,elim_merid];apply cast_ua_fn
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end susp
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attribute susp.north susp.south [constructor]
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attribute susp.rec susp.elim [unfold 6] [recursor 6]
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attribute susp.elim_type [unfold 5]
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attribute susp.rec_on susp.elim_on [unfold 3]
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attribute susp.elim_type_on [unfold 2]
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namespace susp
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open pointed
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variables {X Y Z : Pointed}
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definition pointed_susp [instance] [constructor] (X : Type) : pointed (susp X) :=
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pointed.mk north
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definition Susp [constructor] (X : Type) : Pointed :=
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pointed.mk' (susp X)
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definition Susp_functor (f : X →* Y) : Susp X →* Susp Y :=
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begin
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fconstructor,
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{ intro x, induction x,
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apply north,
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apply south,
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exact merid (f a)},
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{ reflexivity}
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end
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definition Susp_functor_compose (g : Y →* Z) (f : X →* Y)
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: Susp_functor (g ∘* f) ~* Susp_functor g ∘* Susp_functor f :=
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begin
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fconstructor,
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{ intro a, induction a,
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{ reflexivity},
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{ reflexivity},
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{ apply eq_pathover, apply hdeg_square,
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rewrite [▸*,ap_compose' _ (Susp_functor f),↑Susp_functor,+elim_merid]}},
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{ reflexivity}
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end
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-- adjunction from Coq-HoTT
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definition loop_susp_unit [constructor] (X : Pointed) : X →* Ω(Susp X) :=
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begin
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fconstructor,
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{ intro x, exact merid x ⬝ (merid pt)⁻¹},
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{ apply con.right_inv},
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end
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definition loop_susp_unit_natural (f : X →* Y)
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: loop_susp_unit Y ∘* f ~* ap1 (Susp_functor f) ∘* loop_susp_unit X :=
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begin
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induction X with X x, induction Y with Y y, induction f with f pf, esimp at *, induction pf,
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fconstructor,
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{ intro x', esimp [Susp_functor], symmetry,
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exact
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!idp_con ⬝
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(!ap_con ⬝
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whisker_left _ !ap_inv) ⬝
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(!elim_merid ◾ (inverse2 !elim_merid))
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},
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{ rewrite [▸*,idp_con (con.right_inv _)],
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apply inv_con_eq_of_eq_con,
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refine _ ⬝ !con.assoc',
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rewrite inverse2_right_inv,
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refine _ ⬝ !con.assoc',
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rewrite [ap_con_right_inv], unfold Susp_functor, xrewrite [idp_con_idp,-ap_compose], },
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end
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definition loop_susp_counit [constructor] (X : Pointed) : Susp (Ω X) →* X :=
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begin
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fconstructor,
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{ intro x, induction x, exact pt, exact pt, exact a},
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{ reflexivity},
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end
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definition loop_susp_counit_natural (f : X →* Y)
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: f ∘* loop_susp_counit X ~* loop_susp_counit Y ∘* (Susp_functor (ap1 f)) :=
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begin
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induction X with X x, induction Y with Y y, induction f with f pf, esimp at *, induction pf,
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fconstructor,
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{ intro x', induction x' with p,
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{ reflexivity},
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{ reflexivity},
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{ esimp, apply eq_pathover, apply hdeg_square,
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xrewrite [ap_compose f,ap_compose (susp.elim (f x) (f x) (λ (a : f x = f x), a)),▸*],
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xrewrite [+elim_merid,▸*,idp_con]}},
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{ reflexivity}
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end
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definition loop_susp_counit_unit (X : Pointed)
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: ap1 (loop_susp_counit X) ∘* loop_susp_unit (Ω X) ~* pid (Ω X) :=
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begin
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induction X with X x, fconstructor,
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{ intro p, esimp,
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refine !idp_con ⬝
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(!ap_con ⬝
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whisker_left _ !ap_inv) ⬝
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(!elim_merid ◾ inverse2 !elim_merid)},
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{ rewrite [▸*,inverse2_right_inv (elim_merid id idp)],
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refine !con.assoc ⬝ _,
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xrewrite [ap_con_right_inv (susp.elim x x (λa, a)) (merid idp),idp_con_idp,-ap_compose]}
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end
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definition loop_susp_unit_counit (X : Pointed)
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: loop_susp_counit (Susp X) ∘* Susp_functor (loop_susp_unit X) ~* pid (Susp X) :=
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begin
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induction X with X x, fconstructor,
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{ intro x', induction x',
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{ reflexivity},
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{ exact merid pt},
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{ apply eq_pathover,
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xrewrite [▸*, ap_id, ap_compose (susp.elim north north (λa, a)), +elim_merid,▸*],
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apply square_of_eq, exact !idp_con ⬝ !inv_con_cancel_right⁻¹}},
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{ reflexivity}
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end
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definition susp_adjoint_loop (X Y : Pointed) : map₊ (pointed.mk' (susp X)) Y ≃ map₊ X (Ω Y) :=
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begin
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fapply equiv.MK,
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{ intro f, exact ap1 f ∘* loop_susp_unit X},
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{ intro g, exact loop_susp_counit Y ∘* Susp_functor g},
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{ intro g, apply eq_of_phomotopy, esimp,
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refine !pwhisker_right !ap1_compose ⬝* _,
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refine !passoc ⬝* _,
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refine !pwhisker_left !loop_susp_unit_natural⁻¹* ⬝* _,
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refine !passoc⁻¹* ⬝* _,
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refine !pwhisker_right !loop_susp_counit_unit ⬝* _,
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apply pid_comp},
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{ intro f, apply eq_of_phomotopy, esimp,
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refine !pwhisker_left !Susp_functor_compose ⬝* _,
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refine !passoc⁻¹* ⬝* _,
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refine !pwhisker_right !loop_susp_counit_natural⁻¹* ⬝* _,
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refine !passoc ⬝* _,
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refine !pwhisker_left !loop_susp_unit_counit ⬝* _,
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apply comp_pid},
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end
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definition susp_adjoint_loop_nat_right (f : Susp X →* Y) (g : Y →* Z)
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: susp_adjoint_loop X Z (g ∘* f) ~* ap1 g ∘* susp_adjoint_loop X Y f :=
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begin
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esimp [susp_adjoint_loop],
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refine _ ⬝* !passoc,
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apply pwhisker_right,
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apply ap1_compose
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end
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definition susp_adjoint_loop_nat_left (f : Y →* Ω Z) (g : X →* Y)
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: (susp_adjoint_loop X Z)⁻¹ (f ∘* g) ~* (susp_adjoint_loop Y Z)⁻¹ f ∘* Susp_functor g :=
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begin
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esimp [susp_adjoint_loop],
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refine _ ⬝* !passoc⁻¹*,
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apply pwhisker_left,
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apply Susp_functor_compose
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end
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end susp
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