93 lines
3.7 KiB
Text
93 lines
3.7 KiB
Text
import homotopy.susp types.pointed2
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open susp eq pointed function is_equiv lift equiv
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namespace susp
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variables {X X' Y Y' Z : Type*}
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definition susp_functor_pconst_homotopy [unfold 3] {X Y : Type*} (x : psusp X) :
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psusp_functor (pconst X Y) x = pt :=
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begin
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induction x,
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{ reflexivity },
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{ exact (merid pt)⁻¹ },
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{ apply eq_pathover, refine !elim_merid ⬝ph _ ⬝hp !ap_constant⁻¹, exact square_of_eq !con.right_inv⁻¹ }
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end
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definition susp_functor_pconst [constructor] (X Y : Type*) : psusp_functor (pconst X Y) ~* pconst (psusp X) (psusp Y) :=
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begin
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fapply phomotopy.mk,
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{ exact susp_functor_pconst_homotopy },
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{ reflexivity }
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end
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definition psusp_pfunctor [constructor] (X Y : Type*) : ppmap X Y →* ppmap (psusp X) (psusp Y) :=
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pmap.mk psusp_functor (eq_of_phomotopy !susp_functor_pconst)
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definition psusp_pelim [constructor] (X Y : Type*) : ppmap X (Ω Y) →* ppmap (psusp X) Y :=
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ppcompose_left (loop_psusp_counit Y) ∘* psusp_pfunctor X (Ω Y)
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definition loop_psusp_pintro [constructor] (X Y : Type*) : ppmap (psusp X) Y →* ppmap X (Ω Y) :=
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ppcompose_right (loop_psusp_unit X) ∘* pap1 (psusp X) Y
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definition loop_psusp_pintro_natural_left (f : X' →* X) :
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psquare (loop_psusp_pintro X Y) (loop_psusp_pintro X' Y)
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(ppcompose_right (psusp_functor f)) (ppcompose_right f) :=
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!pap1_natural_left ⬝h* ppcompose_right_psquare (loop_psusp_unit_natural f)⁻¹*
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definition loop_psusp_pintro_natural_right (f : Y →* Y') :
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psquare (loop_psusp_pintro X Y) (loop_psusp_pintro X Y')
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(ppcompose_left f) (ppcompose_left (Ω→ f)) :=
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!pap1_natural_right ⬝h* !ppcompose_left_ppcompose_right⁻¹*
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definition is_equiv_loop_psusp_pintro [constructor] (X Y : Type*) :
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is_equiv (loop_psusp_pintro X Y) :=
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begin
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fapply adjointify,
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{ exact psusp_pelim X Y },
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{ intro g, apply eq_of_phomotopy, exact psusp_adjoint_loop_right_inv g },
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{ intro f, apply eq_of_phomotopy, exact psusp_adjoint_loop_left_inv f }
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end
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definition psusp_adjoint_loop' [constructor] (X Y : Type*) : ppmap (psusp X) Y ≃* ppmap X (Ω Y) :=
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pequiv_of_pmap (loop_psusp_pintro X Y) (is_equiv_loop_psusp_pintro X Y)
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definition psusp_adjoint_loop_natural_right (f : Y →* Y') :
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psquare (psusp_adjoint_loop' X Y) (psusp_adjoint_loop' X Y')
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(ppcompose_left f) (ppcompose_left (Ω→ f)) :=
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loop_psusp_pintro_natural_right f
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definition psusp_adjoint_loop_natural_left (f : X' →* X) :
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psquare (psusp_adjoint_loop' X Y) (psusp_adjoint_loop' X' Y)
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(ppcompose_right (psusp_functor f)) (ppcompose_right f) :=
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loop_psusp_pintro_natural_left f
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definition iterate_psusp_iterate_psusp_rev (n m : ℕ) (A : Type*) :
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iterate_psusp n (iterate_psusp m A) ≃* iterate_psusp (m + n) A :=
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begin
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induction n with n e,
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{ reflexivity },
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{ exact psusp_pequiv e }
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end
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definition iterate_psusp_pequiv [constructor] (n : ℕ) {X Y : Type*} (f : X ≃* Y) :
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iterate_psusp n X ≃* iterate_psusp n Y :=
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begin
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induction n with n e,
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{ exact f },
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{ exact psusp_pequiv e }
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end
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open algebra nat
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definition iterate_psusp_iterate_psusp (n m : ℕ) (A : Type*) :
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iterate_psusp n (iterate_psusp m A) ≃* iterate_psusp (n + m) A :=
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iterate_psusp_iterate_psusp_rev n m A ⬝e* pequiv_of_eq (ap (λk, iterate_psusp k A) (add.comm m n))
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definition plift_psusp.{u v} : Π(A : Type*), plift.{u v} (psusp A) ≃* psusp (plift.{u v} A) :=
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begin
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intro A,
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calc
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plift.{u v} (psusp A) ≃* psusp A : by exact (pequiv_plift (psusp A))⁻¹ᵉ*
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... ≃* psusp (plift.{u v} A) : by exact psusp_pequiv (pequiv_plift.{u v} A)
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end
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end susp
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