197 lines
8.6 KiB
Text
197 lines
8.6 KiB
Text
import .pushout types.pointed2 ..move_to_lib
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open susp eq pointed function is_equiv lift equiv is_trunc nat
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namespace susp
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variables {X X' Y Y' Z : Type*}
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definition susp_functor_of_fn [constructor] (f : X → Y) : susp X →* susp Y :=
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pmap.mk (susp_functor' f) idp
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definition susp_pequiv_of_equiv [constructor] (f : X ≃ Y) : susp X ≃* susp Y :=
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pequiv_of_equiv (susp.equiv f) idp
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definition iterate_susp_iterate_susp_rev (n m : ℕ) (A : Type*) :
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iterate_susp n (iterate_susp m A) ≃* iterate_susp (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 susp_pequiv e }
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end
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definition iterate_susp_pequiv [constructor] (n : ℕ) {X Y : Type*} (f : X ≃* Y) :
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iterate_susp n X ≃* iterate_susp 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 susp_pequiv e }
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end
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open algebra nat
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definition iterate_susp_iterate_susp (n m : ℕ) (A : Type*) :
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iterate_susp n (iterate_susp m A) ≃* iterate_susp (n + m) A :=
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iterate_susp_iterate_susp_rev n m A ⬝e* pequiv_of_eq (ap (λk, iterate_susp k A) (add.comm m n))
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definition plift_susp.{u v} : Π(A : Type*), plift.{u v} (susp A) ≃* susp (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} (susp A) ≃* susp A : by exact (pequiv_plift (susp A))⁻¹ᵉ*
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... ≃* susp (plift.{u v} A) : by exact susp_pequiv (pequiv_plift.{u v} A)
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end
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definition is_contr_susp [instance] (A : Type) [H : is_contr A] : is_contr (susp A) :=
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begin
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apply is_contr.mk north,
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intro x, induction x,
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reflexivity,
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exact merid !center,
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apply eq_pathover_constant_left_id_right, apply square_of_eq,
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exact whisker_left idp (ap merid !eq_of_is_contr)
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end
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definition loop_susp_pintro_phomotopy {X Y : Type*} {f g : ⅀ X →* Y} (p : f ~* g) :
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loop_susp_pintro X Y f ~* loop_susp_pintro X Y g :=
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pwhisker_right (loop_susp_unit X) (Ω⇒ p)
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variables {A₀₀ A₂₀ A₀₂ A₂₂ : Type*}
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{f₁₀ : A₀₀ →* A₂₀} {f₁₂ : A₀₂ →* A₂₂}
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{f₀₁ : A₀₀ →* A₀₂} {f₂₁ : A₂₀ →* A₂₂}
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definition susp_functor_psquare (p : psquare f₁₀ f₁₂ f₀₁ f₂₁) :
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psquare (⅀→ f₁₀) (⅀→ f₁₂) (⅀→ f₀₁) (⅀→ f₂₁) :=
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!susp_functor_pcompose⁻¹* ⬝* susp_functor_phomotopy p ⬝* !susp_functor_pcompose
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definition susp_to_loop_psquare (f₁₀ : A₀₀ →* A₂₀) (f₁₂ : A₀₂ →* A₂₂)
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(f₀₁ : susp A₀₀ →* A₀₂) (f₂₁ : susp A₂₀ →* A₂₂) : psquare (⅀→ f₁₀) f₁₂ f₀₁ f₂₁ →
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psquare f₁₀ (Ω→ f₁₂) (loop_susp_pintro A₀₀ A₀₂ f₀₁) (loop_susp_pintro A₂₀ A₂₂ f₂₁) :=
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begin
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intro p,
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refine pvconcat _ (ap1_psquare p),
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exact (loop_susp_unit_natural f₁₀)⁻¹*
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end
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definition loop_to_susp_square (f₁₀ : A₀₀ →* A₂₀) (f₁₂ : A₀₂ →* A₂₂)
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(f₀₁ : A₀₀ →* Ω A₀₂) (f₂₁ : A₂₀ →* Ω A₂₂) : psquare f₁₀ (Ω→ f₁₂) f₀₁ f₂₁ →
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psquare (⅀→ f₁₀) f₁₂ (susp_pelim A₀₀ A₀₂ f₀₁) (susp_pelim A₂₀ A₂₂ f₂₁) :=
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begin
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intro p,
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refine susp_functor_psquare p ⬝v* _,
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exact ptranspose (loop_susp_counit_natural f₁₂)
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end
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open pushout unit prod sigma sigma.ops
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section
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parameters {A : Type*} {n : ℕ} [HA : is_conn n A]
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-- we end up not using this, because to prove that the
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-- composition with the first projection is loop_susp_counit A
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-- is hideous without HIT computations on path constructors
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parameter (A)
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definition pullback_diagonal_prod_of_wedge : susp (Ω A)
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≃ Σ (a : A) (w : wedge A A), prod_of_wedge w = (a, a) :=
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begin
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refine equiv.trans _
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(comm_equiv_unc (λ z, prod_of_wedge (prod.pr1 z) = (prod.pr2 z, prod.pr2 z))),
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apply equiv.symm,
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apply equiv.trans (sigma_equiv_sigma_right
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(λ w, sigma_equiv_sigma_right
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(λ a, prod_eq_equiv (prod_of_wedge w) (a, a)))),
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apply equiv.trans !pushout.flattening', esimp,
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fapply pushout.equiv
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(λ z, ⟨pt, z.2⟩) (λ z, ⟨pt, glue z.1 ▸ z.2⟩) (λ p, star) (λ p, star),
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{ apply equiv.trans !sigma_unit_left, fapply equiv.MK,
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{ intro z, induction z with a w, induction w with p q, exact p ⬝ q⁻¹ },
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{ intro p, exact ⟨pt, (p, idp)⟩ },
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{ intro p, reflexivity },
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{ intro z, induction z with a w, induction w with p q, induction q,
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reflexivity } },
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{ fapply equiv.MK,
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{ intro z, exact star },
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{ intro u, exact ⟨pt, ⟨pt, (idp, idp)⟩ ⟩ },
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{ intro u, induction u, reflexivity },
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{ intro z, induction z with a w, induction w with b z,
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induction z with p q, induction p, esimp at q, induction q,
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reflexivity } },
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{ fapply equiv.MK,
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{ intro z, exact star },
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{ intro u, exact ⟨pt, ⟨pt, (idp, idp)⟩ ⟩ },
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{ intro u, induction u, reflexivity },
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{ intro z, induction z with a w, induction w with b z,
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induction z with p q, induction q, esimp at p, induction p,
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reflexivity } },
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{ intro z, induction z with u w, induction u, induction w with a z,
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induction z with p q, reflexivity },
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{ intro z, induction z with u w, induction u, induction w with a z,
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induction z with p q, reflexivity }
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end
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parameter {A}
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-- instead we directly compare the fibers, using flattening twice
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definition fiber_loop_susp_counit_equiv (a : A)
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: fiber (loop_susp_counit A) a ≃ fiber prod_of_wedge (a, a) :=
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begin
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apply equiv.trans !fiber.sigma_char, apply equiv.trans !pushout.flattening',
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apply equiv.symm, apply equiv.trans !fiber.sigma_char,
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apply equiv.trans (sigma_equiv_sigma_right
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(λ w, prod_eq_equiv (prod_of_wedge w) (a, a))), esimp,
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apply equiv.trans !pushout.flattening',
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esimp,
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fapply pushout.equiv (λ z, ⟨pt, z.2⟩) (λ z, ⟨pt, glue z.1 ▸ z.2⟩)
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(λ z, ⟨star, z.2⟩) (λ z, ⟨star, glue z.1 ▸ z.2⟩),
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{ fapply equiv.MK,
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{ intro w, induction w with u z, induction z with p q,
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exact ⟨q ⬝ p⁻¹, q⟩ },
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{ intro z, induction z with p q, apply dpair star,
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exact (p⁻¹ ⬝ q, q) },
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{ intro z, induction z with p q, esimp, induction q, esimp,
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rewrite [idp_con,inv_inv] },
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{ intro w, induction w with u z, induction u, induction z with p q,
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esimp, induction q, rewrite [idp_con,inv_inv] } },
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{ fapply equiv.MK,
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{ intro w, induction w with b z, induction z with p q, exact ⟨star, q⟩ },
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{ intro z, induction z with u p, induction u, esimp at p, esimp,
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apply dpair a, esimp, exact (idp, p) },
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{ intro z, induction z with u p, induction u, reflexivity },
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{ intro w, induction w with b z, induction z with p q, esimp,
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induction p, reflexivity } },
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{ fapply equiv.MK,
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{ intro w, induction w with b z, induction z with p q, exact ⟨star, p⟩ },
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{ intro z, induction z with u p, induction u, esimp at p, esimp,
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apply dpair a, esimp, exact (p, idp) },
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{ intro z, induction z with u p, induction u, reflexivity },
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{ intro w, induction w with b z, induction z with p q, esimp,
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induction q, reflexivity } },
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{ intro w, induction w with u z, induction u, induction z with p q,
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reflexivity },
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{ intro w, induction w with u z, induction u, induction z with p q,
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esimp, induction q, esimp, krewrite prod_transport, fapply sigma_eq,
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{ exact idp },
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{ esimp, rewrite eq_transport_Fl, rewrite eq_transport_Fl,
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krewrite elim_glue, krewrite [-ap_compose' pr1 prod_of_wedge (glue star)],
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krewrite elim_glue, esimp, apply eq_pathover, rewrite idp_con, esimp,
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apply square_of_eq, rewrite [idp_con,idp_con,inv_inv] } }
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end
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include HA
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open is_conn trunc_index
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parameter (A)
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-- connectivity of loop_susp_counit
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definition is_conn_fun_loop_susp_counit {k : ℕ} (H : k ≤ 2 * n)
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: is_conn_fun k (loop_susp_counit A) :=
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begin
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intro a, apply is_conn.is_conn_equiv_closed_rev k (fiber_loop_susp_counit_equiv a),
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fapply @is_conn.is_conn_of_le (fiber prod_of_wedge (a, a)) k (2 * n)
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(of_nat_le_of_nat H),
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assert H : of_nat (2 * n) = of_nat n + of_nat n,
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{ rewrite (of_nat_add_of_nat n n), apply ap of_nat,
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apply trans (nat.mul_comm 2 n),
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apply ap (λ k, k + n), exact nat.zero_add n },
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rewrite H,
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exact is_conn_fun_prod_of_wedge n n A A (a, a)
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end
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end
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end susp
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