move more stuff
This commit is contained in:
Floris van Doorn
parent
be0d5977f6
commit
e1d2392a13
2 changed files with 80 additions and 231 deletions
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@ -59,6 +59,10 @@ namespace sphere
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attribute untrunc_of_is_trunc [unfold 4]
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definition surf_eq_loop : @surf 1 = circle.loop := sorry
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/-
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Favonia had a good idea, which he got from Ulrik: use the cogroup structure on the suspension to construct a group structure on ΣX →* Y, from which you can easily show that deg(id) = 1. See in the Agda library the files cogroup, cohspace and Group/LoopSuspAdjoint (or something)
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-/
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-- definition π2S2_surf : π2S2 (tr surf) = 1 :> ℤ :=
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-- begin
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@ -19,6 +19,10 @@ le_step_left > le_of_succ_le
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pmap.eta > pmap_eta
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pType.sigma_char' > pType.sigma_char
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cghomotopy_group to aghomotopy_group
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is_trunc_loop_of_is_trunc > is_trunc_loopn_of_is_trunc
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first two arguments reordered in is_trunc_loopn_nat
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-/
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namespace eq
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@ -94,224 +98,11 @@ namespace trunc
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{ exact sorry }
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end
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definition ptrunctype.sigma_char [constructor] (n : ℕ₋₂) :
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n-Type* ≃ Σ(X : Type*), is_trunc n X :=
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equiv.MK (λX, ⟨ptrunctype.to_pType X, _⟩)
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(λX, ptrunctype.mk (carrier X.1) X.2 pt)
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begin intro X, induction X with X HX, induction X, reflexivity end
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begin intro X, induction X, reflexivity end
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definition is_embedding_ptrunctype_to_pType (n : ℕ₋₂) : is_embedding (@ptrunctype.to_pType n) :=
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begin
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intro X Y, fapply is_equiv_of_equiv_of_homotopy,
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{ exact eq_equiv_fn_eq (ptrunctype.sigma_char n) _ _ ⬝e subtype_eq_equiv _ _ },
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intro p, induction p, reflexivity
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end
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definition ptrunctype_eq_equiv {n : ℕ₋₂} (X Y : n-Type*) : (X = Y) ≃ (X ≃* Y) :=
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equiv.mk _ (is_embedding_ptrunctype_to_pType n X Y) ⬝e pType_eq_equiv X Y
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definition Prop_eq {P Q : Prop} (H : P ↔ Q) : P = Q :=
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tua (equiv_of_is_prop (iff.mp H) (iff.mpr H))
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definition is_trunc_ptrunc_of_is_trunc [instance] [priority 500] (A : Type*)
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(n m : ℕ₋₂) [H : is_trunc n A] : is_trunc n (ptrunc m A) :=
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is_trunc_trunc_of_is_trunc A n m
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definition ptrunc_pequiv_ptrunc_of_is_trunc {n m k : ℕ₋₂} {A : Type*}
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(H1 : n ≤ m) (H2 : n ≤ k) (H : is_trunc n A) : ptrunc m A ≃* ptrunc k A :=
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have is_trunc m A, from is_trunc_of_le A H1,
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have is_trunc k A, from is_trunc_of_le A H2,
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pequiv.MK (ptrunc.elim _ (ptr k A)) (ptrunc.elim _ (ptr m A))
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abstract begin
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refine !ptrunc_elim_pcompose⁻¹* ⬝* _,
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exact ptrunc_elim_phomotopy _ !ptrunc_elim_ptr ⬝* !ptrunc_elim_ptr_phomotopy_pid,
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end end
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abstract begin
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refine !ptrunc_elim_pcompose⁻¹* ⬝* _,
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exact ptrunc_elim_phomotopy _ !ptrunc_elim_ptr ⬝* !ptrunc_elim_ptr_phomotopy_pid,
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end end
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definition ptrunc_change_index {k l : ℕ₋₂} (p : k = l) (X : Type*)
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: ptrunc k X ≃* ptrunc l X :=
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pequiv_ap (λ n, ptrunc n X) p
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definition ptrunc_functor_le {k l : ℕ₋₂} (p : l ≤ k) (X : Type*)
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: ptrunc k X →* ptrunc l X :=
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have is_trunc k (ptrunc l X), from is_trunc_of_le _ p,
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ptrunc.elim _ (ptr l X)
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definition trunc_index.pred [unfold 1] (n : ℕ₋₂) : ℕ₋₂ :=
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begin cases n with n, exact -2, exact n end
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/- A more general version of ptrunc_elim_phomotopy, where the proofs of truncatedness might be different -/
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definition ptrunc_elim_phomotopy2 [constructor] (k : ℕ₋₂) {A B : Type*} {f g : A →* B} (H₁ : is_trunc k B)
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(H₂ : is_trunc k B) (p : f ~* g) : @ptrunc.elim k A B H₁ f ~* @ptrunc.elim k A B H₂ g :=
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begin
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fapply phomotopy.mk,
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{ intro x, induction x with a, exact p a },
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{ exact to_homotopy_pt p }
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end
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definition pmap_ptrunc_equiv [constructor] (n : ℕ₋₂) (A B : Type*) [H : is_trunc n B] :
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(ptrunc n A →* B) ≃ (A →* B) :=
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begin
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fapply equiv.MK,
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{ intro g, exact g ∘* ptr n A },
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{ exact ptrunc.elim n },
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{ intro f, apply eq_of_phomotopy, apply ptrunc_elim_ptr },
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{ intro g, apply eq_of_phomotopy,
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exact ptrunc_elim_pcompose n g (ptr n A) ⬝* pwhisker_left g (ptrunc_elim_ptr_phomotopy_pid n A) ⬝*
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pcompose_pid g }
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end
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definition pmap_ptrunc_pequiv [constructor] (n : ℕ₋₂) (A B : Type*) [H : is_trunc n B] :
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ppmap (ptrunc n A) B ≃* ppmap A B :=
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pequiv_of_equiv (pmap_ptrunc_equiv n A B) (eq_of_phomotopy (pconst_pcompose (ptr n A)))
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definition loopn_ptrunc_pequiv_nat (n : ℕ) (k : ℕ) (A : Type*) :
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Ω[k] (ptrunc (n+k) A) ≃* ptrunc n (Ω[k] A) :=
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loopn_pequiv_loopn k (ptrunc_change_index (of_nat_add_of_nat n k)⁻¹ A) ⬝e* loopn_ptrunc_pequiv n k A
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end trunc open trunc
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namespace is_trunc
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open trunc_index is_conn
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lemma is_trunc_loopn_nat (m n : ℕ) (A : Type*) [H : is_trunc (n + m) A] :
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is_trunc n (Ω[m] A) :=
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@is_trunc_loopn n m A (transport (λk, is_trunc k _) (of_nat_add_of_nat n m)⁻¹ H)
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lemma is_trunc_loop_nat (n : ℕ) (A : Type*) [H : is_trunc (n + 1) A] :
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is_trunc n (Ω A) :=
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is_trunc_loop A n
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definition is_trunc_of_eq {n m : ℕ₋₂} (p : n = m) {A : Type} (H : is_trunc n A) : is_trunc m A :=
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transport (λk, is_trunc k A) p H
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definition is_trunc_succ_succ_of_is_trunc_loop (n : ℕ₋₂) (A : Type*) (H : is_trunc (n.+1) (Ω A))
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(H2 : is_conn 0 A) : is_trunc (n.+2) A :=
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begin
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apply is_trunc_succ_of_is_trunc_loop, apply minus_one_le_succ,
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refine is_conn.elim -1 _ _, exact H
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end
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lemma is_trunc_of_is_trunc_loopn (m n : ℕ) (A : Type*) (H : is_trunc n (Ω[m] A))
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(H2 : is_conn (m.-1) A) : is_trunc (m + n) A :=
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begin
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revert A H H2; induction m with m IH: intro A H H2,
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{ rewrite [nat.zero_add], exact H },
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rewrite [succ_add],
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apply is_trunc_succ_succ_of_is_trunc_loop,
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{ apply IH,
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{ apply is_trunc_equiv_closed _ !loopn_succ_in },
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apply is_conn_loop },
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exact is_conn_of_le _ (zero_le_of_nat m)
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end
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lemma is_trunc_of_is_set_loopn (m : ℕ) (A : Type*) (H : is_set (Ω[m] A))
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(H2 : is_conn (m.-1) A) : is_trunc m A :=
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is_trunc_of_is_trunc_loopn m 0 A H H2
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definition is_trunc_of_trivial_homotopy {n : ℕ} {m : ℕ₋₂} {A : Type} (H : is_trunc m A)
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(H2 : Πk a, k > n → is_contr (π[k] (pointed.MK A a))) : is_trunc n A :=
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begin
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fapply is_trunc_is_equiv_closed_rev, { exact @tr n A },
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apply whitehead_principle m,
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{ apply is_equiv_trunc_functor_of_is_conn_fun,
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note z := is_conn_fun_tr n A,
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apply is_conn_fun_of_le _ (of_nat_le_of_nat (zero_le n)), },
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intro a k,
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apply @nat.lt_ge_by_cases n (k+1),
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{ intro H3, apply @is_equiv_of_is_contr, exact H2 _ _ H3,
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refine @trivial_homotopy_group_of_is_trunc _ _ _ _ H3, apply is_trunc_trunc },
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{ intro H3, apply @(is_equiv_π_of_is_connected _ H3), apply is_conn_fun_tr }
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end
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definition is_trunc_of_trivial_homotopy_pointed {n : ℕ} {m : ℕ₋₂} {A : Type*} (H : is_trunc m A)
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(Hconn : is_conn 0 A) (H2 : Πk, k > n → is_contr (π[k] A)) : is_trunc n A :=
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begin
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apply is_trunc_of_trivial_homotopy H,
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intro k a H3, revert a, apply is_conn.elim -1,
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cases A with A a₀, exact H2 k H3
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end
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definition is_trunc_of_is_trunc_succ {n : ℕ} {A : Type} (H : is_trunc (n.+1) A)
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(H2 : Πa, is_contr (π[n+1] (pointed.MK A a))) : is_trunc n A :=
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begin
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apply is_trunc_of_trivial_homotopy H,
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intro k a H3, induction H3 with k H3 IH, exact H2 a,
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apply @trivial_homotopy_group_of_is_trunc _ (n+1) _ H, exact succ_le_succ H3
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end
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definition is_trunc_of_is_trunc_succ_pointed {n : ℕ} {A : Type*} (H : is_trunc (n.+1) A)
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(Hconn : is_conn 0 A) (H2 : is_contr (π[n+1] A)) : is_trunc n A :=
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begin
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apply is_trunc_of_trivial_homotopy_pointed H Hconn,
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intro k H3, induction H3 with k H3 IH, exact H2,
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apply @trivial_homotopy_group_of_is_trunc _ (n+1) _ H, exact succ_le_succ H3
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end
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end is_trunc
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namespace sigma
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open sigma.ops
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definition sigma_functor2 [constructor] {A₁ A₂ A₃ : Type}
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{B₁ : A₁ → Type} {B₂ : A₂ → Type} {B₃ : A₃ → Type}
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(f : A₁ → A₂ → A₃) (g : Π⦃a₁ a₂⦄, B₁ a₁ → B₂ a₂ → B₃ (f a₁ a₂))
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(x₁ : Σa₁, B₁ a₁) (x₂ : Σa₂, B₂ a₂) : Σa₃, B₃ a₃ :=
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⟨f x₁.1 x₂.1, g x₁.2 x₂.2⟩
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definition eq.rec_sigma {A : Type} {B : A → Type} {a : A} {b : B a}
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(P : Π⦃a'⦄ {b' : B a'}, ⟨a, b⟩ = ⟨a', b'⟩ → Type)
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(IH : P idp) ⦃a' : A⦄ {b' : B a'} (p : ⟨a, b⟩ = ⟨a', b'⟩) : P p :=
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begin
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apply transport (λp, P p) (to_left_inv !sigma_eq_equiv p),
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generalize !sigma_eq_equiv p, esimp, intro q,
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induction q with q₁ q₂, induction q₂, exact IH
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end
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definition ap_dpair_eq_dpair_pr {A A' : Type} {B : A → Type} {a a' : A} {b : B a} {b' : B a'} (f : Πa, B a → A') (p : a = a') (q : b =[p] b')
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: ap (λx, f x.1 x.2) (dpair_eq_dpair p q) = apd011 f p q :=
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by induction q; reflexivity
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definition sigma_eq_equiv_of_is_prop_right [constructor] {A : Type} {B : A → Type} (u v : Σa, B a)
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[H : Π a, is_prop (B a)] : u = v ≃ u.1 = v.1 :=
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!sigma_eq_equiv ⬝e !sigma_equiv_of_is_contr_right
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definition ap_sigma_pr1 {A B : Type} {C : B → Type} {a₁ a₂ : A} (f : A → B) (g : Πa, C (f a))
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(p : a₁ = a₂) : (ap (λa, ⟨f a, g a⟩) p)..1 = ap f p :=
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by induction p; reflexivity
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definition ap_sigma_pr2 {A B : Type} {C : B → Type} {a₁ a₂ : A} (f : A → B) (g : Πa, C (f a))
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(p : a₁ = a₂) : (ap (λa, ⟨f a, g a⟩) p)..2 =
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change_path (ap_sigma_pr1 f g p)⁻¹ (pathover_ap C f (apd g p)) :=
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by induction p; reflexivity
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definition ap_sigma_functor_sigma_eq {A A' : Type} {B : A → Type} {B' : A' → Type}
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{a a' : A} {b : B a} {b' : B a'} (f : A → A') (g : Πa, B a → B' (f a)) (p : a = a') (q : b =[p] b') :
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ap (sigma_functor f g) (sigma_eq p q) = sigma_eq (ap f p) (pathover_ap B' f (apo g q)) :=
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by induction q; reflexivity
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definition ap_sigma_functor_id_sigma_eq {A : Type} {B B' : A → Type}
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{a a' : A} {b : B a} {b' : B a'} (g : Πa, B a → B' a) (p : a = a') (q : b =[p] b') :
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ap (sigma_functor id g) (sigma_eq p q) = sigma_eq p (apo g q) :=
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by induction q; reflexivity
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definition sigma_eq_pr2_constant {A B : Type} {a a' : A} {b b' : B} (p : a = a')
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(q : b =[p] b') : ap pr2 (sigma_eq p q) = (eq_of_pathover q) :=
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by induction q; reflexivity
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definition sigma_eq_pr2_constant2 {A B : Type} {a a' : A} {b b' : B} (p : a = a')
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(q : b = b') : ap pr2 (sigma_eq p (pathover_of_eq p q)) = q :=
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by induction p; induction q; reflexivity
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definition sigma_eq_concato_eq {A : Type} {B : A → Type} {a a' : A} {b : B a} {b₁ b₂ : B a'}
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(p : a = a') (q : b =[p] b₁) (q' : b₁ = b₂) : sigma_eq p (q ⬝op q') = sigma_eq p q ⬝ ap (dpair a') q' :=
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by induction q'; reflexivity
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-- open sigma.ops
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-- definition eq.rec_sigma {A : Type} {B : A → Type} {a₀ : A} {b₀ : B a₀}
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-- {P : Π(a : A) (b : B a), ⟨a₀, b₀⟩ = ⟨a, b⟩ → Type} (H : P a₀ b₀ idp) {a : A} {b : B a}
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@ -332,6 +123,60 @@ by induction q'; reflexivity
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-- end
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--rexact @(ap pathover_pr1) _ idpo _,
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definition sigma_functor2 [constructor] {A₁ A₂ A₃ : Type}
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{B₁ : A₁ → Type} {B₂ : A₂ → Type} {B₃ : A₃ → Type}
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(f : A₁ → A₂ → A₃) (g : Π⦃a₁ a₂⦄, B₁ a₁ → B₂ a₂ → B₃ (f a₁ a₂))
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(x₁ : Σa₁, B₁ a₁) (x₂ : Σa₂, B₂ a₂) : Σa₃, B₃ a₃ :=
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⟨f x₁.1 x₂.1, g x₁.2 x₂.2⟩
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definition eq.rec_sigma {A : Type} {B : A → Type} {a : A} {b : B a}
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(P : Π⦃a'⦄ {b' : B a'}, ⟨a, b⟩ = ⟨a', b'⟩ → Type)
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(IH : P idp) ⦃a' : A⦄ {b' : B a'} (p : ⟨a, b⟩ = ⟨a', b'⟩) : P p :=
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begin
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apply transport (λp, P p) (to_left_inv !sigma_eq_equiv p),
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generalize !sigma_eq_equiv p, esimp, intro q,
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induction q with q₁ q₂, induction q₂, exact IH
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end
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definition ap_dpair_eq_dpair_pr {A A' : Type} {B : A → Type} {a a' : A} {b : B a} {b' : B a'} (f : Πa, B a → A') (p : a = a') (q : b =[p] b')
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: ap (λx, f x.1 x.2) (dpair_eq_dpair p q) = apd011 f p q :=
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by induction q; reflexivity
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definition sigma_eq_equiv_of_is_prop_right [constructor] {A : Type} {B : A → Type} (u v : Σa, B a)
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[H : Π a, is_prop (B a)] : u = v ≃ u.1 = v.1 :=
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!sigma_eq_equiv ⬝e !sigma_equiv_of_is_contr_right
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definition ap_sigma_pr1 {A B : Type} {C : B → Type} {a₁ a₂ : A} (f : A → B) (g : Πa, C (f a))
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(p : a₁ = a₂) : (ap (λa, ⟨f a, g a⟩) p)..1 = ap f p :=
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by induction p; reflexivity
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definition ap_sigma_pr2 {A B : Type} {C : B → Type} {a₁ a₂ : A} (f : A → B) (g : Πa, C (f a))
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(p : a₁ = a₂) : (ap (λa, ⟨f a, g a⟩) p)..2 =
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change_path (ap_sigma_pr1 f g p)⁻¹ (pathover_ap C f (apd g p)) :=
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by induction p; reflexivity
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definition ap_sigma_functor_sigma_eq {A A' : Type} {B : A → Type} {B' : A' → Type}
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{a a' : A} {b : B a} {b' : B a'} (f : A → A') (g : Πa, B a → B' (f a)) (p : a = a') (q : b =[p] b') :
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ap (sigma_functor f g) (sigma_eq p q) = sigma_eq (ap f p) (pathover_ap B' f (apo g q)) :=
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by induction q; reflexivity
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definition ap_sigma_functor_id_sigma_eq {A : Type} {B B' : A → Type}
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{a a' : A} {b : B a} {b' : B a'} (g : Πa, B a → B' a) (p : a = a') (q : b =[p] b') :
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ap (sigma_functor id g) (sigma_eq p q) = sigma_eq p (apo g q) :=
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by induction q; reflexivity
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definition sigma_eq_pr2_constant {A B : Type} {a a' : A} {b b' : B} (p : a = a')
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(q : b =[p] b') : ap pr2 (sigma_eq p q) = (eq_of_pathover q) :=
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by induction q; reflexivity
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definition sigma_eq_pr2_constant2 {A B : Type} {a a' : A} {b b' : B} (p : a = a')
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(q : b = b') : ap pr2 (sigma_eq p (pathover_of_eq p q)) = q :=
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by induction p; induction q; reflexivity
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definition sigma_eq_concato_eq {A : Type} {B : A → Type} {a a' : A} {b : B a} {b₁ b₂ : B a'}
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(p : a = a') (q : b =[p] b₁) (q' : b₁ = b₂) : sigma_eq p (q ⬝op q') = sigma_eq p q ⬝ ap (dpair a') q' :=
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by induction q'; reflexivity
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definition sigma_functor_compose {A A' A'' : Type} {B : A → Type} {B' : A' → Type}
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{B'' : A'' → Type} {f' : A' → A''} {f : A → A'} (g' : Πa, B' a → B'' (f' a))
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(g : Πa, B a → B' (f a)) (x : Σa, B a) :
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@ -359,26 +204,26 @@ by induction q'; reflexivity
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sigma_functor_homotopy h k x ⬝
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(sigma_functor_compose g₁₂ g₀₁ x)⁻¹
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definition sigma_equiv_of_is_embedding_left_fun [constructor] {X Y : Type} {P : Y → Type}
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{f : X → Y} (H : Πy, P y → fiber f y) (v : Σy, P y) : Σx, P (f x) :=
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⟨fiber.point (H v.1 v.2), transport P (point_eq (H v.1 v.2))⁻¹ v.2⟩
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definition sigma_equiv_of_is_embedding_left_fun [constructor] {X Y : Type} {P : Y → Type}
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{f : X → Y} (H : Πy, P y → fiber f y) (v : Σy, P y) : Σx, P (f x) :=
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⟨fiber.point (H v.1 v.2), transport P (point_eq (H v.1 v.2))⁻¹ v.2⟩
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definition sigma_equiv_of_is_embedding_left_prop [constructor] {X Y : Type} {P : Y → Type}
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(f : X → Y) (Hf : is_embedding f) (HP : Πx, is_prop (P (f x))) (H : Πy, P y → fiber f y) :
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(Σy, P y) ≃ Σx, P (f x) :=
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begin
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apply equiv.MK (sigma_equiv_of_is_embedding_left_fun H) (sigma_functor f (λa, id)),
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{ intro v, induction v with x p, esimp [sigma_equiv_of_is_embedding_left_fun],
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fapply sigma_eq, apply @is_injective_of_is_embedding _ _ f, exact point_eq (H (f x) p),
|
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apply is_prop.elimo },
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{ intro v, induction v with y p, esimp, fapply sigma_eq, exact point_eq (H y p),
|
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apply tr_pathover }
|
||||
end
|
||||
definition sigma_equiv_of_is_embedding_left_prop [constructor] {X Y : Type} {P : Y → Type}
|
||||
(f : X → Y) (Hf : is_embedding f) (HP : Πx, is_prop (P (f x))) (H : Πy, P y → fiber f y) :
|
||||
(Σy, P y) ≃ Σx, P (f x) :=
|
||||
begin
|
||||
apply equiv.MK (sigma_equiv_of_is_embedding_left_fun H) (sigma_functor f (λa, id)),
|
||||
{ intro v, induction v with x p, esimp [sigma_equiv_of_is_embedding_left_fun],
|
||||
fapply sigma_eq, apply @is_injective_of_is_embedding _ _ f, exact point_eq (H (f x) p),
|
||||
apply is_prop.elimo },
|
||||
{ intro v, induction v with y p, esimp, fapply sigma_eq, exact point_eq (H y p),
|
||||
apply tr_pathover }
|
||||
end
|
||||
|
||||
definition sigma_equiv_of_is_embedding_left_contr [constructor] {X Y : Type} {P : Y → Type}
|
||||
(f : X → Y) (Hf : is_embedding f) (HP : Πx, is_contr (P (f x))) (H : Πy, P y → fiber f y) :
|
||||
(Σy, P y) ≃ X :=
|
||||
sigma_equiv_of_is_embedding_left_prop f Hf _ H ⬝e !sigma_equiv_of_is_contr_right
|
||||
definition sigma_equiv_of_is_embedding_left_contr [constructor] {X Y : Type} {P : Y → Type}
|
||||
(f : X → Y) (Hf : is_embedding f) (HP : Πx, is_contr (P (f x))) (H : Πy, P y → fiber f y) :
|
||||
(Σy, P y) ≃ X :=
|
||||
sigma_equiv_of_is_embedding_left_prop f Hf _ H ⬝e !sigma_equiv_of_is_contr_right
|
||||
|
||||
end sigma open sigma
|
||||
|
||||
|
|
|
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Loading…
Reference in a new issue