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make pointed suspensions, wedges and spheres the default (in contrast to the unpointed ones), remove sphere_index

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All HITs which automatically have a point are pointed without a 'p' in front. HITs which do not automatically have a point do still have a p (e.g. pushout/ppushout).

There were a lot of annoyances with spheres being indexed by N_{-1} with almost no extra generality. We now index the spheres by nat, making sphere 0 = pbool.
This commit is contained in:
Floris van Doorn 2017-07-20 15:01:40 +01:00
parent a02ea6b751
commit ddef24223b
17 changed files with 334 additions and 641 deletions
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12
hott/homotopy/EM.hlean
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@ -226,10 +226,10 @@ namespace EM
/- K(G, n+1) -/
definition EMadd1 : ℕ → Type*
| 0 := EM1 G
| (n+1) := ptrunc (n+2) (psusp (EMadd1 n))
| (n+1) := ptrunc (n+2) (susp (EMadd1 n))
definition EMadd1_succ [unfold_full] (n : ℕ) :
EMadd1 G (succ n) = ptrunc (n.+2) (psusp (EMadd1 G n)) :=
EMadd1 G (succ n) = ptrunc (n.+2) (susp (EMadd1 G n)) :=
idp
definition loop_EM2 : Ω[1] (EMadd1 G 1) ≃* EM1 G :=
@ -239,7 +239,7 @@ namespace EM
begin
induction n with n IH,
{ apply is_conn_EM1 },
{ rewrite EMadd1_succ, esimp, exact _ }
{ rewrite EMadd1_succ, exact _ }
end
definition is_trunc_EMadd1 [instance] (n : ℕ) : is_trunc (n+1) (EMadd1 G n) :=
@ -304,12 +304,12 @@ namespace EM
{ exact EM1_pmap e⁻¹ᵉ* (equiv.inv_preserve_binary e concat mul r) },
rewrite [EMadd1_succ],
exact ptrunc.elim ((succ n).+1)
(psusp.elim (f _ (EM_up e) (is_homomorphism_EM_up e r) _ _)),
(susp_elim (f _ (EM_up e) (is_homomorphism_EM_up e r) _ _)),
end
definition EMadd1_pmap_succ {G : AbGroup} {X : Type*} (n : ℕ) (e : Ω[succ (succ n)] X ≃* G)
r [H1 : is_conn (succ n) X] [H2 : is_trunc ((succ n).+1) X] : EMadd1_pmap (succ n) e r =
ptrunc.elim ((succ n).+1) (psusp.elim (EMadd1_pmap n (EM_up e) (is_homomorphism_EM_up e r))) :=
ptrunc.elim ((succ n).+1) (susp_elim (EMadd1_pmap n (EM_up e) (is_homomorphism_EM_up e r))) :=
by reflexivity
definition loop_EMadd1_pmap {G : AbGroup} {X : Type*} {n : ℕ} (e : Ω[succ (succ n)] X ≃* G)
@ -454,7 +454,7 @@ namespace EM
begin
induction n with n ψ,
{ exact EM1_functor φ },
{ apply ptrunc_functor, apply psusp_functor, exact ψ }
{ apply ptrunc_functor, apply susp_functor, exact ψ }
end
definition EM_functor [unfold 4] {G H : AbGroup} (φ : G →g H) (n : ℕ) :

2
hott/homotopy/cellcomplex.hlean
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@ -5,7 +5,7 @@ Authors: Ulrik Buchholtz
-/
import types.trunc homotopy.sphere hit.pushout
open eq is_trunc is_equiv nat equiv trunc prod pushout sigma sphere_index unit
open eq is_trunc is_equiv nat equiv trunc prod pushout sigma unit pointed
-- where should this be?
definition family : Type := ΣX, X → Type

11
hott/homotopy/circle.hlean
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@ -10,7 +10,7 @@ import .sphere
import types.int.hott
import algebra.homotopy_group .connectedness
open eq susp bool sphere_index is_equiv equiv is_trunc is_conn pi algebra pointed
open eq susp bool is_equiv equiv is_trunc is_conn pi algebra pointed
definition circle : Type₀ := sphere 1
@ -18,8 +18,8 @@ namespace circle
notation `S¹` := circle
definition base1 : S¹ := !north
definition base2 : S¹ := !south
definition seg1 : base1 = base2 := merid !north
definition seg2 : base1 = base2 := merid !south
definition seg1 : base1 = base2 := merid ff
definition seg2 : base1 = base2 := merid tt
definition base : S¹ := base1
definition loop : base = base := seg2 ⬝ seg1⁻¹
@ -32,8 +32,7 @@ namespace circle
{ exact Pb2 },
{ esimp at *, induction b with y,
{ exact Ps1 },
{ exact Ps2},
{ cases y}},
{ exact Ps2 }},
end
definition rec2_on [reducible] {P : S¹ → Type} (x : S¹) (Pb1 : P base1) (Pb2 : P base2)
@ -335,7 +334,7 @@ namespace circle
end
proposition is_conn_circle [instance] : is_conn 0 S¹ :=
sphere.is_conn_sphere -1.+2
sphere.is_conn_sphere 1
definition is_conn_pcircle [instance] : is_conn 0 S¹* := !is_conn_circle
definition is_trunc_pcircle [instance] : is_trunc 1 S¹* := !is_trunc_circle

2
hott/homotopy/cofiber.hlean
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@ -92,7 +92,7 @@ namespace cofiber
{ exact !con_inv_cancel_left⁻¹ ⬝ idp ◾ (!ap_inv⁻¹ ◾ idp) }
end
definition pcofiber_punit (A : Type*) : pcofiber (pconst A punit) ≃* psusp A :=
definition pcofiber_punit (A : Type*) : pcofiber (pconst A punit) ≃* susp A :=
begin
fapply pequiv_of_pmap,
{ fconstructor, intro x, induction x, exact north, exact south, exact merid x,

17
hott/homotopy/complex_hopf.hlean
View file

@ -10,6 +10,7 @@ The H-space structure on S¹ and the complex Hopf fibration
import .hopf .circle types.fin
open eq equiv is_equiv circle is_conn trunc is_trunc sphere susp pointed fiber sphere.ops function
join
namespace hopf
@ -28,23 +29,21 @@ namespace hopf
{ apply is_prop.elimo, apply is_trunc_square }}
end
open sphere_index
definition complex_hopf : S 3 → S 2 :=
definition complex_hopf' : S 3 → S 2 :=
begin
intro x, apply @sigma.pr1 (susp S¹) (hopf S¹),
apply inv (hopf.total S¹), apply inv (join.spheres 1 1), exact x
apply inv (hopf.total S¹), exact (join_sphere 1 1)⁻¹ᵉ x
end
definition complex_phopf [constructor] : S* 3 →* S* 2 :=
proof pmap.mk complex_hopf idp qed
definition complex_hopf [constructor] : S 3 →* S 2 :=
proof pmap.mk complex_hopf' idp qed
definition pfiber_complex_phopf : pfiber complex_phopf ≃* S* 1 :=
definition pfiber_complex_hopf : pfiber complex_hopf ≃* S 1 :=
begin
fapply pequiv_of_equiv,
{ esimp, unfold [complex_hopf],
{ esimp, unfold [complex_hopf'],
refine fiber.equiv_precompose (sigma.pr1 ∘ (hopf.total S¹)⁻¹ᵉ)
(join.spheres (of_nat 1) (of_nat 1))⁻¹ᵉ _ ⬝e _,
(join_sphere 1 1)⁻¹ᵉ _ ⬝e _,
refine fiber.equiv_precompose _ (hopf.total S¹)⁻¹ᵉ _ ⬝e _,
apply fiber_pr1 },
{ reflexivity }

3
hott/homotopy/connectedness.hlean
View file

@ -269,6 +269,9 @@ namespace is_conn
definition is_conn_minus_one (A : Type) (a : ∥ A ∥) : is_conn -1 A :=
is_contr.mk a (is_prop.elim _)
definition is_conn_minus_one_pointed [instance] (A : Type*) : is_conn -1 A :=
is_conn_minus_one A (tr pt)
definition is_conn_trunc [instance] (A : Type) (n k : ℕ₋₂) [H : is_conn n A]
: is_conn n (trunc k A) :=
begin

92
hott/homotopy/freudenthal.hlean
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@ -17,12 +17,12 @@ namespace freudenthal section
/-
This proof is ported from Agda
This is the 95% version of the Freudenthal Suspension Theorem, which means that we don't
prove that loop_psusp_unit : A →* Ω(psusp A) is 2n-connected (if A is n-connected),
prove that loop_susp_unit : A →* Ω(susp A) is 2n-connected (if A is n-connected),
but instead we only prove that it induces an equivalence on the first 2n homotopy groups.
-/
private definition up (a : A) : north = north :> susp A :=
loop_psusp_unit A a
loop_susp_unit A a
definition code_merid : A → ptrunc (n + n) A → ptrunc (n + n) A :=
begin
@ -127,7 +127,7 @@ namespace freudenthal section
theorem decode_coh (a : A) : decode_north =[merid a] decode_south :=
begin
apply arrow_pathover_left, intro c, esimp at *,
apply arrow_pathover_left, intro c,
induction c with a',
rewrite [↑code, elim_type_merid],
refine @wedge_extension.ext _ _ n n _ _ (λ a a', tr (up a') =[merid a] decode_south
@ -155,38 +155,38 @@ namespace freudenthal section
end
parameters (A n)
definition equiv' : trunc (n + n) A ≃ trunc (n + n) (Ω (psusp A)) :=
definition equiv' : trunc (n + n) A ≃ trunc (n + n) (Ω (susp A)) :=
equiv.MK decode_north encode decode_encode encode_decode_north
definition pequiv' : ptrunc (n + n) A ≃* ptrunc (n + n) (Ω (psusp A)) :=
definition pequiv' : ptrunc (n + n) A ≃* ptrunc (n + n) (Ω (susp A)) :=
pequiv_of_equiv equiv' decode_north_pt
-- We don't prove this:
-- theorem freudenthal_suspension : is_conn_fun (n+n) (loop_psusp_unit A) := sorry
-- theorem freudenthal_suspension : is_conn_fun (n+n) (loop_susp_unit A) := sorry
end end freudenthal
open algebra group
definition freudenthal_pequiv (A : Type*) {n k : ℕ} [is_conn n A] (H : k ≤ 2 * n)
: ptrunc k A ≃* ptrunc k (Ω (psusp A)) :=
: ptrunc k A ≃* ptrunc k (Ω (susp A)) :=
have H' : k ≤[ℕ₋₂] n + n,
by rewrite [mul.comm at H, -algebra.zero_add n at {1}]; exact of_nat_le_of_nat H,
ptrunc_pequiv_ptrunc_of_le H' (freudenthal.pequiv' A n)
definition freudenthal_equiv {A : Type*} {n k : ℕ} [is_conn n A] (H : k ≤ 2 * n)
: trunc k A ≃ trunc k (Ω (psusp A)) :=
: trunc k A ≃ trunc k (Ω (susp A)) :=
freudenthal_pequiv A H
definition freudenthal_homotopy_group_pequiv (A : Type*) {n k : ℕ} [is_conn n A] (H : k ≤ 2 * n)
: π[k + 1] (psusp A) ≃* π[k] A :=
: π[k + 1] (susp A) ≃* π[k] A :=
calc
π[k + 1] (psusp A) ≃* π[k] (Ω (psusp A)) : homotopy_group_succ_in (psusp A) k
... ≃* Ω[k] (ptrunc k (Ω (psusp A))) : homotopy_group_pequiv_loop_ptrunc k (Ω (psusp A))
π[k + 1] (susp A) ≃* π[k] (Ω (susp A)) : homotopy_group_succ_in (susp A) k
... ≃* Ω[k] (ptrunc k (Ω (susp A))) : homotopy_group_pequiv_loop_ptrunc k (Ω (susp A))
... ≃* Ω[k] (ptrunc k A) : loopn_pequiv_loopn k (freudenthal_pequiv A H)
... ≃* π[k] A : (homotopy_group_pequiv_loop_ptrunc k A)⁻¹ᵉ*
definition freudenthal_homotopy_group_isomorphism (A : Type*) {n k : ℕ} [is_conn n A]
(H : k + 1 ≤ 2 * n) : πg[k+2] (psusp A) ≃g πg[k + 1] A :=
(H : k + 1 ≤ 2 * n) : πg[k+2] (susp A) ≃g πg[k + 1] A :=
begin
fapply isomorphism_of_equiv,
{ exact equiv_of_pequiv (freudenthal_homotopy_group_pequiv A H)},
@ -199,7 +199,7 @@ begin
end
definition to_pmap_freudenthal_pequiv {A : Type*} (n k : ℕ) [is_conn n A] (H : k ≤ 2 * n)
: freudenthal_pequiv A H ~* ptrunc_functor k (loop_psusp_unit A) :=
: freudenthal_pequiv A H ~* ptrunc_functor k (loop_susp_unit A) :=
begin
fapply phomotopy.mk,
{ intro x, induction x with a, reflexivity },
@ -208,24 +208,24 @@ end
definition ptrunc_elim_freudenthal_pequiv {A B : Type*} (n k : ℕ) [is_conn n A] (H : k ≤ 2 * n)
(f : A →* Ω B) [is_trunc (k.+1) (B)] :
ptrunc.elim k (Ω→ (psusp.elim f)) ∘* freudenthal_pequiv A H ~* ptrunc.elim k f :=
ptrunc.elim k (Ω→ (susp_elim f)) ∘* freudenthal_pequiv A H ~* ptrunc.elim k f :=
begin
refine pwhisker_left _ !to_pmap_freudenthal_pequiv ⬝* _,
refine !ptrunc_elim_ptrunc_functor ⬝* _,
exact ptrunc_elim_phomotopy k !ap1_psusp_elim,
exact ptrunc_elim_phomotopy k !ap1_susp_elim,
end
namespace susp
definition iterate_psusp_stability_pequiv (A : Type*) {k n : ℕ} [is_conn 0 A]
(H : k ≤ 2 * n) : π[k + 1] (iterate_psusp (n + 1) A) ≃* π[k] (iterate_psusp n A) :=
have is_conn n (iterate_psusp n A), by rewrite [-zero_add n]; exact _,
freudenthal_homotopy_group_pequiv (iterate_psusp n A) H
definition iterate_susp_stability_pequiv_of_is_conn_0 (A : Type*) {k n : ℕ} [is_conn 0 A]
(H : k ≤ 2 * n) : π[k + 1] (iterate_susp (n + 1) A) ≃* π[k] (iterate_susp n A) :=
have is_conn n (iterate_susp n A), by rewrite [-zero_add n]; exact _,
freudenthal_homotopy_group_pequiv (iterate_susp n A) H
definition iterate_psusp_stability_isomorphism (A : Type*) {k n : ℕ} [is_conn 0 A]
(H : k + 1 ≤ 2 * n) : πg[k+2] (iterate_psusp (n + 1) A) ≃g πg[k+1] (iterate_psusp n A) :=
have is_conn n (iterate_psusp n A), by rewrite [-zero_add n]; exact _,
freudenthal_homotopy_group_isomorphism (iterate_psusp n A) H
definition iterate_susp_stability_isomorphism_of_is_conn_0 (A : Type*) {k n : ℕ} [is_conn 0 A]
(H : k + 1 ≤ 2 * n) : πg[k+2] (iterate_susp (n + 1) A) ≃g πg[k+1] (iterate_susp n A) :=
have is_conn n (iterate_susp n A), by rewrite [-zero_add n]; exact _,
freudenthal_homotopy_group_isomorphism (iterate_susp n A) H
definition stability_helper1 {k n : ℕ} (H : k + 2 ≤ 2 * n) : k ≤ 2 * pred n :=
begin
@ -233,34 +233,28 @@ namespace susp
apply pred_le_pred, apply pred_le_pred, exact H
end
definition stability_helper2 (A : Type) {k n : ℕ} (H : k + 2 ≤ 2 * n) :
is_conn (pred n) (iterate_susp (n + 1) A) :=
have Π(n : ℕ), n = -2 + (succ n + 1),
definition stability_helper2 (A : Type*) {k n : ℕ} (H : k + 2 ≤ 2 * n) :
is_conn (pred n) (iterate_susp n A) :=
have Π(n : ℕ), n = -1 + (n + 1),
begin intro n, induction n with n IH, reflexivity, exact ap succ IH end,
begin
cases n with n,
{ exfalso, exact not_succ_le_zero _ H },
{ esimp, rewrite [this n], apply is_conn_iterate_susp}
{ esimp, rewrite [this n], exact is_conn_iterate_susp -1 _ A }
end
definition iterate_susp_stability_pequiv (A : Type) {k n : ℕ}
(H : k + 2 ≤ 2 * n) : π[k + 1] (pointed.MK (iterate_susp (n + 2) A) !north) ≃*
π[k ] (pointed.MK (iterate_susp (n + 1) A) !north) :=
have is_conn (pred n) (carrier (pointed.MK (iterate_susp (n + 1) A) !north)), from
stability_helper2 A H,
freudenthal_homotopy_group_pequiv (pointed.MK (iterate_susp (n + 1) A) !north)
(stability_helper1 H)
definition iterate_susp_stability_pequiv (A : Type*) {k n : ℕ}
(H : k + 2 ≤ 2 * n) : π[k + 1] (iterate_susp (n + 1) A) ≃* π[k] (iterate_susp n A) :=
have is_conn (pred n) (iterate_susp n A), from stability_helper2 A H,
freudenthal_homotopy_group_pequiv (iterate_susp n A) (stability_helper1 H)
definition iterate_susp_stability_isomorphism (A : Type) {k n : ℕ}
(H : k + 3 ≤ 2 * n) : πg[k+1 +1] (pointed.MK (iterate_susp (n + 2) A) !north) ≃g
πg[k+1] (pointed.MK (iterate_susp (n + 1) A) !north) :=
have is_conn (pred n) (carrier (pointed.MK (iterate_susp (n + 1) A) !north)), from
@stability_helper2 A (k+1) n H,
freudenthal_homotopy_group_isomorphism (pointed.MK (iterate_susp (n + 1) A) !north)
(stability_helper1 H)
definition iterate_susp_stability_isomorphism (A : Type*) {k n : ℕ}
(H : k + 3 ≤ 2 * n) : πg[k+2] (iterate_susp (n + 1) A) ≃g πg[k+1] (iterate_susp n A) :=
have is_conn (pred n) (iterate_susp n A), from @stability_helper2 A (k+1) n H,
freudenthal_homotopy_group_isomorphism (iterate_susp n A) (stability_helper1 H)
definition iterated_freudenthal_pequiv (A : Type*) {n k m : ℕ} [HA : is_conn n A] (H : k ≤ 2 * n)
: ptrunc k A ≃* ptrunc k (Ω[m] (iterate_psusp m A)) :=
: ptrunc k A ≃* ptrunc k (Ω[m] (iterate_susp m A)) :=
begin
revert A n k HA H, induction m with m IH: intro A n k HA H,
{ reflexivity },
@ -269,13 +263,13 @@ namespace susp
succ k ≤ succ (2 * n) : succ_le_succ H
... ≤ 2 * succ n : self_le_succ,
exact calc
ptrunc k A ≃* ptrunc k (Ω (psusp A)) : freudenthal_pequiv A H
... ≃* Ω (ptrunc (succ k) (psusp A)) : loop_ptrunc_pequiv
... ≃* Ω (ptrunc (succ k) (Ω[m] (iterate_psusp m (psusp A)))) :
loop_pequiv_loop (IH (psusp A) (succ n) (succ k) _ H2)
... ≃* ptrunc k (Ω[succ m] (iterate_psusp m (psusp A))) : loop_ptrunc_pequiv
... ≃* ptrunc k (Ω[succ m] (iterate_psusp (succ m) A)) :
ptrunc_pequiv_ptrunc _ (loopn_pequiv_loopn _ !iterate_psusp_succ_in)}
ptrunc k A ≃* ptrunc k (Ω (susp A)) : freudenthal_pequiv A H
... ≃* Ω (ptrunc (succ k) (susp A)) : loop_ptrunc_pequiv
... ≃* Ω (ptrunc (succ k) (Ω[m] (iterate_susp m (susp A)))) :
loop_pequiv_loop (IH (susp A) (succ n) (succ k) _ H2)
... ≃* ptrunc k (Ω[succ m] (iterate_susp m (susp A))) : loop_ptrunc_pequiv
... ≃* ptrunc k (Ω[succ m] (iterate_susp (succ m) A)) :
ptrunc_pequiv_ptrunc _ (loopn_pequiv_loopn _ !iterate_susp_succ_in)}
end
end susp

6
hott/homotopy/homotopy_group.hlean
View file

@ -38,16 +38,14 @@ namespace is_trunc
end
-- Corollary 8.3.3
section
open sphere sphere.ops sphere_index
theorem homotopy_group_sphere_le (n k : ℕ) (H : k < n) : is_contr (π[k] (S* n)) :=
open sphere sphere.ops
theorem homotopy_group_sphere_le (n k : ℕ) (H : k < n) : is_contr (π[k] (S n)) :=
begin
cases n with n,
{ exfalso, apply not_lt_zero, exact H},
{ have H2 : k ≤ n, from le_of_lt_succ H,
apply @(trivial_homotopy_group_of_is_conn _ H2) }
end
end
theorem is_contr_HG_fiber_of_is_connected {A B : Type*} (k n : ℕ) (f : A →* B)
[H : is_conn_fun n f] (H2 : k ≤ n) : is_contr (π[k] (pfiber f)) :=

16
hott/homotopy/hopf.hlean
View file

@ -8,7 +8,7 @@ H-spaces and the Hopf construction
import types.equiv .wedge .join
open eq eq.ops equiv is_equiv is_conn is_trunc trunc susp join
open eq eq.ops equiv is_equiv is_conn is_trunc trunc susp join pointed
namespace hopf
@ -186,15 +186,15 @@ section
(equiv.MK decode' encode decode_encode encode_decode')⁻¹ᵉ
definition main_lemma_point
: ptrunc 1 (Ω(psusp A)) ≃* pointed.MK A 1 :=
: ptrunc 1 (Ω(susp A)) ≃* pointed.MK A 1 :=
pointed.pequiv_of_equiv main_lemma idp
protected definition delooping : Ω (ptrunc 2 (psusp A)) ≃* pointed.MK A 1 :=
loop_ptrunc_pequiv 1 (psusp A) ⬝e* main_lemma_point
protected definition delooping : Ω (ptrunc 2 (susp A)) ≃* pointed.MK A 1 :=
loop_ptrunc_pequiv 1 (susp A) ⬝e* main_lemma_point
/- characterization of the underlying pointed maps -/
definition to_pmap_main_lemma_point_pinv
: main_lemma_point⁻¹ᵉ* ~* !ptr ∘* loop_psusp_unit (pointed.MK A 1) :=
: main_lemma_point⁻¹ᵉ* ~* !ptr ∘* loop_susp_unit (pointed.MK A 1) :=
begin
fapply phomotopy.mk,
{ intro a, reflexivity },
@ -202,7 +202,7 @@ section
end
definition to_pmap_delooping_pinv :
delooping⁻¹ᵉ* ~* Ω→ !ptr ∘* loop_psusp_unit (pointed.MK A 1) :=
delooping⁻¹ᵉ* ~* Ω→ !ptr ∘* loop_susp_unit (pointed.MK A 1) :=
begin
refine !trans_pinv ⬝* _,
refine pwhisker_left _ !to_pmap_main_lemma_point_pinv ⬝* _,
@ -211,12 +211,12 @@ section
end
definition hopf_delooping_elim {B : Type*} (f : pointed.MK A 1 →* Ω B) [H2 : is_trunc 2 B] :
Ω→(ptrunc.elim 2 (psusp.elim f)) ∘* (hopf.delooping A coh)⁻¹ᵉ* ~* f :=
Ω→(ptrunc.elim 2 (susp_elim f)) ∘* (hopf.delooping A coh)⁻¹ᵉ* ~* f :=
begin
refine pwhisker_left _ !to_pmap_delooping_pinv ⬝* _,
refine !passoc⁻¹* ⬝* _,
refine pwhisker_right _ (!ap1_pcompose⁻¹* ⬝* ap1_phomotopy !ptrunc_elim_ptr) ⬝* _,
apply ap1_psusp_elim
apply ap1_susp_elim
end
end

2
hott/homotopy/imaginaroid.hlean
View file

@ -8,7 +8,7 @@ Cayley-Dickson construction via imaginaroids
import algebra.group cubical.square types.pi .hopf
open eq eq.ops equiv susp hopf
open eq eq.ops equiv susp hopf pointed
open [notation] sum
namespace imaginaroid

2
hott/homotopy/interval.hlean
View file

@ -7,7 +7,7 @@ Declaration of the interval
-/
import .susp types.eq types.prod cubical.square
open eq susp unit equiv is_trunc nat prod
open eq susp unit equiv is_trunc nat prod pointed
definition interval : Type₀ := susp unit

74
hott/homotopy/join.hlean
View file

@ -57,11 +57,11 @@ namespace join
protected definition hsquare {a a' : A} {b b' : B} (p : a = a') (q : b = b') :
square (ap inl p) (ap inr q) (glue a b) (glue a' b') :=
eq.rec_on p (eq.rec_on q hrfl)
by induction p; induction q; exact hrfl
protected definition vsquare {a a' : A} {b b' : B} (p : a = a') (q : b = b') :
square (glue a b) (glue a' b') (ap inl p) (ap inr q) :=
eq.rec_on p (eq.rec_on q vrfl)
by induction p; induction q; exact vrfl
end
@ -120,7 +120,7 @@ end join
namespace join
variables {A₁ A₂ B₁ B₂ : Type}
protected definition functor [reducible]
definition join_functor [reducible]
(f : A₁ → A₂) (g : B₁ → B₂) : join A₁ B₁ → join A₂ B₂ :=
begin
intro x, induction x with a b a b,
@ -132,12 +132,12 @@ namespace join
: join.diamond a a' b b' → join.diamond (f a) (f a') (g b) (g b') :=
begin
unfold join.diamond, intro s,
note s' := aps (join.functor f g) s,
note s' := aps (join_functor f g) s,
do 2 rewrite eq.ap_inv at s',
do 4 rewrite join.elim_glue at s', exact s'
end
protected definition equiv_closed
definition join_equiv_join
: A₁ ≃ A₂ → B₁ ≃ B₂ → join A₁ B₁ ≃ join A₂ B₂ :=
begin
intros H K,
@ -170,7 +170,7 @@ namespace join
cases p, apply pathover_idp_of_eq, apply join.symm_diamond
end
protected definition empty (A : Type) : join empty A ≃ A :=
definition join_empty (A : Type) : join empty A ≃ A :=
begin
fapply equiv.MK,
{ intro x, induction x with z a z a,
@ -185,7 +185,7 @@ namespace join
{ induction z } }
end
protected definition bool (A : Type) : join bool A ≃ susp A :=
definition join_bool (A : Type) : join bool A ≃ susp A :=
begin
fapply equiv.MK,
{ intro ba, induction ba with [b, a, b, a],
@ -229,7 +229,7 @@ end join
namespace join
variables (A B C : Type)
protected definition is_contr [HA : is_contr A] :
definition is_contr_join [HA : is_contr A] :
is_contr (join A B) :=
begin
fapply is_contr.mk, exact inl (center A),
@ -239,24 +239,24 @@ namespace join
generalize center_eq a, intro p, cases p, apply idp_con,
end
protected definition swap : join A B → join B A :=
definition join_swap : join A B → join B A :=
begin
intro x, induction x with a b a b, exact inr a, exact inl b,
apply !glue⁻¹
end
protected definition swap_involutive (x : join A B) :
join.swap B A (join.swap A B x) = x :=
definition join_swap_involutive (x : join A B) :
join_swap B A (join_swap A B x) = x :=
begin
induction x with a b a b, do 2 reflexivity,
apply eq_pathover, rewrite ap_id,
apply hdeg_square, esimp[join.swap],
apply hdeg_square,
apply concat, apply ap_compose' (join.elim _ _ _),
krewrite [join.elim_glue, ap_inv, join.elim_glue], apply inv_inv,
end
protected definition symm : join A B ≃ join B A :=
by fapply equiv.MK; do 2 apply join.swap; do 2 apply join.swap_involutive
definition join_symm : join A B ≃ join B A :=
by fapply equiv.MK; do 2 apply join_swap; do 2 apply join_swap_involutive
end join
@ -482,49 +482,49 @@ section join_switch
end join_switch
protected definition switch_equiv (A B C : Type) : join (join A B) C ≃ join (join C B) A :=
definition join_switch_equiv (A B C : Type) : join (join A B) C ≃ join (join C B) A :=
by apply equiv.MK; do 2 apply join.switch_involutive
protected definition assoc (A B C : Type) : join (join A B) C ≃ join A (join B C) :=
calc join (join A B) C ≃ join (join C B) A : join.switch_equiv
... ≃ join A (join C B) : join.symm
... ≃ join A (join B C) : join.equiv_closed erfl (join.symm C B)
definition join_assoc (A B C : Type) : join (join A B) C ≃ join A (join B C) :=
calc join (join A B) C ≃ join (join C B) A : join_switch_equiv
... ≃ join A (join C B) : join_symm
... ≃ join A (join B C) : join_equiv_join erfl (join_symm C B)
protected definition ap_assoc_inv_glue_inl {A B : Type} (C : Type) (a : A) (b : B)
: ap (to_inv (join.assoc A B C)) (glue a (inl b)) = ap inl (glue a b) :=
definition ap_join_assoc_inv_glue_inl {A B : Type} (C : Type) (a : A) (b : B)
: ap (to_inv (join_assoc A B C)) (glue a (inl b)) = ap inl (glue a b) :=
begin
unfold join.assoc, rewrite ap_compose, krewrite join.elim_glue,
unfold join_assoc, rewrite ap_compose, krewrite join.elim_glue,
rewrite ap_compose, krewrite join.elim_glue, rewrite ap_inv, krewrite join.elim_glue,
unfold switch_coh, unfold join.symm, unfold join.swap, esimp, rewrite eq.inv_inv
unfold switch_coh, unfold join_symm, unfold join_swap, esimp, rewrite inv_inv
end
protected definition ap_assoc_inv_glue_inr {A C : Type} (B : Type) (a : A) (c : C)
: ap (to_inv (join.assoc A B C)) (glue a (inr c)) = glue (inl a) c :=
: ap (to_inv (join_assoc A B C)) (glue a (inr c)) = glue (inl a) c :=
begin
unfold join.assoc, rewrite ap_compose, krewrite join.elim_glue,
unfold join_assoc, rewrite ap_compose, krewrite join.elim_glue,
rewrite ap_compose, krewrite join.elim_glue, rewrite ap_inv, krewrite join.elim_glue,
unfold switch_coh, unfold join.symm, unfold join.swap, esimp, rewrite eq.inv_inv
unfold switch_coh, unfold join_symm, unfold join_swap, esimp, rewrite inv_inv
end
end join
namespace join
open sphere sphere_index sphere.ops
protected definition spheres (n m : ℕ₋₁) : join (S n) (S m) ≃ S (n+1+m) :=
open sphere sphere.ops
definition join_sphere (n m : ℕ) : join (S n) (S m) ≃ S (n+m+1) :=
begin
apply equiv.trans (join.symm (S n) (S m)),
refine join_symm (S n) (S m) ⬝e _,
induction m with m IH,
{ exact join.empty (S n) },
{ calc join (S m.+1) (S n)
{ exact join_bool (S n) },
{ calc join (S (m+1)) (S n)
≃ join (join bool (S m)) (S n)
: join.equiv_closed (equiv.symm (join.bool (S m))) erfl
: join_equiv_join (join_bool (S m))⁻¹ᵉ erfl
... ≃ join bool (join (S m) (S n))
: join.assoc
... ≃ join bool (S (n+1+m))
: join.equiv_closed erfl IH
... ≃ sphere (n+1+m.+1)
: join.bool (S (n+1+m)) }
: join_assoc
... ≃ join bool (S (n+m+1))
: join_equiv_join erfl IH
... ≃ sphere (n+m+2)
: join_bool (S (n+m+1)) }
end
end join

32
hott/homotopy/quaternionic_hopf.hlean
View file

@ -9,19 +9,18 @@ The H-space structure on S³ and the quaternionic Hopf fibration
import .complex_hopf .imaginaroid
open eq equiv is_equiv circle is_conn trunc is_trunc sphere_index sphere susp
open imaginaroid
open eq equiv is_equiv circle is_conn trunc is_trunc sphere susp imaginaroid pointed bool join
namespace hopf
definition involutive_neg_empty [instance] : involutive_neg empty :=
⦃ involutive_neg, neg := empty.elim, neg_neg := by intro a; induction a ⦄
definition involutive_neg_bool [instance] : involutive_neg bool :=
⦃ involutive_neg, neg := bnot, neg_neg := by intro a; induction a: reflexivity ⦄
definition involutive_neg_circle [instance] : involutive_neg circle :=
by change involutive_neg (susp (susp empty)); exact _
by change involutive_neg (susp bool); exact _
definition has_star_circle [instance] : has_star circle :=
by change has_star (susp (susp empty)); exact _
by change has_star (susp bool); exact _
-- this is the "natural" conjugation defined using the base-loop recursor
definition circle_star [reducible] : S¹ → S¹ :=
@ -52,7 +51,7 @@ namespace hopf
(ap (λw, w ⬝ (tr_constant seg1 base)⁻¹) (con.right_inv seg2)⁻¹),
apply con.assoc },
{ apply eq_pathover, krewrite elim_merid, rewrite elim_seg2,
apply square_of_eq, rewrite [↑loop,con_inv,inv_inv,idp_con],
apply square_of_eq, rewrite [↑circle.loop,con_inv,inv_inv,idp_con],
apply con.assoc }
end
@ -85,10 +84,11 @@ namespace hopf
{ apply is_prop.elimo } }
end
open sphere.ops
definition imaginaroid_sphere_zero [instance]
: imaginaroid (sphere (-1.+1)) :=
⦃ imaginaroid,
neg_neg := susp_neg_neg,
: imaginaroid (S 0) :=
⦃ imaginaroid, involutive_neg_bool,
mul := circle_mul,
one_mul := circle_base_mul,
mul_one := circle_mul_base,
@ -96,12 +96,9 @@ namespace hopf
norm := circle_norm,
star_mul := circle_star_mul ⦄
local attribute sphere [reducible]
open sphere.ops
definition sphere_three_h_space [instance] : h_space (S 3) :=
@h_space_equiv_closed (join S¹ S¹)
(cd_h_space (S -1.+1) circle_assoc) (S 3) (join.spheres 1 1)
(cd_h_space (S 0) circle_assoc) (S 3) (join_sphere 1 1)
definition is_conn_sphere_three : is_conn 0 (S 3) :=
begin
@ -115,10 +112,13 @@ namespace hopf
local attribute is_conn_sphere_three [instance]
definition quaternionic_hopf : S 7 → S 4 :=
definition quaternionic_hopf' : S 7 → S 4 :=
begin
intro x, apply @sigma.pr1 (susp (S 3)) (hopf (S 3)),
apply inv (hopf.total (S 3)), apply inv (join.spheres 3 3), exact x
apply inv (hopf.total (S 3)), apply inv (join_sphere 3 3), exact x
end
definition quaternionic_hopf [constructor] : S 7 →* S 4 :=
pmap.mk quaternionic_hopf' idp
end hopf

343
hott/homotopy/sphere.hlean
View file

@ -8,7 +8,7 @@ Declaration of the n-spheres
import .susp types.trunc
open eq nat susp bool is_trunc unit pointed algebra
open eq nat susp bool is_trunc unit pointed algebra equiv
/-
We can define spheres with the following possible indices:
@ -16,310 +16,49 @@ open eq nat susp bool is_trunc unit pointed algebra
- nat (forgetting that S^-1 = empty)
- nat, but counting wrong (S^0 = empty, S^1 = bool, ...)
- some new type "integers >= -1"
We choose the last option here.
We choose the second option here.
-/
/- Sphere levels -/
inductive sphere_index : Type₀ :=
| minus_one : sphere_index
| succ : sphere_index → sphere_index
notation `ℕ₋₁` := sphere_index
namespace trunc_index
definition sub_one [reducible] (n : ℕ₋₁) : ℕ₋₂ :=
sphere_index.rec_on n -2 (λ n k, k.+1)
postfix `..-1`:(max+1) := sub_one
definition of_sphere_index [reducible] (n : ℕ₋₁) : ℕ₋₂ :=
n..-1.+1
-- we use a double dot to distinguish with the notation .-1 in trunc_index (of type ℕ → ℕ₋₂)
end trunc_index
namespace sphere_index
/-
notation for sphere_index is -1, 0, 1, ...
from 0 and up this comes from a coercion from num to sphere_index (via nat)
-/
postfix `.+1`:(max+1) := sphere_index.succ
postfix `.+2`:(max+1) := λ(n : sphere_index), (n .+1 .+1)
notation `-1` := minus_one
definition has_zero_sphere_index [instance] : has_zero ℕ₋₁ :=
has_zero.mk (succ minus_one)
definition has_one_sphere_index [instance] : has_one ℕ₋₁ :=
has_one.mk (succ (succ minus_one))
definition add_plus_one (n m : ℕ₋₁) : ℕ₋₁ :=
sphere_index.rec_on m n (λ k l, l .+1)
-- addition of sphere_indices, where (-1 + -1) is defined to be -1.
protected definition add (n m : ℕ₋₁) : ℕ₋₁ :=
sphere_index.cases_on m
(sphere_index.cases_on n -1 id)
(sphere_index.rec n (λn' r, succ r))
inductive le (a : ℕ₋₁) : ℕ₋₁ → Type :=
| sp_refl : le a a
| step : Π {b}, le a b → le a (b.+1)
infix ` +1+ `:65 := sphere_index.add_plus_one
definition has_add_sphere_index [instance] [priority 2000] [reducible] : has_add ℕ₋₁ :=
has_add.mk sphere_index.add
definition has_le_sphere_index [instance] : has_le ℕ₋₁ :=
has_le.mk sphere_index.le
definition sub_one [reducible] (n : ℕ) : ℕ₋₁ :=
nat.rec_on n -1 (λ n k, k.+1)
postfix `..-1`:(max+1) := sub_one
definition of_nat [coercion] [reducible] (n : ℕ) : ℕ₋₁ :=
n..-1.+1
-- we use a double dot to distinguish with the notation .-1 in trunc_index (of type ℕ → ℕ₋₂)
definition add_one [reducible] (n : ℕ₋₁) : ℕ :=
sphere_index.rec_on n 0 (λ n k, nat.succ k)
definition add_plus_one_of_nat (n m : ℕ) : (n +1+ m) = sphere_index.of_nat (n + m + 1) :=
begin
induction m with m IH,
{ reflexivity },
{ exact ap succ IH}
end
definition succ_sub_one (n : ℕ) : (nat.succ n)..-1 = n :> ℕ₋₁ :=
idp
definition add_sub_one (n m : ℕ) : (n + m)..-1 = n..-1 +1+ m..-1 :> ℕ₋₁ :=
begin
induction m with m IH,
{ reflexivity },
{ exact ap succ IH }
end
definition succ_le_succ {n m : ℕ₋₁} (H : n ≤ m) : n.+1 ≤[ℕ₋₁] m.+1 :=
by induction H with m H IH; apply le.sp_refl; exact le.step IH
definition minus_one_le (n : ℕ₋₁) : -1 ≤[ℕ₋₁] n :=
by induction n with n IH; apply le.sp_refl; exact le.step IH
open decidable
protected definition has_decidable_eq [instance] : Π(n m : ℕ₋₁), decidable (n = m)
| has_decidable_eq -1 -1 := inl rfl
| has_decidable_eq (n.+1) -1 := inr (by contradiction)
| has_decidable_eq -1 (m.+1) := inr (by contradiction)
| has_decidable_eq (n.+1) (m.+1) :=
match has_decidable_eq n m with
| inl xeqy := inl (by rewrite xeqy)
| inr xney := inr (λ h : succ n = succ m, by injection h with xeqy; exact absurd xeqy xney)
end
definition not_succ_le_minus_two {n : sphere_index} (H : n .+1 ≤[ℕ₋₁] -1) : empty :=
by cases H
protected definition le_trans {n m k : ℕ₋₁} (H1 : n ≤[ℕ₋₁] m) (H2 : m ≤[ℕ₋₁] k) : n ≤[ℕ₋₁] k :=
begin
induction H2 with k H2 IH,
{ exact H1},
{ exact le.step IH}
end
definition le_of_succ_le_succ {n m : ℕ₋₁} (H : n.+1 ≤[ℕ₋₁] m.+1) : n ≤[ℕ₋₁] m :=
begin
cases H with m H',
{ apply le.sp_refl},
{ exact sphere_index.le_trans (le.step !le.sp_refl) H'}
end
theorem not_succ_le_self {n : ℕ₋₁} : ¬n.+1 ≤[ℕ₋₁] n :=
begin
induction n with n IH: intro H,
{ exact not_succ_le_minus_two H},
{ exact IH (le_of_succ_le_succ H)}
end
protected definition le_antisymm {n m : ℕ₋₁} (H1 : n ≤[ℕ₋₁] m) (H2 : m ≤[ℕ₋₁] n) : n = m :=
begin
induction H2 with n H2 IH,
{ reflexivity},
{ exfalso, apply @not_succ_le_self n, exact sphere_index.le_trans H1 H2}
end
protected definition le_succ {n m : ℕ₋₁} (H1 : n ≤[ℕ₋₁] m): n ≤[ℕ₋₁] m.+1 :=
le.step H1
definition add_plus_one_minus_one (n : ℕ₋₁) : n +1+ -1 = n := idp
definition add_plus_one_succ (n m : ℕ₋₁) : n +1+ (m.+1) = (n +1+ m).+1 := idp
definition minus_one_add_plus_one (n : ℕ₋₁) : -1 +1+ n = n :=
begin induction n with n IH, reflexivity, exact ap succ IH end
definition succ_add_plus_one (n m : ℕ₋₁) : (n.+1) +1+ m = (n +1+ m).+1 :=
begin induction m with m IH, reflexivity, exact ap succ IH end
definition sphere_index_of_nat_add_one (n : ℕ₋₁) : sphere_index.of_nat (add_one n) = n.+1 :=
begin induction n with n IH, reflexivity, exact ap succ IH end
definition add_one_succ (n : ℕ₋₁) : add_one (n.+1) = succ (add_one n) :=
by reflexivity
definition add_one_sub_one (n : ℕ) : add_one (n..-1) = n :=
begin induction n with n IH, reflexivity, exact ap nat.succ IH end
definition add_one_of_nat (n : ℕ) : add_one n = nat.succ n :=
ap nat.succ (add_one_sub_one n)
definition sphere_index.of_nat_succ (n : ℕ)
: sphere_index.of_nat (nat.succ n) = (sphere_index.of_nat n).+1 :=
begin induction n with n IH, reflexivity, exact ap succ IH end
/-
warning: if this coercion is available, the coercion ℕ → ℕ₋₂ is the composition of the coercions
ℕ → ℕ₋₁ → ℕ₋₂. We don't want this composition as coercion, because it has worse computational
properties. You can rewrite it with trans_to_of_sphere_index_eq defined below.
-/
attribute trunc_index.of_sphere_index [coercion]
end sphere_index open sphere_index
definition weak_order_sphere_index [trans_instance] [reducible] : weak_order sphere_index :=
weak_order.mk le sphere_index.le.sp_refl @sphere_index.le_trans @sphere_index.le_antisymm
namespace trunc_index
definition sub_two_eq_sub_one_sub_one (n : ℕ) : n.-2 = n..-1..-1 :=
begin
induction n with n IH,
{ reflexivity},
{ exact ap trunc_index.succ IH}
end
definition of_nat_sub_one (n : ℕ)
: (sphere_index.of_nat n)..-1 = (trunc_index.sub_two n).+1 :=
begin
induction n with n IH,
{ reflexivity},
{ exact ap trunc_index.succ IH}
end
definition sub_one_of_sphere_index (n : ℕ)
: of_sphere_index n..-1 = (trunc_index.sub_two n).+1 :=
begin
induction n with n IH,
{ reflexivity},
{ exact ap trunc_index.succ IH}
end
definition succ_sub_one (n : ℕ₋₁) : n.+1..-1 = n :> ℕ₋₂ :=
idp
definition of_sphere_index_of_nat (n : ℕ)
: of_sphere_index (sphere_index.of_nat n) = of_nat n :> ℕ₋₂ :=
begin
induction n with n IH,
{ reflexivity},
{ exact ap trunc_index.succ IH}
end
definition trans_to_of_sphere_index_eq (n : ℕ)
: trunc_index._trans_to_of_sphere_index n = of_nat n :> ℕ₋₂ :=
of_sphere_index_of_nat n
definition trunc_index_of_nat_add_one (n : ℕ₋₁)
: trunc_index.of_nat (add_one n) = (of_sphere_index n).+1 :=
begin induction n with n IH, reflexivity, exact ap succ IH end
definition of_sphere_index_succ (n : ℕ₋₁) : of_sphere_index (n.+1) = (of_sphere_index n).+1 :=
begin induction n with n IH, reflexivity, exact ap succ IH end
end trunc_index
open sphere_index equiv
definition sphere (n : ℕ₋₁) : Type₀ := iterate_susp (add_one n) empty
definition sphere (n : ℕ) : Type* := iterate_susp n pbool
namespace sphere
export [notation] sphere_index
definition base {n : ℕ} : sphere n := north
definition pointed_sphere [instance] [constructor] (n : ℕ) : pointed (sphere n) :=
pointed.mk base
definition psphere [constructor] (n : ℕ) : Type* := pointed.mk' (sphere n)
namespace ops
abbreviation S := sphere
notation `S*` := psphere
end ops
open sphere.ops
definition sphere_minus_one : S -1 = empty := idp
definition sphere_succ [unfold_full] (n : ℕ₋₁) : S n.+1 = susp (S n) := idp
definition psphere_succ [unfold_full] (n : ℕ) : S* (n + 1) = psusp (S* n) := idp
definition psphere_eq_iterate_susp (n : ℕ)
: S* n = pointed.MK (iterate_susp (succ n) empty) !north :=
begin
esimp,
apply ap (λx, pointed.MK (susp x) (@north x)); apply ap (λx, iterate_susp x empty),
apply add_one_sub_one
end
definition sphere_succ [unfold_full] (n : ℕ) : S (n+1) = susp (S n) := idp
definition sphere_eq_iterate_susp (n : ℕ) : S n = iterate_susp n pbool := idp
definition equator [constructor] (n : ℕ) : S* n →* Ω (S* (succ n)) :=
loop_psusp_unit (S* n)
definition equator [constructor] (n : ℕ) : S n →* Ω (S (succ n)) :=
loop_susp_unit (S n)
definition surf {n : ℕ} : Ω[n] (S* n) :=
definition surf {n : ℕ} : Ω[n] (S n) :=
begin
induction n with n s,
{ exact south },
{ exact (loopn_succ_in (S* (succ n)) n)⁻¹ᵉ* (apn n (equator n) s) }
{ exact tt },
{ exact (loopn_succ_in (S (succ n)) n)⁻¹ᵉ* (apn n (equator n) s) }
end
definition bool_of_sphere [unfold 1] : S 0 → bool :=
proof susp.rec ff tt (λx, empty.elim x) qed
definition sphere_equiv_bool [constructor] : S 0 ≃ bool := by reflexivity
definition sphere_of_bool [unfold 1] : bool → S 0
| ff := proof north qed
| tt := proof south qed
definition sphere_pequiv_pbool [constructor] : S 0 ≃* pbool := by reflexivity
definition sphere_equiv_bool [constructor] : S 0 ≃ bool :=
equiv.MK bool_of_sphere
sphere_of_bool
(λb, match b with | tt := idp | ff := idp end)
(λx, proof susp.rec_on x idp idp (empty.rec _) qed)
definition sphere_pequiv_iterate_susp (n : ℕ) : sphere n ≃* iterate_susp n pbool :=
by reflexivity
definition psphere_pequiv_pbool [constructor] : S* 0 ≃* pbool :=
pequiv_of_equiv sphere_equiv_bool idp
definition sphere_eq_bool : S 0 = bool :=
ua sphere_equiv_bool
definition sphere_eq_pbool : S* 0 = pbool :=
pType_eq sphere_equiv_bool idp
definition psphere_pequiv_iterate_psusp (n : ℕ) : psphere n ≃* iterate_psusp n pbool :=
begin
induction n with n e,
{ exact psphere_pequiv_pbool },
{ exact psusp_pequiv e }
end
definition psphere_pmap_pequiv' (A : Type*) (n : ℕ) : ppmap (S* n) A ≃* Ω[n] A :=
definition sphere_pmap_pequiv' (A : Type*) (n : ℕ) : ppmap (S n) A ≃* Ω[n] A :=
begin
revert A, induction n with n IH: intro A,
{ refine _ ⬝e* !ppmap_pbool_pequiv, exact pequiv_ppcompose_right psphere_pequiv_pbool⁻¹ᵉ* },
{ refine psusp_adjoint_loop (S* n) A ⬝e* IH (Ω A) ⬝e* !loopn_succ_in⁻¹ᵉ* }
{ refine !ppmap_pbool_pequiv },
{ refine susp_adjoint_loop (S n) A ⬝e* IH (Ω A) ⬝e* !loopn_succ_in⁻¹ᵉ* }
end
definition psphere_pmap_pequiv (A : Type*) (n : ℕ) : ppmap (S* n) A ≃* Ω[n] A :=
definition sphere_pmap_pequiv (A : Type*) (n : ℕ) : ppmap (S n) A ≃* Ω[n] A :=
begin
fapply pequiv_change_fun,
{ exact psphere_pmap_pequiv' A n },
{ exact sphere_pmap_pequiv' A n },
{ exact papn_fun A surf },
{ revert A, induction n with n IH: intro A,
{ reflexivity },
@ -327,33 +66,29 @@ namespace sphere
exact !loopn_succ_in_inv_natural⁻¹* _ }}
end
protected definition elim {n : ℕ} {P : Type*} (p : Ω[n] P) : S* n →* P :=
!psphere_pmap_pequiv⁻¹ᵉ* p
protected definition elim {n : ℕ} {P : Type*} (p : Ω[n] P) : S n →* P :=
!sphere_pmap_pequiv⁻¹ᵉ* p
-- definition elim_surf {n : ℕ} {P : Type*} (p : Ω[n] P) : apn n (sphere.elim p) surf = p :=
-- begin
-- induction n with n IH,
-- { esimp [apn,surf,sphere.elim,psphere_pmap_equiv], apply sorry},
-- { esimp [apn,surf,sphere.elim,sphere_pmap_equiv], apply sorry},
-- { apply sorry}
-- end
end sphere
namespace sphere
open is_conn trunc_index sphere_index sphere.ops
open is_conn trunc_index sphere.ops
-- Corollary 8.2.2
theorem is_conn_sphere [instance] (n : ℕ₋₁) : is_conn (n..-1) (S n) :=
theorem is_conn_sphere [instance] (n : ℕ) : is_conn (n.-1) (S n) :=
begin
induction n with n IH,
{ apply is_conn_minus_two },
{ rewrite [trunc_index.succ_sub_one n, sphere.sphere_succ],
apply is_conn_susp }
{ apply is_conn_minus_one_pointed },
{ apply is_conn_susp, exact IH }
end
theorem is_conn_psphere [instance] (n : ℕ) : is_conn (n.-1) (S* n) :=
transport (λx, is_conn x (sphere n)) (of_nat_sub_one n) (is_conn_sphere n)
end sphere
open sphere sphere.ops
@ -361,12 +96,12 @@ open sphere sphere.ops
namespace is_trunc
open trunc_index
variables {n : ℕ} {A : Type}
definition is_trunc_of_psphere_pmap_equiv_constant
(H : Π(a : A) (f : S* n →* pointed.Mk a) (x : S n), f x = f base) : is_trunc (n.-2.+1) A :=
definition is_trunc_of_sphere_pmap_equiv_constant
(H : Π(a : A) (f : S n →* pointed.Mk a) (x : S n), f x = f pt) : is_trunc (n.-2.+1) A :=
begin
apply iff.elim_right !is_trunc_iff_is_contr_loop,
intro a,
apply is_trunc_equiv_closed, exact !psphere_pmap_pequiv,
apply is_trunc_equiv_closed, exact !sphere_pmap_pequiv,
fapply is_contr.mk,
{ exact pmap.mk (λx, a) idp},
{ intro f, fapply pmap_eq,
@ -375,30 +110,30 @@ namespace is_trunc
end
definition is_trunc_iff_map_sphere_constant
(H : Π(f : S n → A) (x : S n), f x = f base) : is_trunc (n.-2.+1) A :=
(H : Π(f : S n → A) (x : S n), f x = f pt) : is_trunc (n.-2.+1) A :=
begin
apply is_trunc_of_psphere_pmap_equiv_constant,
apply is_trunc_of_sphere_pmap_equiv_constant,
intros, cases f with f p, esimp at *, apply H
end
definition psphere_pmap_equiv_constant_of_is_trunc' [H : is_trunc (n.-2.+1) A]
(a : A) (f : S* n →* pointed.Mk a) (x : S n) : f x = f base :=
definition sphere_pmap_equiv_constant_of_is_trunc' [H : is_trunc (n.-2.+1) A]
(a : A) (f : S n →* pointed.Mk a) (x : S n) : f x = f pt :=
begin
let H' := iff.elim_left (is_trunc_iff_is_contr_loop n A) H a,
note H'' := @is_trunc_equiv_closed_rev _ _ _ !psphere_pmap_pequiv H',
note H'' := @is_trunc_equiv_closed_rev _ _ _ !sphere_pmap_pequiv H',
esimp at H'',
have p : f = pmap.mk (λx, f base) (respect_pt f),
have p : f = pmap.mk (λx, f pt) (respect_pt f),
by apply is_prop.elim,
exact ap10 (ap pmap.to_fun p) x
end
definition psphere_pmap_equiv_constant_of_is_trunc [H : is_trunc (n.-2.+1) A]
(a : A) (f : S* n →* pointed.Mk a) (x y : S n) : f x = f y :=
let H := psphere_pmap_equiv_constant_of_is_trunc' a f in !H ⬝ !H⁻¹
definition sphere_pmap_equiv_constant_of_is_trunc [H : is_trunc (n.-2.+1) A]
(a : A) (f : S n →* pointed.Mk a) (x y : S n) : f x = f y :=
let H := sphere_pmap_equiv_constant_of_is_trunc' a f in !H ⬝ !H⁻¹
definition map_sphere_constant_of_is_trunc [H : is_trunc (n.-2.+1) A]
(f : S n → A) (x y : S n) : f x = f y :=
psphere_pmap_equiv_constant_of_is_trunc (f base) (pmap.mk f idp) x y
sphere_pmap_equiv_constant_of_is_trunc (f pt) (pmap.mk f idp) x y
definition map_sphere_constant_of_is_trunc_self [H : is_trunc (n.-2.+1) A]
(f : S n → A) (x : S n) : map_sphere_constant_of_is_trunc f x x = idp :=

61
hott/homotopy/sphere2.hlean
View file

@ -14,56 +14,56 @@ In this file we calculate
import .homotopy_group .freudenthal
open eq group algebra is_equiv equiv fin prod chain_complex pointed fiber nat is_trunc trunc_index
sphere.ops trunc is_conn susp
sphere.ops trunc is_conn susp bool
namespace sphere
/- Corollaries of the complex hopf fibration combined with the LES of homotopy groups -/
open sphere sphere.ops int circle hopf
definition π2S2 : πg[1+1] (S* 2) ≃g gℤ :=
definition π2S2 : πg[1+1] (S 2) ≃g gℤ :=
begin
refine _ ⬝g fundamental_group_of_circle,
refine _ ⬝g homotopy_group_isomorphism_of_pequiv _ pfiber_complex_phopf,
refine _ ⬝g homotopy_group_isomorphism_of_pequiv _ pfiber_complex_hopf,
fapply isomorphism_of_equiv,
{ fapply equiv.mk,
{ exact cc_to_fn (LES_of_homotopy_groups complex_phopf) (1, 2)},
{ refine LES_is_equiv_of_trivial complex_phopf 1 2 _ _,
{ exact cc_to_fn (LES_of_homotopy_groups complex_hopf) (1, 2)},
{ refine LES_is_equiv_of_trivial complex_hopf 1 2 _ _,
{ have H : 1 ≤[ℕ] 2, from !one_le_succ,
apply trivial_homotopy_group_of_is_conn, exact H, rexact is_conn_psphere 3 },
apply trivial_homotopy_group_of_is_conn, exact H, rexact is_conn_sphere 3 },
{ refine tr_rev (λx, is_contr (ptrunctype._trans_of_to_pType x))
(LES_of_homotopy_groups_1 complex_phopf 2) _,
apply trivial_homotopy_group_of_is_conn, apply le.refl, rexact is_conn_psphere 3 }}},
(LES_of_homotopy_groups_1 complex_hopf 2) _,
apply trivial_homotopy_group_of_is_conn, apply le.refl, rexact is_conn_sphere 3 }}},
{ exact homomorphism.struct (homomorphism_LES_of_homotopy_groups_fun _ (0, 2))}
end
open circle
definition πnS3_eq_πnS2 (n : ℕ) : πg[n+2 +1] (S* 3) ≃g πg[n+2 +1] (S* 2) :=
definition πnS3_eq_πnS2 (n : ℕ) : πg[n+2 +1] (S 3) ≃g πg[n+2 +1] (S 2) :=
begin
fapply isomorphism_of_equiv,
{ fapply equiv.mk,
{ exact cc_to_fn (LES_of_homotopy_groups complex_phopf) (n+3, 0)},
{ have H : is_trunc 1 (pfiber complex_phopf),
from @(is_trunc_equiv_closed_rev _ pfiber_complex_phopf) is_trunc_circle,
refine LES_is_equiv_of_trivial complex_phopf (n+3) 0 _ _,
{ exact cc_to_fn (LES_of_homotopy_groups complex_hopf) (n+3, 0)},
{ have H : is_trunc 1 (pfiber complex_hopf),
from @(is_trunc_equiv_closed_rev _ pfiber_complex_hopf) is_trunc_circle,
refine LES_is_equiv_of_trivial complex_hopf (n+3) 0 _ _,
{ have H2 : 1 ≤[ℕ] n + 1, from !one_le_succ,
exact @trivial_ghomotopy_group_of_is_trunc _ _ _ H H2 },
{ refine tr_rev (λx, is_contr (ptrunctype._trans_of_to_pType x))
(LES_of_homotopy_groups_2 complex_phopf _) _,
(LES_of_homotopy_groups_2 complex_hopf _) _,
have H2 : 1 ≤[ℕ] n + 2, from !one_le_succ,
apply trivial_ghomotopy_group_of_is_trunc _ _ _ H2 }}},
{ exact homomorphism.struct (homomorphism_LES_of_homotopy_groups_fun _ (n+2, 0))}
end
definition sphere_stability_pequiv (k n : ℕ) (H : k + 2 ≤ 2 * n) :
π[k + 1] (S* (n+1)) ≃* π[k] (S* n) :=
begin rewrite [+ psphere_eq_iterate_susp], exact iterate_susp_stability_pequiv empty H end
π[k + 1] (S (n+1)) ≃* π[k] (S n) :=
iterate_susp_stability_pequiv pbool H
definition stability_isomorphism (k n : ℕ) (H : k + 3 ≤ 2 * n)
: πg[k+1 +1] (S* (n+1)) ≃g πg[k+1] (S* n) :=
begin rewrite [+ psphere_eq_iterate_susp], exact iterate_susp_stability_isomorphism empty H end
: πg[k+1 +1] (S (n+1)) ≃g πg[k+1] (S n) :=
iterate_susp_stability_isomorphism pbool H
open int circle hopf
definition πnSn (n : ℕ) : πg[n+1] (S* (succ n)) ≃g gℤ :=
definition πnSn (n : ℕ) : πg[n+1] (S (n+1)) ≃g gℤ :=
begin
cases n with n IH,
{ exact fundamental_group_of_circle },
@ -73,34 +73,17 @@ namespace sphere
rexact add_mul_le_mul_add n 1 2 }}
end
theorem not_is_trunc_sphere (n : ℕ) : ¬is_trunc n (S* (succ n)) :=
theorem not_is_trunc_sphere (n : ℕ) : ¬is_trunc n (S (n+1)) :=
begin
intro H,
note H2 := trivial_ghomotopy_group_of_is_trunc (S* (succ n)) n n !le.refl,
note H2 := trivial_ghomotopy_group_of_is_trunc (S (n+1)) n n !le.refl,
have H3 : is_contr ℤ, from is_trunc_equiv_closed _ (equiv_of_isomorphism (πnSn n)),
have H4 : (0 : ℤ) ≠ (1 : ℤ), from dec_star,
apply H4,
apply is_prop.elim,
end
section
open sphere_index
definition not_is_trunc_sphere' (n : ℕ₋₁) : ¬is_trunc n (S (n.+1)) :=
begin
cases n with n,
{ esimp [sphere.ops.S, sphere], intro H,
have H2 : is_prop bool, from @(is_trunc_equiv_closed -1 sphere_equiv_bool) H,
have H3 : bool.tt ≠ bool.ff, from dec_star, apply H3, apply is_prop.elim},
{ intro H, apply not_is_trunc_sphere (add_one n),
rewrite [▸*, trunc_index_of_nat_add_one, -add_one_succ,
sphere_index_of_nat_add_one],
exact H}
end
end
definition π3S2 : πg[2+1] (S* 2) ≃g gℤ :=
definition π3S2 : πg[2+1] (S 2) ≃g gℤ :=
(πnS3_eq_πnS2 0)⁻¹ᵍ ⬝g πnSn 2
end sphere

255
hott/homotopy/susp.hlean
View file

@ -8,15 +8,31 @@ Declaration of suspension
import hit.pushout types.pointed2 cubical.square .connectedness
open pushout unit eq equiv
open pushout unit eq equiv pointed
definition susp (A : Type) : Type := pushout (λ(a : A), star) (λ(a : A), star)
definition susp' (A : Type) : Type := pushout (λ(a : A), star) (λ(a : A), star)
namespace susp
definition north' {A : Type} : susp' A :=
inl star
definition pointed_susp [instance] [constructor] (X : Type)
: pointed (susp' X) :=
pointed.mk north'
end susp open susp
definition susp [constructor] (X : Type) : Type* :=
pointed.MK (susp' X) north'
notation `⅀` := susp
namespace susp
variable {A : Type}
definition north {A : Type} : susp A :=
inl star
north'
definition south {A : Type} : susp A :=
inr star
@ -25,7 +41,7 @@ namespace susp
glue a
protected definition rec {P : susp A → Type} (PN : P north) (PS : P south)
(Pm : Π(a : A), PN =[merid a] PS) (x : susp A) : P x :=
(Pm : Π(a : A), PN =[merid a] PS) (x : susp' A) : P x :=
begin
induction x with u u,
{ cases u, exact PN},
@ -33,7 +49,7 @@ namespace susp
{ apply Pm},
end
protected definition rec_on [reducible] {P : susp A → Type} (y : susp A)
protected definition rec_on [reducible] {P : susp A → Type} (y : susp' A)
(PN : P north) (PS : P south) (Pm : Π(a : A), PN =[merid a] PS) : P y :=
susp.rec PN PS Pm y
@ -43,10 +59,10 @@ namespace susp
!rec_glue
protected definition elim {P : Type} (PN : P) (PS : P) (Pm : A → PN = PS)
(x : susp A) : P :=
(x : susp' A) : P :=
susp.rec PN PS (λa, pathover_of_eq _ (Pm a)) x
protected definition elim_on [reducible] {P : Type} (x : susp A)
protected definition elim_on [reducible] {P : Type} (x : susp' A)
(PN : P) (PS : P) (Pm : A → PN = PS) : P :=
susp.elim PN PS Pm x
@ -58,10 +74,10 @@ namespace susp
end
protected definition elim_type (PN : Type) (PS : Type) (Pm : A → PN ≃ PS)
(x : susp A) : Type :=
(x : susp' A) : Type :=
pushout.elim_type (λx, PN) (λx, PS) Pm x
protected definition elim_type_on [reducible] (x : susp A)
protected definition elim_type_on [reducible] (x : susp' A)
(PN : Type) (PS : Type) (Pm : A → PN ≃ PS) : Type :=
susp.elim_type PN PS Pm x
@ -79,7 +95,7 @@ namespace susp
end susp
attribute susp.north susp.south [constructor]
attribute susp.north' susp.north susp.south [constructor]
attribute susp.rec susp.elim [unfold 6] [recursor 6]
attribute susp.elim_type [unfold 5]
attribute susp.rec_on susp.elim_on [unfold 3]
@ -94,28 +110,23 @@ namespace susp
[H : is_conn n A] : is_conn (n .+1) (susp A) :=
is_contr.mk (tr north)
begin
apply trunc.rec,
fapply susp.rec,
intro x, induction x with x, induction x,
{ reflexivity },
{ exact (trunc.rec (λa, ap tr (merid a)) (center (trunc n A))) },
{ intro a,
generalize (center (trunc n A)),
apply trunc.rec,
intro a',
{ generalize (center (trunc n A)),
intro x, induction x with a',
apply pathover_of_tr_eq,
rewrite [eq_transport_Fr,idp_con],
revert H, induction n with [n, IH],
{ intro H, apply is_prop.elim },
{ intros H,
change ap (@tr n .+2 (susp A)) (merid a) = ap tr (merid a'),
revert H, induction n with n IH: intro H,
{ apply is_prop.elim },
{ change ap (@tr n .+2 (susp A)) (merid a) = ap tr (merid a'),
generalize a',
apply is_conn_fun.elim n
(is_conn_fun_from_unit n A a)
(λx : A, trunctype.mk' n (ap (@tr n .+2 (susp A)) (merid a) = ap tr (merid x))),
intros,
change ap (@tr n .+2 (susp A)) (merid a) = ap tr (merid a),
reflexivity
}
reflexivity }
}
end
@ -152,7 +163,7 @@ namespace susp
variables {A B : Type} (f : A → B)
include f
protected definition functor [unfold 4] : susp A → susp B :=
definition susp_functor' [unfold 4] : susp A → susp B :=
begin
intro x, induction x with a,
{ exact north },
@ -164,19 +175,19 @@ namespace susp
include Hf
open is_equiv
protected definition is_equiv_functor [instance] [constructor] : is_equiv (susp.functor f) :=
adjointify (susp.functor f) (susp.functor f⁻¹)
protected definition is_equiv_functor [instance] [constructor] : is_equiv (susp_functor' f) :=
adjointify (susp_functor' f) (susp_functor' f⁻¹)
abstract begin
intro sb, induction sb with b, do 2 reflexivity,
apply eq_pathover,
rewrite [ap_id,ap_compose' (susp.functor f) (susp.functor f⁻¹)],
rewrite [ap_id,ap_compose' (susp_functor' f) (susp_functor' f⁻¹)],
krewrite [susp.elim_merid,susp.elim_merid], apply transpose,
apply susp.merid_square (right_inv f b)
end end
abstract begin
intro sa, induction sa with a, do 2 reflexivity,
apply eq_pathover,
rewrite [ap_id,ap_compose' (susp.functor f⁻¹) (susp.functor f)],
rewrite [ap_id,ap_compose' (susp_functor' f⁻¹) (susp_functor' f)],
krewrite [susp.elim_merid,susp.elim_merid], apply transpose,
apply susp.merid_square (left_inv f a)
end end
@ -188,59 +199,42 @@ namespace susp
variables {A B : Type} (f : A ≃ B)
protected definition equiv : susp A ≃ susp B :=
equiv.mk (susp.functor f) _
equiv.mk (susp_functor' f) _
end susp
namespace susp
open pointed
definition pointed_susp [instance] [constructor] (X : Type)
: pointed (susp X) :=
pointed.mk north
end susp
open susp
definition psusp [constructor] (X : Type) : Type* :=
pointed.mk' (susp X)
notation `⅀` := psusp
namespace susp
open pointed is_trunc
variables {X Y Z : Type*}
definition is_conn_psusp [instance] (n : trunc_index) (A : Type*)
[H : is_conn n A] : is_conn (n .+1) (psusp A) :=
is_conn_susp n A
definition psusp_functor [constructor] (f : X →* Y) : psusp X →* psusp Y :=
definition susp_functor [constructor] (f : X →* Y) : susp X →* susp Y :=
begin
fconstructor,
{ exact susp.functor f },
{ exact susp_functor' f },
{ reflexivity }
end
definition is_equiv_psusp_functor [constructor] (f : X →* Y) [Hf : is_equiv f]
: is_equiv (psusp_functor f) :=
definition is_equiv_susp_functor [constructor] (f : X →* Y) [Hf : is_equiv f]
: is_equiv (susp_functor f) :=
susp.is_equiv_functor f
definition psusp_pequiv [constructor] (f : X ≃* Y) : psusp X ≃* psusp Y :=
definition susp_pequiv [constructor] (f : X ≃* Y) : susp X ≃* susp Y :=
pequiv_of_equiv (susp.equiv f) idp
definition psusp_functor_pcompose (g : Y →* Z) (f : X →* Y) :
psusp_functor (g ∘* f) ~* psusp_functor g ∘* psusp_functor f :=
definition susp_functor_pcompose (g : Y →* Z) (f : X →* Y) :
susp_functor (g ∘* f) ~* susp_functor g ∘* susp_functor f :=
begin
fapply phomotopy.mk,
{ intro x, induction x,
{ reflexivity },
{ reflexivity },
{ apply eq_pathover, apply hdeg_square, esimp,
refine !elim_merid ⬝ _ ⬝ (ap_compose (psusp_functor g) _ _)⁻¹ᵖ,
{ apply eq_pathover, apply hdeg_square,
refine !elim_merid ⬝ _ ⬝ (ap_compose (susp_functor g) _ _)⁻¹ᵖ,
refine _ ⬝ ap02 _ !elim_merid⁻¹, exact !elim_merid⁻¹ }},
{ reflexivity },
end
definition psusp_functor_phomotopy {f g : X →* Y} (p : f ~* g) :
psusp_functor f ~* psusp_functor g :=
definition susp_functor_phomotopy {f g : X →* Y} (p : f ~* g) :
susp_functor f ~* susp_functor g :=
begin
fapply phomotopy.mk,
{ intro x, induction x,
@ -251,7 +245,7 @@ namespace susp
{ reflexivity },
end
definition psusp_functor_pid (A : Type*) : psusp_functor (pid A) ~* pid (psusp A) :=
definition susp_functor_pid (A : Type*) : susp_functor (pid A) ~* pid (susp A) :=
begin
fapply phomotopy.mk,
{ intro x, induction x,
@ -264,15 +258,15 @@ namespace susp
/- adjunction originally ported from Coq-HoTT,
but we proved some additional naturality conditions -/
definition loop_psusp_unit [constructor] (X : Type*) : X →* Ω(psusp X) :=
definition loop_susp_unit [constructor] (X : Type*) : X →* Ω(susp X) :=
begin
fconstructor,
{ intro x, exact merid x ⬝ (merid pt)⁻¹ },
{ apply con.right_inv },
end
definition loop_psusp_unit_natural (f : X →* Y)
: loop_psusp_unit Y ∘* f ~* Ω→ (psusp_functor f) ∘* loop_psusp_unit X :=
definition loop_susp_unit_natural (f : X →* Y)
: loop_susp_unit Y ∘* f ~* Ω→ (susp_functor f) ∘* loop_susp_unit X :=
begin
induction X with X x, induction Y with Y y, induction f with f pf, esimp at *, induction pf,
fapply phomotopy.mk,
@ -292,15 +286,15 @@ namespace susp
rewrite [-ap_compose (concat idp)] },
end
definition loop_psusp_counit [constructor] (X : Type*) : psusp (Ω X) →* X :=
definition loop_susp_counit [constructor] (X : Type*) : susp (Ω X) →* X :=
begin
fconstructor,
{ intro x, induction x, exact pt, exact pt, exact a },
{ reflexivity },
end
definition loop_psusp_counit_natural (f : X →* Y)
: f ∘* loop_psusp_counit X ~* loop_psusp_counit Y ∘* (psusp_functor (ap1 f)) :=
definition loop_susp_counit_natural (f : X →* Y)
: f ∘* loop_susp_counit X ~* loop_susp_counit Y ∘* (susp_functor (ap1 f)) :=
begin
induction X with X x, induction Y with Y y, induction f with f pf, esimp at *, induction pf,
fconstructor,
@ -313,8 +307,8 @@ namespace susp
{ reflexivity }
end
definition loop_psusp_counit_unit (X : Type*)
: ap1 (loop_psusp_counit X) ∘* loop_psusp_unit (Ω X) ~* pid (Ω X) :=
definition loop_susp_counit_unit (X : Type*)
: ap1 (loop_susp_counit X) ∘* loop_susp_unit (Ω X) ~* pid (Ω X) :=
begin
induction X with X x, fconstructor,
{ intro p, esimp,
@ -328,8 +322,8 @@ namespace susp
-ap_compose] }
end
definition loop_psusp_unit_counit (X : Type*)
: loop_psusp_counit (psusp X) ∘* psusp_functor (loop_psusp_unit X) ~* pid (psusp X) :=
definition loop_susp_unit_counit (X : Type*)
: loop_susp_counit (susp X) ∘* susp_functor (loop_susp_unit X) ~* pid (susp X) :=
begin
induction X with X x, fconstructor,
{ intro x', induction x',
@ -341,152 +335,141 @@ namespace susp
{ reflexivity }
end
definition psusp.elim [constructor] {X Y : Type*} (f : X →* Ω Y) : psusp X →* Y :=
loop_psusp_counit Y ∘* psusp_functor f
definition susp_elim [constructor] {X Y : Type*} (f : X →* Ω Y) : susp X →* Y :=
loop_susp_counit Y ∘* susp_functor f
definition loop_psusp_intro [constructor] {X Y : Type*} (f : psusp X →* Y) : X →* Ω Y :=
ap1 f ∘* loop_psusp_unit X
definition loop_susp_intro [constructor] {X Y : Type*} (f : susp X →* Y) : X →* Ω Y :=
ap1 f ∘* loop_susp_unit X
definition psusp_elim_psusp_functor {A B C : Type*} (g : B →* Ω C) (f : A →* B) :
psusp.elim g ∘* psusp_functor f ~* psusp.elim (g ∘* f) :=
definition susp_elim_susp_functor {A B C : Type*} (g : B →* Ω C) (f : A →* B) :
susp_elim g ∘* susp_functor f ~* susp_elim (g ∘* f) :=
begin
refine !passoc ⬝* _, exact pwhisker_left _ !psusp_functor_pcompose⁻¹*
refine !passoc ⬝* _, exact pwhisker_left _ !susp_functor_pcompose⁻¹*
end
definition psusp_elim_phomotopy {A B : Type*} {f g : A →* Ω B} (p : f ~* g) : psusp.elim f ~* psusp.elim g :=
pwhisker_left _ (psusp_functor_phomotopy p)
definition susp_elim_phomotopy {A B : Type*} {f g : A →* Ω B} (p : f ~* g) : susp_elim f ~* susp_elim g :=
pwhisker_left _ (susp_functor_phomotopy p)
definition psusp_elim_natural {X Y Z : Type*} (g : Y →* Z) (f : X →* Ω Y)
: g ∘* psusp.elim f ~* psusp.elim (Ω→ g ∘* f) :=
definition susp_elim_natural {X Y Z : Type*} (g : Y →* Z) (f : X →* Ω Y)
: g ∘* susp_elim f ~* susp_elim (Ω→ g ∘* f) :=
begin
refine _ ⬝* pwhisker_left _ !psusp_functor_pcompose⁻¹*,
refine _ ⬝* pwhisker_left _ !susp_functor_pcompose⁻¹*,
refine !passoc⁻¹* ⬝* _ ⬝* !passoc,
exact pwhisker_right _ !loop_psusp_counit_natural
exact pwhisker_right _ !loop_susp_counit_natural
end
definition loop_psusp_intro_natural {X Y Z : Type*} (g : psusp Y →* Z) (f : X →* Y) :
loop_psusp_intro (g ∘* psusp_functor f) ~* loop_psusp_intro g ∘* f :=
pwhisker_right _ !ap1_pcompose ⬝* !passoc ⬝* pwhisker_left _ !loop_psusp_unit_natural⁻¹* ⬝*
definition loop_susp_intro_natural {X Y Z : Type*} (g : susp Y →* Z) (f : X →* Y) :
loop_susp_intro (g ∘* susp_functor f) ~* loop_susp_intro g ∘* f :=
pwhisker_right _ !ap1_pcompose ⬝* !passoc ⬝* pwhisker_left _ !loop_susp_unit_natural⁻¹* ⬝*
!passoc⁻¹*
definition psusp_adjoint_loop_right_inv {X Y : Type*} (g : X →* Ω Y) :
loop_psusp_intro (psusp.elim g) ~* g :=
definition susp_adjoint_loop_right_inv {X Y : Type*} (g : X →* Ω Y) :
loop_susp_intro (susp_elim g) ~* g :=
begin
refine !pwhisker_right !ap1_pcompose ⬝* _,
refine !passoc ⬝* _,
refine !pwhisker_left !loop_psusp_unit_natural⁻¹* ⬝* _,
refine !pwhisker_left !loop_susp_unit_natural⁻¹* ⬝* _,
refine !passoc⁻¹* ⬝* _,
refine !pwhisker_right !loop_psusp_counit_unit ⬝* _,
refine !pwhisker_right !loop_susp_counit_unit ⬝* _,
apply pid_pcompose
end
definition psusp_adjoint_loop_left_inv {X Y : Type*} (f : psusp X →* Y) :
psusp.elim (loop_psusp_intro f) ~* f :=
definition susp_adjoint_loop_left_inv {X Y : Type*} (f : susp X →* Y) :
susp_elim (loop_susp_intro f) ~* f :=
begin
refine !pwhisker_left !psusp_functor_pcompose ⬝* _,
refine !pwhisker_left !susp_functor_pcompose ⬝* _,
refine !passoc⁻¹* ⬝* _,
refine !pwhisker_right !loop_psusp_counit_natural⁻¹* ⬝* _,
refine !pwhisker_right !loop_susp_counit_natural⁻¹* ⬝* _,
refine !passoc ⬝* _,
refine !pwhisker_left !loop_psusp_unit_counit ⬝* _,
refine !pwhisker_left !loop_susp_unit_counit ⬝* _,
apply pcompose_pid
end
definition psusp_adjoint_loop_unpointed [constructor] (X Y : Type*) : psusp X →* Y ≃ X →* Ω Y :=
definition susp_adjoint_loop_unpointed [constructor] (X Y : Type*) : susp X →* Y ≃ X →* Ω Y :=
begin
fapply equiv.MK,
{ exact loop_psusp_intro },
{ exact psusp.elim },
{ intro g, apply eq_of_phomotopy, exact psusp_adjoint_loop_right_inv g },
{ intro f, apply eq_of_phomotopy, exact psusp_adjoint_loop_left_inv f }
{ exact loop_susp_intro },
{ exact susp_elim },
{ intro g, apply eq_of_phomotopy, exact susp_adjoint_loop_right_inv g },
{ intro f, apply eq_of_phomotopy, exact susp_adjoint_loop_left_inv f }
end
definition psusp_adjoint_loop_pconst (X Y : Type*) :
psusp_adjoint_loop_unpointed X Y (pconst (psusp X) Y) ~* pconst X (Ω Y) :=
definition susp_adjoint_loop_pconst (X Y : Type*) :
susp_adjoint_loop_unpointed X Y (pconst (susp X) Y) ~* pconst X (Ω Y) :=
begin
refine pwhisker_right _ !ap1_pconst ⬝* _,
apply pconst_pcompose
end
definition psusp_adjoint_loop [constructor] (X Y : Type*) : ppmap (psusp X) Y ≃* ppmap X (Ω Y) :=
definition susp_adjoint_loop [constructor] (X Y : Type*) : ppmap (susp X) Y ≃* ppmap X (Ω Y) :=
begin
apply pequiv_of_equiv (psusp_adjoint_loop_unpointed X Y),
apply pequiv_of_equiv (susp_adjoint_loop_unpointed X Y),
apply eq_of_phomotopy,
apply psusp_adjoint_loop_pconst
apply susp_adjoint_loop_pconst
end
definition ap1_psusp_elim {A : Type*} {X : Type*} (p : A →* Ω X) :
Ω→(psusp.elim p) ∘* loop_psusp_unit A ~* p :=
psusp_adjoint_loop_right_inv p
definition ap1_susp_elim {A : Type*} {X : Type*} (p : A →* Ω X) :
Ω→(susp_elim p) ∘* loop_susp_unit A ~* p :=
susp_adjoint_loop_right_inv p
definition psusp_adjoint_loop_nat_right (f : psusp X →* Y) (g : Y →* Z)
: psusp_adjoint_loop X Z (g ∘* f) ~* ap1 g ∘* psusp_adjoint_loop X Y f :=
definition susp_adjoint_loop_nat_right (f : susp X →* Y) (g : Y →* Z)
: susp_adjoint_loop X Z (g ∘* f) ~* ap1 g ∘* susp_adjoint_loop X Y f :=
begin
esimp [psusp_adjoint_loop],
esimp [susp_adjoint_loop],
refine _ ⬝* !passoc,
apply pwhisker_right,
apply ap1_pcompose
end
definition psusp_adjoint_loop_nat_left (f : Y →* Ω Z) (g : X →* Y)
: (psusp_adjoint_loop X Z)⁻¹ᵉ (f ∘* g) ~* (psusp_adjoint_loop Y Z)⁻¹ᵉ f ∘* psusp_functor g :=
definition susp_adjoint_loop_nat_left (f : Y →* Ω Z) (g : X →* Y)
: (susp_adjoint_loop X Z)⁻¹ᵉ (f ∘* g) ~* (susp_adjoint_loop Y Z)⁻¹ᵉ f ∘* susp_functor g :=
begin
esimp [psusp_adjoint_loop],
esimp [susp_adjoint_loop],
refine _ ⬝* !passoc⁻¹*,
apply pwhisker_left,
apply psusp_functor_pcompose
apply susp_functor_pcompose
end
/- iterated suspension -/
definition iterate_susp (n : ℕ) (A : Type) : Type := iterate susp n A
definition iterate_psusp (n : ℕ) (A : Type*) : Type* := iterate (λX, psusp X) n A
definition iterate_susp (n : ℕ) (A : Type*) : Type* := iterate (λX, susp X) n A
open is_conn trunc_index nat
definition iterate_susp_succ (n : ℕ) (A : Type) :
definition iterate_susp_succ (n : ℕ) (A : Type*) :
iterate_susp (succ n) A = susp (iterate_susp n A) :=
idp
definition is_conn_iterate_susp [instance] (n : ℕ₋₂) (m : ℕ) (A : Type)
definition is_conn_iterate_susp [instance] (n : ℕ₋₂) (m : ℕ) (A : Type*)
[H : is_conn n A] : is_conn (n + m) (iterate_susp m A) :=
begin induction m with m IH, exact H, exact @is_conn_susp _ _ IH end
definition is_conn_iterate_psusp [instance] (n : ℕ₋₂) (m : ℕ) (A : Type*)
[H : is_conn n A] : is_conn (n + m) (iterate_psusp m A) :=
begin induction m with m IH, exact H, exact @is_conn_susp _ _ IH end
-- Separate cases for n = 0, which comes up often
definition is_conn_iterate_susp_zero [instance] (m : ℕ) (A : Type)
definition is_conn_iterate_susp_zero [instance] (m : ℕ) (A : Type*)
[H : is_conn 0 A] : is_conn m (iterate_susp m A) :=
begin induction m with m IH, exact H, exact @is_conn_susp _ _ IH end
definition is_conn_iterate_psusp_zero [instance] (m : ℕ) (A : Type*)
[H : is_conn 0 A] : is_conn m (iterate_psusp m A) :=
begin induction m with m IH, exact H, exact @is_conn_susp _ _ IH end
definition iterate_psusp_functor (n : ℕ) {A B : Type*} (f : A →* B) :
iterate_psusp n A →* iterate_psusp n B :=
definition iterate_susp_functor (n : ℕ) {A B : Type*} (f : A →* B) :
iterate_susp n A →* iterate_susp n B :=
begin
induction n with n g,
{ exact f },
{ exact psusp_functor g }
{ exact susp_functor g }
end
definition iterate_psusp_succ_in (n : ℕ) (A : Type*) :
iterate_psusp (succ n) A ≃* iterate_psusp n (psusp A) :=
definition iterate_susp_succ_in (n : ℕ) (A : Type*) :
iterate_susp (succ n) A ≃* iterate_susp n (susp A) :=
begin
induction n with n IH,
{ reflexivity},
{ exact psusp_pequiv IH}
{ exact susp_pequiv IH}
end
definition iterate_psusp_adjoint_loopn [constructor] (X Y : Type*) (n : ℕ) :
ppmap (iterate_psusp n X) Y ≃* ppmap X (Ω[n] Y) :=
definition iterate_susp_adjoint_loopn [constructor] (X Y : Type*) (n : ℕ) :
ppmap (iterate_susp n X) Y ≃* ppmap X (Ω[n] Y) :=
begin
revert X Y, induction n with n IH: intro X Y,
{ reflexivity },
{ refine !psusp_adjoint_loop ⬝e* !IH ⬝e* _, apply pequiv_ppcompose_left,
{ refine !susp_adjoint_loop ⬝e* !IH ⬝e* _, apply pequiv_ppcompose_left,
symmetry, apply loopn_succ_in }
end
end susp

19
hott/homotopy/wedge.hlean
View file

@ -9,10 +9,10 @@ import hit.pushout .connectedness types.unit
open eq pushout pointed unit trunc_index
definition wedge (A B : Type*) : Type := ppushout (pconst punit A) (pconst punit B)
local attribute wedge [reducible]
definition pwedge [constructor] (A B : Type*) : Type* := pointed.mk' (wedge A B)
infixr ` ∨ ` := pwedge
definition wedge' (A B : Type*) : Type := ppushout (pconst punit A) (pconst punit B)
local attribute wedge' [reducible]
definition wedge [constructor] (A B : Type*) : Type* := pointed.mk' (wedge' A B)
infixr ` ∨ ` := wedge
namespace wedge
@ -21,11 +21,11 @@ namespace wedge
protected definition rec {A B : Type*} {P : wedge A B → Type} (Pinl : Π(x : A), P (inl x))
(Pinr : Π(x : B), P (inr x)) (Pglue : pathover P (Pinl pt) wedge.glue (Pinr pt))
(y : wedge A B) : P y :=
(y : wedge' A B) : P y :=
by induction y; apply Pinl; apply Pinr; induction x; exact Pglue
protected definition elim {A B : Type*} {P : Type} (Pinl : A → P)
(Pinr : B → P) (Pglue : Pinl pt = Pinr pt) (y : wedge A B) : P :=
(Pinr : B → P) (Pglue : Pinl pt = Pinr pt) (y : wedge' A B) : P :=
by induction y with a b x; exact Pinl a; exact Pinr b; induction x; exact Pglue
protected definition rec_glue {A B : Type*} {P : wedge A B → Type} (Pinl : Π(x : A), P (inl x))
@ -33,11 +33,10 @@ namespace wedge
apd (wedge.rec Pinl Pinr Pglue) wedge.glue = Pglue :=
!pushout.rec_glue
protected definition elim_glue {A B : Type*} {P : Type} (Pinl : A → P)
(Pinr : B → P) (Pglue : Pinl pt = Pinr pt) : ap (wedge.elim Pinl Pinr Pglue) wedge.glue = Pglue :=
protected definition elim_glue {A B : Type*} {P : Type} (Pinl : A → P) (Pinr : B → P)
(Pglue : Pinl pt = Pinr pt) : ap (wedge.elim Pinl Pinr Pglue) wedge.glue = Pglue :=
!pushout.elim_glue
end wedge
attribute wedge.rec wedge.elim [recursor 7] [unfold 7]
@ -45,7 +44,7 @@ attribute wedge.rec wedge.elim [recursor 7] [unfold 7]
namespace wedge
-- TODO maybe find a cleaner proof
protected definition unit (A : Type*) : A ≃* pwedge punit A :=
protected definition unit (A : Type*) : A ≃* wedge punit A :=
begin
fapply pequiv_of_pmap,
{ fapply pmap.mk, intro a, apply pinr a, apply respect_pt },