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feat(hott): Port files from other repositories to the HoTT library. This commit adds truncated 2-quotients, groupoid quotients, Eilenberg MacLane spaces, chain complexes, the long exact sequence of homotopy groups, the Freudenthal Suspension Theorem, Whitehead's principle, and the computation of homotopy groups of almost all spheres which are known in HoTT.
2016-04-22 19:12:25 +00:00
/-
Copyright (c) 2016 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
The Freudenthal Suspension Theorem
-/
import homotopy.wedge homotopy.circle
open eq is_conn is_trunc pointed susp nat pi equiv is_equiv trunc fiber trunc_index
namespace freudenthal section
parameters {A : Type*} {n : ℕ} [is_conn n A]
/-
This proof is ported from Agda
This is the 95% version of the Freudenthal Suspension Theorem, which means that we don't
make pointed suspensions, wedges and spheres the default (in contrast to the unpointed ones), remove sphere_index 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.
2017-07-20 14:01:40 +00:00
prove that loop_susp_unit : A →* Ω(susp A) is 2n-connected (if A is n-connected),
feat(hott): Port files from other repositories to the HoTT library. This commit adds truncated 2-quotients, groupoid quotients, Eilenberg MacLane spaces, chain complexes, the long exact sequence of homotopy groups, the Freudenthal Suspension Theorem, Whitehead's principle, and the computation of homotopy groups of almost all spheres which are known in HoTT.
2016-04-22 19:12:25 +00:00
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 :=
make pointed suspensions, wedges and spheres the default (in contrast to the unpointed ones), remove sphere_index 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.
2017-07-20 14:01:40 +00:00
loop_susp_unit A a
feat(hott): Port files from other repositories to the HoTT library. This commit adds truncated 2-quotients, groupoid quotients, Eilenberg MacLane spaces, chain complexes, the long exact sequence of homotopy groups, the Freudenthal Suspension Theorem, Whitehead's principle, and the computation of homotopy groups of almost all spheres which are known in HoTT.
2016-04-22 19:12:25 +00:00
definition code_merid : A → ptrunc (n + n) A → ptrunc (n + n) A :=
begin
have is_conn n (ptrunc (n + n) A), from !is_conn_trunc,
feat(hott/algebra): define additive structures to be multiplicative structures
2016-09-19 18:41:21 +00:00
refine @wedge_extension.ext _ _ n n _ _ (λ x y, ttrunc (n + n) A) _ _ _ _,
{ intros, apply is_trunc_trunc}, -- this subgoal might become unnecessary if
-- type class inference catches it
feat(hott): Port files from other repositories to the HoTT library. This commit adds truncated 2-quotients, groupoid quotients, Eilenberg MacLane spaces, chain complexes, the long exact sequence of homotopy groups, the Freudenthal Suspension Theorem, Whitehead's principle, and the computation of homotopy groups of almost all spheres which are known in HoTT.
2016-04-22 19:12:25 +00:00
{ exact tr},
{ exact id},
{ reflexivity}
end
definition code_merid_β_left (a : A) : code_merid a pt = tr a :=
by apply wedge_extension.β_left
definition code_merid_β_right (b : ptrunc (n + n) A) : code_merid pt b = b :=
by apply wedge_extension.β_right
definition code_merid_coh : code_merid_β_left pt = code_merid_β_right pt :=
begin
symmetry, apply eq_of_inv_con_eq_idp, apply wedge_extension.coh
end
definition is_equiv_code_merid (a : A) : is_equiv (code_merid a) :=
begin
have Πa, is_trunc n.-2.+1 (is_equiv (code_merid a)),
move more results and make arguments explicit More results from the Spectral repository are moved to this library Also make various type-class arguments of truncatedness and equivalences which were hard to synthesize explicit
2018-09-11 14:45:30 +00:00
from λa, is_trunc_of_le _ !minus_one_le_succ _,
feat(hott): Port files from other repositories to the HoTT library. This commit adds truncated 2-quotients, groupoid quotients, Eilenberg MacLane spaces, chain complexes, the long exact sequence of homotopy groups, the Freudenthal Suspension Theorem, Whitehead's principle, and the computation of homotopy groups of almost all spheres which are known in HoTT.
2016-04-22 19:12:25 +00:00
refine is_conn.elim (n.-1) _ _ a,
move more results and make arguments explicit More results from the Spectral repository are moved to this library Also make various type-class arguments of truncatedness and equivalences which were hard to synthesize explicit
2018-09-11 14:45:30 +00:00
{ esimp, exact homotopy_closed id code_merid_β_right⁻¹ʰᵗʸ _ }
feat(hott): Port files from other repositories to the HoTT library. This commit adds truncated 2-quotients, groupoid quotients, Eilenberg MacLane spaces, chain complexes, the long exact sequence of homotopy groups, the Freudenthal Suspension Theorem, Whitehead's principle, and the computation of homotopy groups of almost all spheres which are known in HoTT.
2016-04-22 19:12:25 +00:00
end
definition code_merid_equiv [constructor] (a : A) : trunc (n + n) A ≃ trunc (n + n) A :=
equiv.mk _ (is_equiv_code_merid a)
definition code_merid_inv_pt (x : trunc (n + n) A) : (code_merid_equiv pt)⁻¹ x = x :=
begin
refine ap010 @(is_equiv.inv _) _ x ⬝ _,
move more results and make arguments explicit More results from the Spectral repository are moved to this library Also make various type-class arguments of truncatedness and equivalences which were hard to synthesize explicit
2018-09-11 14:45:30 +00:00
{ exact homotopy_closed id code_merid_β_right⁻¹ʰᵗʸ _ },
feat(hott): Port files from other repositories to the HoTT library. This commit adds truncated 2-quotients, groupoid quotients, Eilenberg MacLane spaces, chain complexes, the long exact sequence of homotopy groups, the Freudenthal Suspension Theorem, Whitehead's principle, and the computation of homotopy groups of almost all spheres which are known in HoTT.
2016-04-22 19:12:25 +00:00
{ apply is_conn.elim_β},
{ reflexivity}
end
definition code [unfold 4] : susp A → Type :=
susp.elim_type (trunc (n + n) A) (trunc (n + n) A) code_merid_equiv
definition is_trunc_code (x : susp A) : is_trunc (n + n) (code x) :=
begin
induction x with a: esimp,
{ exact _},
{ exact _},
{ apply is_prop.elimo}
end
local attribute is_trunc_code [instance]
definition decode_north [unfold 4] : code north → trunc (n + n) (north = north :> susp A) :=
trunc_functor (n + n) up
definition decode_north_pt : decode_north (tr pt) = tr idp :=
ap tr !con.right_inv
definition decode_south [unfold 4] : code south → trunc (n + n) (north = south :> susp A) :=
trunc_functor (n + n) merid
definition encode' {x : susp A} (p : north = x) : code x :=
transport code p (tr pt)
definition encode [unfold 5] {x : susp A} (p : trunc (n + n) (north = x)) : code x :=
begin
induction p with p,
exact transport code p (tr pt)
end
theorem encode_decode_north (c : code north) : encode (decode_north c) = c :=
begin
have H : Πc, is_trunc (n + n) (encode (decode_north c) = c), from _,
esimp at *,
induction c with a,
rewrite [↑[encode, decode_north, up, code], con_tr, elim_type_merid, ▸*,
code_merid_β_left, elim_type_merid_inv, ▸*, code_merid_inv_pt]
end
definition decode_coh_f (a : A) : tr (up pt) =[merid a] decode_south (code_merid a (tr pt)) :=
begin
refine _ ⬝op ap decode_south (code_merid_β_left a)⁻¹,
apply trunc_pathover,
apply eq_pathover_constant_left_id_right,
apply square_of_eq,
fix(hott/init/path): reorder arguments of whisker_right
2016-11-24 05:13:05 +00:00
exact whisker_right (merid a) !con.right_inv
feat(hott): Port files from other repositories to the HoTT library. This commit adds truncated 2-quotients, groupoid quotients, Eilenberg MacLane spaces, chain complexes, the long exact sequence of homotopy groups, the Freudenthal Suspension Theorem, Whitehead's principle, and the computation of homotopy groups of almost all spheres which are known in HoTT.
2016-04-22 19:12:25 +00:00
end
definition decode_coh_g (a' : A) : tr (up a') =[merid pt] decode_south (code_merid pt (tr a')) :=
begin
refine _ ⬝op ap decode_south (code_merid_β_right (tr a'))⁻¹,
apply trunc_pathover,
apply eq_pathover_constant_left_id_right,
apply square_of_eq, refine !inv_con_cancel_right ⬝ !idp_con⁻¹
end
definition decode_coh_lem {A : Type} {a a' : A} (p : a = a')
fix(hott/init/path): reorder arguments of whisker_right
2016-11-24 05:13:05 +00:00
: whisker_right p (con.right_inv p) = inv_con_cancel_right p p ⬝ (idp_con p)⁻¹ :=
feat(hott): Port files from other repositories to the HoTT library. This commit adds truncated 2-quotients, groupoid quotients, Eilenberg MacLane spaces, chain complexes, the long exact sequence of homotopy groups, the Freudenthal Suspension Theorem, Whitehead's principle, and the computation of homotopy groups of almost all spheres which are known in HoTT.
2016-04-22 19:12:25 +00:00
by induction p; reflexivity
theorem decode_coh (a : A) : decode_north =[merid a] decode_south :=
begin
make pointed suspensions, wedges and spheres the default (in contrast to the unpointed ones), remove sphere_index 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.
2017-07-20 14:01:40 +00:00
apply arrow_pathover_left, intro c,
feat(hott): Port files from other repositories to the HoTT library. This commit adds truncated 2-quotients, groupoid quotients, Eilenberg MacLane spaces, chain complexes, the long exact sequence of homotopy groups, the Freudenthal Suspension Theorem, Whitehead's principle, and the computation of homotopy groups of almost all spheres which are known in HoTT.
2016-04-22 19:12:25 +00:00
induction c with a',
feat(hott/algebra): define additive structures to be multiplicative structures
2016-09-19 18:41:21 +00:00
rewrite [↑code, elim_type_merid],
refine @wedge_extension.ext _ _ n n _ _ (λ a a', tr (up a') =[merid a] decode_south
(to_fun (code_merid_equiv a) (tr a'))) _ _ _ _ a a',
{ intros, apply is_trunc_pathover, apply is_trunc_succ, apply is_trunc_trunc},
feat(hott): Port files from other repositories to the HoTT library. This commit adds truncated 2-quotients, groupoid quotients, Eilenberg MacLane spaces, chain complexes, the long exact sequence of homotopy groups, the Freudenthal Suspension Theorem, Whitehead's principle, and the computation of homotopy groups of almost all spheres which are known in HoTT.
2016-04-22 19:12:25 +00:00
{ exact decode_coh_f},
{ exact decode_coh_g},
{ clear a a', unfold [decode_coh_f, decode_coh_g], refine ap011 concato_eq _ _,
{ refine ap (λp, trunc_pathover (eq_pathover_constant_left_id_right (square_of_eq p))) _,
apply decode_coh_lem},
{ apply ap (λp, ap decode_south p⁻¹), apply code_merid_coh}}
end
definition decode [unfold 4] {x : susp A} (c : code x) : trunc (n + n) (north = x) :=
begin
induction x with a,
{ exact decode_north c},
{ exact decode_south c},
{ exact decode_coh a}
end
theorem decode_encode {x : susp A} (p : trunc (n + n) (north = x)) : decode (encode p) = p :=
begin
induction p with p, induction p, esimp, apply decode_north_pt
end
parameters (A n)
make pointed suspensions, wedges and spheres the default (in contrast to the unpointed ones), remove sphere_index 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.
2017-07-20 14:01:40 +00:00
definition equiv' : trunc (n + n) A ≃ trunc (n + n) (Ω (susp A)) :=
feat(hott): Port files from other repositories to the HoTT library. This commit adds truncated 2-quotients, groupoid quotients, Eilenberg MacLane spaces, chain complexes, the long exact sequence of homotopy groups, the Freudenthal Suspension Theorem, Whitehead's principle, and the computation of homotopy groups of almost all spheres which are known in HoTT.
2016-04-22 19:12:25 +00:00
equiv.MK decode_north encode decode_encode encode_decode_north
make pointed suspensions, wedges and spheres the default (in contrast to the unpointed ones), remove sphere_index 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.
2017-07-20 14:01:40 +00:00
definition pequiv' : ptrunc (n + n) A ≃* ptrunc (n + n) (Ω (susp A)) :=
feat(hott): Port files from other repositories to the HoTT library. This commit adds truncated 2-quotients, groupoid quotients, Eilenberg MacLane spaces, chain complexes, the long exact sequence of homotopy groups, the Freudenthal Suspension Theorem, Whitehead's principle, and the computation of homotopy groups of almost all spheres which are known in HoTT.
2016-04-22 19:12:25 +00:00
pequiv_of_equiv equiv' decode_north_pt
-- We don't prove this:
make pointed suspensions, wedges and spheres the default (in contrast to the unpointed ones), remove sphere_index 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.
2017-07-20 14:01:40 +00:00
-- theorem freudenthal_suspension : is_conn_fun (n+n) (loop_susp_unit A) := sorry
feat(hott): Port files from other repositories to the HoTT library. This commit adds truncated 2-quotients, groupoid quotients, Eilenberg MacLane spaces, chain complexes, the long exact sequence of homotopy groups, the Freudenthal Suspension Theorem, Whitehead's principle, and the computation of homotopy groups of almost all spheres which are known in HoTT.
2016-04-22 19:12:25 +00:00
end end freudenthal
open algebra group
various cleanup changes in library some of the changes are backported from the hott3 library pi_pathover and pi_pathover' are interchanged (same for variants and for sigma) various definitions received explicit arguments: pinverse and eq_equiv_homotopy and ***.sigma_char eq_of_fn_eq_fn is renamed to inj in definitions about higher loop spaces and homotopy groups, the natural number arguments are now consistently before the type arguments
2018-09-07 14:30:58 +00:00
definition freudenthal_pequiv {n k : ℕ} (H : k ≤ 2 * n) (A : Type*) [is_conn n A]
make pointed suspensions, wedges and spheres the default (in contrast to the unpointed ones), remove sphere_index 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.
2017-07-20 14:01:40 +00:00
: ptrunc k A ≃* ptrunc k (Ω (susp A)) :=
feat(hott): Port files from other repositories to the HoTT library. This commit adds truncated 2-quotients, groupoid quotients, Eilenberg MacLane spaces, chain complexes, the long exact sequence of homotopy groups, the Freudenthal Suspension Theorem, Whitehead's principle, and the computation of homotopy groups of almost all spheres which are known in HoTT.
2016-04-22 19:12:25 +00:00
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)
various cleanup changes in library some of the changes are backported from the hott3 library pi_pathover and pi_pathover' are interchanged (same for variants and for sigma) various definitions received explicit arguments: pinverse and eq_equiv_homotopy and ***.sigma_char eq_of_fn_eq_fn is renamed to inj in definitions about higher loop spaces and homotopy groups, the natural number arguments are now consistently before the type arguments
2018-09-07 14:30:58 +00:00
definition freudenthal_equiv {n k : ℕ} (H : k ≤ 2 * n) (A : Type*) [is_conn n A]
make pointed suspensions, wedges and spheres the default (in contrast to the unpointed ones), remove sphere_index 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.
2017-07-20 14:01:40 +00:00
: trunc k A ≃ trunc k (Ω (susp A)) :=
various cleanup changes in library some of the changes are backported from the hott3 library pi_pathover and pi_pathover' are interchanged (same for variants and for sigma) various definitions received explicit arguments: pinverse and eq_equiv_homotopy and ***.sigma_char eq_of_fn_eq_fn is renamed to inj in definitions about higher loop spaces and homotopy groups, the natural number arguments are now consistently before the type arguments
2018-09-07 14:30:58 +00:00
freudenthal_pequiv H A
feat(hott): Port files from other repositories to the HoTT library. This commit adds truncated 2-quotients, groupoid quotients, Eilenberg MacLane spaces, chain complexes, the long exact sequence of homotopy groups, the Freudenthal Suspension Theorem, Whitehead's principle, and the computation of homotopy groups of almost all spheres which are known in HoTT.
2016-04-22 19:12:25 +00:00
various cleanup changes in library some of the changes are backported from the hott3 library pi_pathover and pi_pathover' are interchanged (same for variants and for sigma) various definitions received explicit arguments: pinverse and eq_equiv_homotopy and ***.sigma_char eq_of_fn_eq_fn is renamed to inj in definitions about higher loop spaces and homotopy groups, the natural number arguments are now consistently before the type arguments
2018-09-07 14:30:58 +00:00
definition freudenthal_homotopy_group_pequiv {n k : ℕ} (H : k ≤ 2 * n) (A : Type*) [is_conn n A]
make pointed suspensions, wedges and spheres the default (in contrast to the unpointed ones), remove sphere_index 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.
2017-07-20 14:01:40 +00:00
: π[k + 1] (susp A) ≃* π[k] A :=
feat(hott): Port files from other repositories to the HoTT library. This commit adds truncated 2-quotients, groupoid quotients, Eilenberg MacLane spaces, chain complexes, the long exact sequence of homotopy groups, the Freudenthal Suspension Theorem, Whitehead's principle, and the computation of homotopy groups of almost all spheres which are known in HoTT.
2016-04-22 19:12:25 +00:00
calc
various cleanup changes in library some of the changes are backported from the hott3 library pi_pathover and pi_pathover' are interchanged (same for variants and for sigma) various definitions received explicit arguments: pinverse and eq_equiv_homotopy and ***.sigma_char eq_of_fn_eq_fn is renamed to inj in definitions about higher loop spaces and homotopy groups, the natural number arguments are now consistently before the type arguments
2018-09-07 14:30:58 +00:00
π[k + 1] (susp A) ≃* π[k] (Ω (susp A)) : homotopy_group_succ_in k (susp A)
make pointed suspensions, wedges and spheres the default (in contrast to the unpointed ones), remove sphere_index 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.
2017-07-20 14:01:40 +00:00
... ≃* Ω[k] (ptrunc k (Ω (susp A))) : homotopy_group_pequiv_loop_ptrunc k (Ω (susp A))
various cleanup changes in library some of the changes are backported from the hott3 library pi_pathover and pi_pathover' are interchanged (same for variants and for sigma) various definitions received explicit arguments: pinverse and eq_equiv_homotopy and ***.sigma_char eq_of_fn_eq_fn is renamed to inj in definitions about higher loop spaces and homotopy groups, the natural number arguments are now consistently before the type arguments
2018-09-07 14:30:58 +00:00
... ≃* Ω[k] (ptrunc k A) : loopn_pequiv_loopn k (freudenthal_pequiv H A)
make pointed suspensions, wedges and spheres the default (in contrast to the unpointed ones), remove sphere_index 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.
2017-07-20 14:01:40 +00:00
... ≃* π[k] A : (homotopy_group_pequiv_loop_ptrunc k A)⁻¹ᵉ*
feat(hott): Port files from other repositories to the HoTT library. This commit adds truncated 2-quotients, groupoid quotients, Eilenberg MacLane spaces, chain complexes, the long exact sequence of homotopy groups, the Freudenthal Suspension Theorem, Whitehead's principle, and the computation of homotopy groups of almost all spheres which are known in HoTT.
2016-04-22 19:12:25 +00:00
various cleanup changes in library some of the changes are backported from the hott3 library pi_pathover and pi_pathover' are interchanged (same for variants and for sigma) various definitions received explicit arguments: pinverse and eq_equiv_homotopy and ***.sigma_char eq_of_fn_eq_fn is renamed to inj in definitions about higher loop spaces and homotopy groups, the natural number arguments are now consistently before the type arguments
2018-09-07 14:30:58 +00:00
definition freudenthal_homotopy_group_isomorphism {n k : ℕ} (H : k + 1 ≤ 2 * n)
(A : Type*) [is_conn n A] : πg[k+2] (susp A) ≃g πg[k + 1] A :=
feat(hott): Port files from other repositories to the HoTT library. This commit adds truncated 2-quotients, groupoid quotients, Eilenberg MacLane spaces, chain complexes, the long exact sequence of homotopy groups, the Freudenthal Suspension Theorem, Whitehead's principle, and the computation of homotopy groups of almost all spheres which are known in HoTT.
2016-04-22 19:12:25 +00:00
begin
fapply isomorphism_of_equiv,
various cleanup changes in library some of the changes are backported from the hott3 library pi_pathover and pi_pathover' are interchanged (same for variants and for sigma) various definitions received explicit arguments: pinverse and eq_equiv_homotopy and ***.sigma_char eq_of_fn_eq_fn is renamed to inj in definitions about higher loop spaces and homotopy groups, the natural number arguments are now consistently before the type arguments
2018-09-07 14:30:58 +00:00
{ exact equiv_of_pequiv (freudenthal_homotopy_group_pequiv H A)},
feat(hott): Port files from other repositories to the HoTT library. This commit adds truncated 2-quotients, groupoid quotients, Eilenberg MacLane spaces, chain complexes, the long exact sequence of homotopy groups, the Freudenthal Suspension Theorem, Whitehead's principle, and the computation of homotopy groups of almost all spheres which are known in HoTT.
2016-04-22 19:12:25 +00:00
{ intro g h,
feat(pointed): make the naming in the pointed library more consistent. Also start on a naming conventions file
2016-09-22 19:42:46 +00:00
refine _ ⬝ !homotopy_group_pequiv_loop_ptrunc_inv_con,
fix two errors in the hott library
2018-09-19 23:48:55 +00:00
refine ap !homotopy_group_pequiv_loop_ptrunc⁻¹ᵉ* _,
feat(hott): Port files from other repositories to the HoTT library. This commit adds truncated 2-quotients, groupoid quotients, Eilenberg MacLane spaces, chain complexes, the long exact sequence of homotopy groups, the Freudenthal Suspension Theorem, Whitehead's principle, and the computation of homotopy groups of almost all spheres which are known in HoTT.
2016-04-22 19:12:25 +00:00
refine ap (loopn_pequiv_loopn _ _) _ ⬝ !loopn_pequiv_loopn_con,
feat(pointed): make the naming in the pointed library more consistent. Also start on a naming conventions file
2016-09-22 19:42:46 +00:00
refine ap !homotopy_group_pequiv_loop_ptrunc _ ⬝ !homotopy_group_pequiv_loop_ptrunc_con,
apply homotopy_group_succ_in_con}
feat(hott): Port files from other repositories to the HoTT library. This commit adds truncated 2-quotients, groupoid quotients, Eilenberg MacLane spaces, chain complexes, the long exact sequence of homotopy groups, the Freudenthal Suspension Theorem, Whitehead's principle, and the computation of homotopy groups of almost all spheres which are known in HoTT.
2016-04-22 19:12:25 +00:00
end
move files from Spectral
2018-09-11 16:57:19 +00:00
definition to_pmap_freudenthal_pequiv (n k : ℕ) (H : k ≤ 2 * n) (A : Type*) [is_conn n A]
: freudenthal_pequiv H A ~* ptrunc_functor k (loop_susp_unit A) :=
begin
fapply phomotopy.mk,
{ intro x, induction x with a, reflexivity },
{ refine !idp_con ⬝ _, refine _ ⬝ ap02 _ !idp_con⁻¹, refine _ ⬝ !ap_compose, apply ap_compose }
end
feat(hott) move many lemmas to library, and cleanup various parts
2016-11-23 22:59:13 +00:00
move files from Spectral
2018-09-11 16:57:19 +00:00
definition ptrunc_elim_freudenthal_pequiv (n k : ℕ) (H : k ≤ 2 * n) {A B : Type*} [is_conn n A]
(f : A →* Ω B) [is_trunc (k.+1) (B)] :
ptrunc.elim k (Ω→ (susp_elim f)) ∘* freudenthal_pequiv H A ~* ptrunc.elim k f :=
begin
refine pwhisker_left _ !to_pmap_freudenthal_pequiv ⬝* _,
refine !ptrunc_elim_ptrunc_functor ⬝* _,
exact ptrunc_elim_phomotopy k !ap1_susp_elim,
end
definition freudenthal_pequiv_trunc_index' (A : Type*) (n : ℕ) (k : ℕ₋₂) [HA : is_conn n A]
(H : k ≤ of_nat (2 * n)) : ptrunc k A ≃* ptrunc k (Ω (susp A)) :=
begin
assert lem : Π(l : ℕ₋₂), l ≤ 0 → ptrunc l A ≃* ptrunc l (Ω (susp A)),
{ intro l H', exact ptrunc_pequiv_ptrunc_of_le H' (freudenthal_pequiv (zero_le (2 * n)) A) },
cases k with k, { exact lem -2 (minus_two_le 0) },
cases k with k, { exact lem -1 (succ_le_succ (minus_two_le -1)) },
rewrite [-of_nat_add_two at *, add_two_sub_two at HA],
exact freudenthal_pequiv (le_of_of_nat_le_of_nat H) A
end
feat(hott) move many lemmas to library, and cleanup various parts
2016-11-23 22:59:13 +00:00
feat(hott): Port files from other repositories to the HoTT library. This commit adds truncated 2-quotients, groupoid quotients, Eilenberg MacLane spaces, chain complexes, the long exact sequence of homotopy groups, the Freudenthal Suspension Theorem, Whitehead's principle, and the computation of homotopy groups of almost all spheres which are known in HoTT.
2016-04-22 19:12:25 +00:00
namespace susp
move files from Spectral
2018-09-11 16:57:19 +00:00
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 H (iterate_susp n A)
feat(hott): Port files from other repositories to the HoTT library. This commit adds truncated 2-quotients, groupoid quotients, Eilenberg MacLane spaces, chain complexes, the long exact sequence of homotopy groups, the Freudenthal Suspension Theorem, Whitehead's principle, and the computation of homotopy groups of almost all spheres which are known in HoTT.
2016-04-22 19:12:25 +00:00
move files from Spectral
2018-09-11 16:57:19 +00:00
definition iterate_susp_stability_isomorphism_of_is_conn_0 {k n : ℕ} (H : k + 1 ≤ 2 * n)
(A : Type*) [is_conn 0 A] : 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 H (iterate_susp n A)
feat(hott): Port files from other repositories to the HoTT library. This commit adds truncated 2-quotients, groupoid quotients, Eilenberg MacLane spaces, chain complexes, the long exact sequence of homotopy groups, the Freudenthal Suspension Theorem, Whitehead's principle, and the computation of homotopy groups of almost all spheres which are known in HoTT.
2016-04-22 19:12:25 +00:00
move files from Spectral
2018-09-11 16:57:19 +00:00
definition stability_helper1 {k n : ℕ} (H : k + 2 ≤ 2 * n) : k ≤ 2 * pred n :=
begin
rewrite [mul_pred_right], change pred (pred (k + 2)) ≤ pred (pred (2 * n)),
apply pred_le_pred, apply pred_le_pred, exact H
end
feat(hott): Port files from other repositories to the HoTT library. This commit adds truncated 2-quotients, groupoid quotients, Eilenberg MacLane spaces, chain complexes, the long exact sequence of homotopy groups, the Freudenthal Suspension Theorem, Whitehead's principle, and the computation of homotopy groups of almost all spheres which are known in HoTT.
2016-04-22 19:12:25 +00:00
move files from Spectral
2018-09-11 16:57:19 +00:00
definition stability_helper2 {k n : ℕ} (H : k + 2 ≤ 2 * n) (A : Type*) :
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], exact is_conn_iterate_susp -1 _ A }
end
feat(hott): Port files from other repositories to the HoTT library. This commit adds truncated 2-quotients, groupoid quotients, Eilenberg MacLane spaces, chain complexes, the long exact sequence of homotopy groups, the Freudenthal Suspension Theorem, Whitehead's principle, and the computation of homotopy groups of almost all spheres which are known in HoTT.
2016-04-22 19:12:25 +00:00
move files from Spectral
2018-09-11 16:57:19 +00:00
definition iterate_susp_stability_pequiv {k n : ℕ} (H : k + 2 ≤ 2 * n) (A : Type*) :
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 H A,
freudenthal_homotopy_group_pequiv (stability_helper1 H) (iterate_susp n A)
make pointed suspensions, wedges and spheres the default (in contrast to the unpointed ones), remove sphere_index 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.
2017-07-20 14:01:40 +00:00
move files from Spectral
2018-09-11 16:57:19 +00:00
definition iterate_susp_stability_isomorphism {k n : ℕ} (H : k + 3 ≤ 2 * n) (A : Type*) :
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 (k+1) n H A,
freudenthal_homotopy_group_isomorphism (stability_helper1 H) (iterate_susp n A)
definition iterated_freudenthal_pequiv {n k m : ℕ} (H : k ≤ 2 * n) (A : Type*) [HA : is_conn n 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 },
{ have H2 : succ k ≤ 2 * succ n,
from calc
succ k ≤ succ (2 * n) : succ_le_succ H
... ≤ 2 * succ n : self_le_succ,
exact calc
ptrunc k A ≃* ptrunc k (Ω (susp A)) : freudenthal_pequiv H A
... ≃* 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
feat(hott): Port files from other repositories to the HoTT library. This commit adds truncated 2-quotients, groupoid quotients, Eilenberg MacLane spaces, chain complexes, the long exact sequence of homotopy groups, the Freudenthal Suspension Theorem, Whitehead's principle, and the computation of homotopy groups of almost all spheres which are known in HoTT.
2016-04-22 19:12:25 +00:00
feat(hott) move many lemmas to library, and cleanup various parts
2016-11-23 22:59:13 +00:00
feat(hott): Port files from other repositories to the HoTT library. This commit adds truncated 2-quotients, groupoid quotients, Eilenberg MacLane spaces, chain complexes, the long exact sequence of homotopy groups, the Freudenthal Suspension Theorem, Whitehead's principle, and the computation of homotopy groups of almost all spheres which are known in HoTT.
2016-04-22 19:12:25 +00:00
end susp
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