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{-# OPTIONS --type-in-type --without-K #-}
open import lib.Prelude
open Truncation
open LoopSpace
open Int
module homotopy.Whitehead where
module LoopEquivToPathEquiv {A B : Type}
(f : AB)
(zero : IsEquiv {Trunc (tl 0) A} {Trunc (tl 0) B} (Trunc-func f))
(loops : (x : A) → IsEquiv{(Path{A} x x)} {(Path{B} (f x) (f x))} (ap f)) where
eqv : (x : A) (x' : A) (α : x ≃ x') IsEquiv{(Path{A} x x')}{(Path{B} (f x) (f x'))} (ap f)
eqv x .x id = loops x
paths' : (x : A) (x' : A) IsWEq{(Path{A} x x')}{(Path{B} (f x) (f x'))} (ap f)
paths' x x' β = Trunc-rec (Contractible-is-HProp _)
(λ α coe (! (IsWeq≃IsEquiv (ap f))) (eqv x x' α) β)
fact2 where
fact1 : Path{Trunc (tl 0) A} ([ x ]) ([ x' ])
fact1 = IsEquiv.α zero [ x' ] ∘ ap (IsEquiv.g zero) (ap [_] β) ∘ ! (IsEquiv.α zero [ x ])
fact2 : Trunc -1 (Path x x')
fact2 = coe (! (TruncPath.path -1)) fact1
abstract
paths : (x : A) (x' : A) IsEquiv{(Path{A} x x')}{(Path{B} (f x) (f x'))} (ap f)
paths x x' = coe (IsWeq≃IsEquiv (ap f)) (paths' x x')
module SplitEquiv {A B : Type}
(f : AB)
(zero : IsEquiv {Trunc (tl 0) A} {Trunc (tl 0) B} (Trunc-func f))
(one : (x : A) → IsEquiv {(Path{A} x x)} {(Path{B} (f x) (f x))} (ap f))
where
one' : (x x' : A) IsEquiv {(Path{A} x x')} {(Path{B} (f x) (f x'))} (ap f)
one' = LoopEquivToPathEquiv.paths f zero one
iseqv' : IsWEq f
iseqv' y = gen tx tβ where
tx : Trunc (tl 0) A
tx = IsEquiv.g zero [ y ]
: Path{Trunc (tl 0) B} (Trunc-func f (IsEquiv.g zero [ y ])) [ y ]
= IsEquiv.β zero [ y ]
gen : (x : Trunc (tl 0) A) Path{Trunc (tl 0) B} (Trunc-func f x) [ y ]
-> Contractible (HFiber f y)
gen = Trunc-elim _ (λ _ increment-level (Πlevel (λ _ Contractible-is-HProp _)))
(λ x tα
Trunc-rec (Contractible-is-HProp _)
(λ α (x , α) ,
(λ {(x' , α') pair≃ (IsEquiv.g (one' x x') (! α' ∘ α))
(transport (λ v Path (f v) y) (IsEquiv.g (one' x x') (! α' ∘ α)) α ≃〈 transport-Path-pre' f (IsEquiv.g (one' x x') (! α' ∘ α)) α 〉
α ∘ ! (ap f (IsEquiv.g (one' x x') (! α' ∘ α))) ≃〈 ap (λ x0 α ∘ ! x0) (IsEquiv.β (one' x x') (! α' ∘ α)) 〉
α ∘ ! (! α' ∘ α) ≃〈 ap (λ x0 α ∘ x0) (!-∘ (! α') α) 〉
α ∘ ! α ∘ ! (! α') ≃〈 !-inv-r-front α (! (! α')) 〉
! (! α') ≃〈 !-invol α' 〉
α' ∎)}))
(coe (! (TruncPath.path -1)) tα))
iseqv : IsEquiv f
iseqv = coe (IsWeq≃IsEquiv f) iseqv'
module Whitehead-NType where
whitehead : {A B : Type} (n : Positive)
(nA : NType (tlp n) A) (nB : NType (tlp n) B)
(f : A B)
(zero : IsEquiv {Trunc (tl 0) A} {Trunc (tl 0) B} (Trunc-func f))
(pis : k x IsEquiv{π k A x}{π k B (f x)} (Trunc-func (ap^ k f)))
-> IsEquiv f
whitehead One nA nB f zero pis =
SplitEquiv.iseqv f zero
(λ x snd (UnTrunc.eqv (tl 0) _ (use-level nB (f x) (f x))
∘equiv (Trunc-func (ap f) , pis One x)
∘equiv (!equiv (UnTrunc.eqv (tl 0) _ (use-level nA x x)))))
whitehead {A}{B} (S n) nA nB f zero pis = SplitEquiv.iseqv f zero (λ x IH x) where
IH : (x : A) IsEquiv {Path x x}{Path (f x) (f x)} (ap f)
IH x = whitehead n (use-level nA x x) (use-level nB (f x) (f x)) (ap f)
(pis One x)
(λ k α transport IsEquiv (coe-incr-pis k α) (coe-is-equiv (incr-pis k α))) where
incr-pis : k α (π k (Path {A} x x) α) ≃ (π k (Path {B} (f x) (f x)) (ap f α))
incr-pis k α =
-- optimized proof-term
ap (Trunc (tl 0)) ((! (rebase-Loop-Path k (ap f α))) ∘ (LoopPath.path k))
∘ ua (Trunc-func (ap^ (S k) f) , pis (S k) x)
∘ ap (Trunc (tl 0)) ((! (LoopPath.path {A} {x} k)) ∘ (rebase-Loop-Path k α))
{- legible version:
π k (Path {A} x x) α ≃〈 ap (Trunc (tl 0)) (rebase-Loop-Path k α) 〉
π k (Path {A} x x) id ≃〈 ap (Trunc (tl 0)) (! (LoopPath.path {A} {x} k)) 〉
π (S k) A x ≃〈 ua (Trunc-func (ap^ (S k) f) , pis (S k) x) 〉
π (S k) B (f x) ≃〈 ap (Trunc (tl 0)) (LoopPath.path k) 〉
π k (Path {B} (f x) (f x)) id ≃〈 ap (Trunc (tl 0)) (! (rebase-Loop-Path k (ap f α))) 〉
π k (Path {B} (f x) (f x)) (ap f α) ∎
-}
coe-incr-pis : k α -> coe (incr-pis k α) ≃ Trunc-func (ap^ k (ap f))
coe-incr-pis k α = coe (incr-pis k α) ≃〈 rearrange (ap^ (S k) f) (pis (S k) x) (! (rebase-Loop-Path k (ap f α))) (LoopPath.path k) (! (LoopPath.path {A} {x} k)) (rebase-Loop-Path k α)〉
Trunc-func ( coe (! (rebase-Loop-Path k (ap f α)))
o coe (LoopPath.path k)
o (ap^ (S k) f)
o coe (! (LoopPath.path {A} {x} k))
o coe (rebase-Loop-Path k α)) ≃〈 ap Trunc-func STS 〉
Trunc-func (ap^ k (ap f)) ∎ where
rearrange : {A B C C' D E : Type} (f : C C') (e : IsEquiv (Trunc-func f))
1 : D ≃ E) (α2 : C' ≃ D) (α3 : B ≃ C) (α4 : A ≃ B)
coe (ap (Trunc (tl 0)) (α1 ∘ α2)
∘ ua (Trunc-func f , e)
∘ ap (Trunc (tl 0)) (α3 ∘ α4))
≃ Trunc-func (coe α1 o coe α2 o f o coe α3 o coe α4)
rearrange f e id id id id = type≃β (Trunc-func f , e) ∘ ap coe (∘-unit-l (ua (Trunc-func f , e)))
reduce-rebase-Loop-Path :
{x' : A} (α : x ≃ x')
(fl : {x'} (α : x ≃ x')
Loop k (Path x x') α
Loop k (Path (f x) (f x')) (ap f α))
-> Path
(coe (! (rebase-Loop-Path k (ap f α))) o
fl id o
coe (rebase-Loop-Path k α))
(fl α)
reduce-rebase-Loop-Path id fl = id
STS : ( coe (! (rebase-Loop-Path k (ap f α)))
o coe (LoopPath.path k)
o (ap^ (S k) f)
o coe (! (LoopPath.path {A} {x} k))
o coe (rebase-Loop-Path k α))
≃ (ap^ k (ap f))
STS = coe (! (rebase-Loop-Path k (ap f α))) o
coe (LoopPath.path k) o
ap^ (S k) f o
coe (! (LoopPath.path {A} {x} k)) o
coe (rebase-Loop-Path k α) ≃〈 ap2 (λ x' y coe (! (rebase-Loop-Path k (ap f α))) o x' o ap^ (S k) f o y o coe (rebase-Loop-Path k α)) (type≃β (LoopPath.eqv k)) (type≃β! (LoopPath.eqv k)) 〉
coe (! (rebase-Loop-Path k (ap f α))) o
loopSN1 k o ap^ (S k) f o loopN1S k o
coe (rebase-Loop-Path k α) ≃〈 ap (λ x' coe (! (rebase-Loop-Path k (ap f α))) o x' o coe (rebase-Loop-Path k α)) (! (λ≃ (ap^-ap-assoc k f))) 〉
coe (! (rebase-Loop-Path k (ap f α))) o
(ap^ k (ap f)) o
coe (rebase-Loop-Path k α) ≃〈 reduce-rebase-Loop-Path α (λ {x' : A} (α' : Path x x') (l : Loop k (Id x x') α') ap^ k (ap f) l) 〉
(ap^ k (ap f) ∎)
-- ENH strengthen to
-- (pis : ∀ k x → k <=tl n → IsEquiv{(π k A x)}{(π k B (f x))} (Trunc-func (ap^ k f)))
-- by truncation the rest are trivial
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