michael/Spectral
1
0
Fork 0
Code Issues Pull requests Projects Releases Packages Wiki Activity Actions

index spectra by a general succ_str, +Z, or +N, as appropriate

Browse source
This commit is contained in:
Mike Shulman 2016-03-22 08:10:10 -07:00
parent 5379c2e253
commit 559777e45c
1 changed files with 124 additions and 61 deletions
Show stats Download patch file Download diff file Expand all files Collapse all files

185
homotopy/spectrum.hlean
View file

@ -5,7 +5,7 @@ Authors: Michael Shulman
-/ -/
import types.int types.pointed2 types.trunc homotopy.susp algebra.homotopy_group import types.int types.pointed2 types.trunc homotopy.susp algebra.homotopy_group .chain_complex
open eq nat int susp pointed pmap sigma is_equiv equiv fiber algebra trunc trunc_index open eq nat int susp pointed pmap sigma is_equiv equiv fiber algebra trunc trunc_index
/----------------------------------------- /-----------------------------------------
@ -136,99 +136,160 @@ open pointed
Basic definitions Basic definitions
---------------------/ ---------------------/
structure prespectrum := open succ_str
(deloop : ℕ → Type*)
(glue : Πn, (deloop n) →* (Ω (deloop (succ n))))
attribute prespectrum.deloop [coercion] /- The basic definitions of spectra and prespectra make sense for any successor-structure. -/
structure is_spectrum [class] (E : prespectrum) := structure gen_prespectrum (N : succ_str) :=
(is_equiv_glue : Πn, is_equiv (prespectrum.glue E n)) (deloop : N → Type*)
(glue : Π(n:N), (deloop n) →* (Ω (deloop (S n))))
attribute gen_prespectrum.deloop [coercion]
structure is_spectrum [class] {N : succ_str} (E : gen_prespectrum N) :=
(is_equiv_glue : Πn, is_equiv (gen_prespectrum.glue E n))
attribute is_spectrum.is_equiv_glue [instance] attribute is_spectrum.is_equiv_glue [instance]
definition equiv_glue (E : prespectrum) [H : is_spectrum E] (n:ℕ) : (E n) ≃* (Ω (E (succ n))) := structure gen_spectrum (N : succ_str) :=
pequiv_of_pmap (prespectrum.glue E n) _ (to_prespectrum : gen_prespectrum N)
structure spectrum :=
(to_prespectrum : prespectrum)
(to_is_spectrum : is_spectrum to_prespectrum) (to_is_spectrum : is_spectrum to_prespectrum)
attribute spectrum.to_prespectrum [coercion] attribute gen_spectrum.to_prespectrum [coercion]
attribute spectrum.to_is_spectrum [instance] attribute gen_spectrum.to_is_spectrum [instance]
-- Classically, spectra and prespectra use the successor structure +ℕ.
-- But we will use +ℤ instead, to reduce case analysis later on.
abbreviation spectrum := gen_spectrum +ℤ
abbreviation spectrum.mk := @gen_spectrum.mk +ℤ
namespace spectrum namespace spectrum
abbreviation glue := prespectrum.glue definition glue {{N : succ_str}} := @gen_prespectrum.glue N
--definition glue := (@gen_prespectrum.glue +ℤ)
definition equiv_glue {N : succ_str} (E : gen_prespectrum N) [H : is_spectrum E] (n:N) : (E n) ≃* (Ω (E (S n))) :=
pequiv_of_pmap (glue E n) (is_spectrum.is_equiv_glue E n)
-- An easy way to define a spectrum. -- Sometimes an ℕ-indexed version does arise naturally, however, so
definition MK (deloop : ℕ → Type*) (glue : Πn, (deloop n) ≃* (Ω (deloop (succ n)))) : spectrum := -- we give a standard way to extend an ℕ-indexed (pre)spectrum to a
spectrum.mk (prespectrum.mk deloop (λn, glue n)) (is_spectrum.mk (λn, _)) -- ℤ-indexed one.
/- Spectrum maps -/ definition psp_of_nat_indexed [constructor] (E : gen_prespectrum +ℕ) : gen_prespectrum +ℤ :=
structure smap (E F : prespectrum) := gen_prespectrum.mk
(to_fun : Πn, E n →* F n) (λ(n:ℤ), match n with
(glue_square : Πn, glue F n ∘* to_fun n ~* Ω→ (to_fun (succ n)) ∘* glue E n) | of_nat k := E k
| neg_succ_of_nat k := Ω[succ k] (E 0)
end)
begin
intros n, cases n with n n: esimp,
{ exact (gen_prespectrum.glue E n) },
cases n with n,
{ exact (pid _) },
{ exact (pid _) }
end
definition is_spectrum_of_nat_indexed [instance] (E : gen_prespectrum +ℕ) [H : is_spectrum E] : is_spectrum (psp_of_nat_indexed E) :=
begin
apply is_spectrum.mk, intros n, cases n with n n: esimp,
{ apply is_spectrum.is_equiv_glue },
cases n with n: apply is_equiv_id
end
protected definition of_nat_indexed (E : gen_prespectrum +ℕ) [H : is_spectrum E] : spectrum
:= spectrum.mk (psp_of_nat_indexed E) (is_spectrum_of_nat_indexed E)
-- In fact, a (pre)spectrum indexed on any pointed successor structure
-- gives rise to one indexed on +ℕ, so in this sense +ℤ is a
-- "universal" successor structure for indexing spectra.
definition succ_str.of_nat {N : succ_str} (z : N) : ℕ → N
| succ_str.of_nat zero := z
| succ_str.of_nat (succ k) := S (succ_str.of_nat k)
definition psp_of_gen_indexed [constructor] {N : succ_str} (z : N) (E : gen_prespectrum N) : gen_prespectrum +ℤ :=
psp_of_nat_indexed (gen_prespectrum.mk (λn, E (succ_str.of_nat z n)) (λn, gen_prespectrum.glue E (succ_str.of_nat z n)))
definition is_spectrum_of_gen_indexed [instance] {N : succ_str} (z : N) (E : gen_prespectrum N) [H : is_spectrum E]
: is_spectrum (psp_of_gen_indexed z E) :=
begin
apply is_spectrum_of_nat_indexed, apply is_spectrum.mk, intros n, esimp, apply is_spectrum.is_equiv_glue
end
protected definition of_gen_indexed [constructor] {N : succ_str} (z : N) (E : gen_spectrum N) : spectrum :=
spectrum.mk (psp_of_gen_indexed z E) (is_spectrum_of_gen_indexed z E)
-- Generally it's easiest to define a spectrum by giving 'equiv's
-- directly. This works for any indexing succ_str.
protected definition MK {N : succ_str} (deloop : N → Type*) (glue : Π(n:N), (deloop n) ≃* (Ω (deloop (S n)))) : gen_spectrum N :=
gen_spectrum.mk (gen_prespectrum.mk deloop (λ(n:N), glue n))
(begin
apply is_spectrum.mk, intros n, esimp,
apply pequiv.to_is_equiv -- Why doesn't typeclass inference find this?
end)
-- Finally, we combine them and give a way to produce a (ℤ-)spectrum from a ℕ-indexed family of 'equiv's.
protected definition Mk (deloop : ℕ → Type*) (glue : Π(n:ℕ), (deloop n) ≃* (Ω (deloop (nat.succ n)))) : spectrum :=
spectrum.of_nat_indexed (spectrum.MK deloop glue)
-- (Pre)spectrum maps. These make sense for any succ_str.
structure smap {N : succ_str} (E F : gen_prespectrum N) :=
(to_fun : Π(n:N), E n →* F n)
(glue_square : Π(n:N), glue F n ∘* to_fun n ~* Ω→ (to_fun (S n)) ∘* glue E n)
open smap open smap
infix ` →ₛ `:30 := smap infix ` →ₛ `:30 := smap
attribute smap.to_fun [coercion] attribute smap.to_fun [coercion]
definition sglue_square {E F : spectrum} (f : E →ₛ F) (n : ℕ) -- A version of 'glue_square' in the spectrum case that uses 'equiv_glue'
: equiv_glue F n ∘* f n ~* Ω→ (f (succ n)) ∘* equiv_glue E n definition sglue_square {N : succ_str} {E F : gen_spectrum N} (f : E →ₛ F) (n : N)
:= glue_square f n : equiv_glue F n ∘* f n ~* Ω→ (f (S n)) ∘* equiv_glue E n
-- I guess this is necessary because structures lack definitional eta?
:= phomotopy.mk (glue_square f n) (to_homotopy_pt (glue_square f n))
definition scompose {X Y Z : prespectrum} (g : Y →ₛ Z) (f : X →ₛ Y) : X →ₛ Z := definition scompose {N : succ_str} {X Y Z : gen_prespectrum N} (g : Y →ₛ Z) (f : X →ₛ Y) : X →ₛ Z :=
smap.mk (λn, g n ∘* f n) smap.mk (λn, g n ∘* f n)
(λn, calc glue Z n ∘* to_fun g n ∘* to_fun f n (λn, calc glue Z n ∘* to_fun g n ∘* to_fun f n
~* (glue Z n ∘* to_fun g n) ∘* to_fun f n : passoc ~* (glue Z n ∘* to_fun g n) ∘* to_fun f n : passoc
... ~* (Ω→(to_fun g (succ n)) ∘* glue Y n) ∘* to_fun f n : pwhisker_right (to_fun f n) (glue_square g n) ... ~* (Ω→(to_fun g (S n)) ∘* glue Y n) ∘* to_fun f n : pwhisker_right (to_fun f n) (glue_square g n)
... ~* Ω→(to_fun g (succ n)) ∘* (glue Y n ∘* to_fun f n) : passoc ... ~* Ω→(to_fun g (S n)) ∘* (glue Y n ∘* to_fun f n) : passoc
... ~* Ω→(to_fun g (succ n)) ∘* (Ω→ (f (succ n)) ∘* glue X n) : pwhisker_left Ω→(to_fun g (succ n)) (glue_square f n) ... ~* Ω→(to_fun g (S n)) ∘* (Ω→ (f (S n)) ∘* glue X n) : pwhisker_left Ω→(to_fun g (S n)) (glue_square f n)
... ~* (Ω→(to_fun g (succ n)) ∘* Ω→(f (succ n))) ∘* glue X n : passoc ... ~* (Ω→(to_fun g (S n)) ∘* Ω→(f (S n))) ∘* glue X n : passoc
... ~* Ω→(to_fun g (succ n) ∘* to_fun f (succ n)) ∘* glue X n : pwhisker_right (glue X n) (ap1_compose _ _)) ... ~* Ω→(to_fun g (S n) ∘* to_fun f (S n)) ∘* glue X n : pwhisker_right (glue X n) (ap1_compose _ _))
infixr ` ∘ₛ `:60 := scompose infixr ` ∘ₛ `:60 := scompose
/- Suspension prespectra -/ /- Suspension prespectra -/
definition psp_suspn : ℕ → Type* → Type* -- This should probably go in 'susp'
| psp_suspn 0 X := X definition psuspn : ℕ → Type* → Type*
| psp_suspn (succ n) X := psusp (psp_suspn n X) | psuspn 0 X := X
| psuspn (succ n) X := psusp (psuspn n X)
definition psp_susp_oo (X : Type*) := -- Suspension prespectra are one that's naturally indexed on the natural numbers
prespectrum.mk (λn, psp_suspn n X) (λn, loop_susp_unit (psp_suspn n X)) definition psp_susp (X : Type*) : gen_prespectrum +ℕ :=
gen_prespectrum.mk (λn, psuspn n X) (λn, loop_susp_unit (psuspn n X))
/- Truncations -/ /- Truncations -/
-- We could truncate prespectra too, but since the operation
-- preserves spectra and isn't "correct" acting on prespectra
-- without spectrifying them first anyway, why bother?
definition strunc (k : ℕ₋₂) (E : spectrum) : spectrum := definition strunc (k : ℕ₋₂) (E : spectrum) : spectrum :=
spectrum.MK (λ(n:ℕ), ptrunc (k + n) (E n)) spectrum.Mk (λ(n:ℕ), ptrunc (k + n) (E n))
(λ(n:ℕ), (loop_ptrunc_pequiv (k + n) (E (succ n)))⁻¹ᵉ* ∘*ᵉ (ptrunc_pequiv_ptrunc (k + n) (equiv_glue E n))) (λ(n:ℕ), (loop_ptrunc_pequiv (k + n) (E (succ n)))⁻¹ᵉ*
∘*ᵉ (ptrunc_pequiv_ptrunc (k + n) (equiv_glue E (int.of_nat n))))
/--------------------- /---------------------
Homotopy groups Homotopy groups
---------------------/ ---------------------/
/- A spectrum has homotopy groups indexed by all integers. The naive -- Here we start to reap the rewards of using ℤ-indexing: we can
definition would be -- read off the homotopy groups without any tedious case-analysis of
-- n. We increment by 2 in order to ensure that they are all
match n with -- automatically abelian groups.
| neg_succ_of_nat k := π[0] (E (1+k)) definition shomotopy_group [constructor] (n : ℤ) (E : spectrum) : CommGroup := πag[0+2] (E (2 + n))
| of_nat k := π[k] (E 0)
end
but in order to ensure easily that they are all abelian groups, we
start shifting out earlier. Since homotopy groups commute
appropriately with loop spaces, this is equivalent.
-/
definition shomotopy_group [constructor] (n : ℤ) (E : spectrum) : CommGroup :=
match n with
| neg_succ_of_nat k := πag[0+2] (E (3 + k))
| of_nat 0 := πag[0+2] (E 2)
| of_nat 1 := πag[0+2] (E 1)
| of_nat (succ (succ k)) := πag[k+2] (E 0)
end
notation `πₛ[`:95 n:0 `] `:0 E:95 := shomotopy_group n E notation `πₛ[`:95 n:0 `] `:0 E:95 := shomotopy_group n E
@ -236,17 +297,19 @@ namespace spectrum
Cotensor of spectra by types Cotensor of spectra by types
-------------------------------/ -------------------------------/
definition sp_cotensor (A : Type*) (B : spectrum) : spectrum := -- Makes sense for any indexing succ_str. Could be done for
-- prespectra too, but as with truncation, why bother?
definition sp_cotensor {N : succ_str} (A : Type*) (B : gen_spectrum N) : gen_spectrum N :=
spectrum.MK (λn, ppmap A (B n)) spectrum.MK (λn, ppmap A (B n))
(λn, (loop_pmap_commute A (B (succ n)))⁻¹ᵉ* ∘*ᵉ (equiv_ppcompose_left (equiv_glue B n))) (λn, (loop_pmap_commute A (B (S n)))⁻¹ᵉ* ∘*ᵉ (equiv_ppcompose_left (equiv_glue B n)))
/----------------------------------------- /-----------------------------------------
Fibers and long exact sequences Fibers and long exact sequences
-----------------------------------------/ -----------------------------------------/
definition sfiber (E F : spectrum) (f : E →ₛ F) : spectrum := definition sfiber {N : succ_str} (E F : gen_spectrum N) (f : E →ₛ F) : gen_spectrum N :=
spectrum.MK (λn, pfiber (f n)) spectrum.MK (λn, pfiber (f n))
(λn, pfiber_loop_space (f (succ n)) ∘*ᵉ pfiber_equiv_of_square (sglue_square f n)) (λn, pfiber_loop_space (f (S n)) ∘*ᵉ pfiber_equiv_of_square (sglue_square f n))
/- Mapping spectra -/ /- Mapping spectra -/