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fix indexing for homotopy group of presprectrum

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Floris van Doorn 2018-03-24 17:08:41 -04:00
parent bcb78b4575
commit 9b624edb9f
1 changed files with 7 additions and 7 deletions
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14
spectrum/basic.hlean
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@ -729,28 +729,28 @@ namespace spectrum
/- homotopy group of a prespectrum -/ /- homotopy group of a prespectrum -/
definition pshomotopy_group_hom (n : ℤ) (E : prespectrum) (k : ℕ) definition pshomotopy_group_hom (n : ℤ) (E : prespectrum) (k : ℕ)
: πag[k + 2] (E (-n - 2 + k)) →g πag[k + 3] (E (-n - 2 + (k + 1))) := : πag[k + 2] (E (-n + 2 + k)) →g πag[k + 3] (E (-n + 2 + (k + 1))) :=
begin begin
refine _ ∘g π→g[k+2] (glue E _), refine _ ∘g π→g[k+2] (glue E _),
refine (ghomotopy_group_succ_in _ (k+1))⁻¹ᵍ ∘g _, refine (ghomotopy_group_succ_in _ (k+1))⁻¹ᵍ ∘g _,
refine homotopy_group_isomorphism_of_pequiv (k+1) refine homotopy_group_isomorphism_of_pequiv (k+1)
(loop_pequiv_loop (pequiv_of_eq (ap E (add.assoc (-n - 2) k 1)))) (loop_pequiv_loop (pequiv_of_eq (ap E (add.assoc (-n + 2) k 1))))
end end
definition pshomotopy_group (n : ℤ) (E : prespectrum) : AbGroup := definition pshomotopy_group (n : ℤ) (E : prespectrum) : AbGroup :=
group.seq_colim (λ(k : ℕ), πag[k+2] (E (-n - 2 + k))) (pshomotopy_group_hom n E) group.seq_colim (λ(k : ℕ), πag[k+2] (E (-n + 2 + k))) (pshomotopy_group_hom n E)
notation `πₚₛ[`:95 n:0 `]`:0 := pshomotopy_group n notation `πₚₛ[`:95 n:0 `]`:0 := pshomotopy_group n
definition pshomotopy_group_fun (n : ℤ) {E F : prespectrum} (f : E →ₛ F) : definition pshomotopy_group_fun (n : ℤ) {E F : prespectrum} (f : E →ₛ F) :
πₚₛ[n] E →g πₚₛ[n] F := πₚₛ[n] E →g πₚₛ[n] F :=
proof proof
group.seq_colim_functor (λk, π→g[k+2] (f (-n - 2 +[ℤ] k))) group.seq_colim_functor (λk, π→g[k+2] (f (-n + 2 +[ℤ] k)))
begin begin
intro k, intro k,
note sq1 := homotopy_group_homomorphism_psquare (k+2) (ptranspose (smap.glue_square f (-n - 2 +[ℤ] k))), note sq1 := homotopy_group_homomorphism_psquare (k+2) (ptranspose (smap.glue_square f (-n + 2 +[ℤ] k))),
note sq2 := homotopy_group_functor_psquare (k+2) (ap1_psquare (ptransport_natural E F f (add.assoc (-n - 2) k 1))), note sq2 := homotopy_group_functor_psquare (k+2) (ap1_psquare (ptransport_natural E F f (add.assoc (-n + 2) k 1))),
note sq3 := (homotopy_group_succ_in_natural (k+2) (f (-n - 2 +[ℤ] (k+1))))⁻¹ʰᵗʸʰ, note sq3 := (homotopy_group_succ_in_natural (k+2) (f (-n + 2 +[ℤ] (k+1))))⁻¹ʰᵗʸʰ,
note sq4 := hsquare_of_psquare sq2, note sq4 := hsquare_of_psquare sq2,
note rect := sq1 ⬝htyh sq4 ⬝htyh sq3, note rect := sq1 ⬝htyh sq4 ⬝htyh sq3,
exact sorry --sq1 ⬝htyh sq4 ⬝htyh sq3, exact sorry --sq1 ⬝htyh sq4 ⬝htyh sq3,