Merge branch 'master' of https://github.com/cmu-phil/Spectral
This commit is contained in:
Egbert Rijke
commit
b027186436
6 changed files with 112 additions and 30 deletions
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@ -98,4 +98,4 @@ These projects are mostly done
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- We will try to make sure that this repository compiles with the newest version of Lean 2.
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- Installation instructions for Lean 2 can be found [here](https://github.com/leanprover/lean2).
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- Some notes on the Emacs mode can be found [here](https://github.com/leanprover/lean2/blob/master/src/emacs/README.md) (for example if some unicode characters don't show up, or increase the spacing between lines by a lot).
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- If you contribute, please use rebase instead off merge (e.g. `git pull -r`).
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- If you contribute, please use rebase instead of merge (e.g. `git pull -r`).
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@ -1,7 +1,11 @@
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/- various groups of maps. Most importantly we define a group structure on trunc 0 (A →* Ω B),
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which is used in the definition of cohomology -/
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/-
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Copyright (c) 2016-2017 Floris van Doorn. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Floris van Doorn, Ulrik Buchholtz
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--author: Floris van Doorn
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Various groups of maps. Most importantly we define a group structure
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on trunc 0 (A →* Ω B), which is used in the definition of cohomology.
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-/
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import algebra.group_theory ..pointed ..pointed_pi eq2 homotopy.susp
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open pi pointed algebra group eq equiv is_trunc trunc susp
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@ -242,5 +246,59 @@ namespace group
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definition AbGroup_trunc_ppi [reducible] [constructor] {A : Type*} (B : A → Type*) : AbGroup :=
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AbGroup.mk (trunc 0 (Π*a, Ω (Ω (B a)))) _
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definition trunc_ppi_isomorphic_pmap (A B : Type*)
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: Group.mk (trunc 0 (Π*(a : A), Ω B)) !trunc_group
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≃g Group.mk (trunc 0 (A →* Ω B)) !trunc_group :=
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begin
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apply trunc_isomorphism_of_equiv (ppi_equiv_pmap A (Ω B)),
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intro h k, induction h with h h_pt, induction k with k k_pt, reflexivity
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end
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section
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universe variables u v
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variables {A : pType.{u}} {B : A → Type.{v}} {x₀ : B pt} {k l m : ppi_gen B x₀}
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definition ppi_homotopy_of_eq_homomorphism (p : k = l) (q : l = m)
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: ppi_homotopy_of_eq (p ⬝ q) = ppi_homotopy_of_eq p ⬝*' ppi_homotopy_of_eq q :=
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begin
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induction q, induction p, induction k with k q, induction q, reflexivity
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end
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protected definition ppi_mul_loop.lemma1 {X : Type} {x : X} (p q : x = x) (p_pt : idp = p) (q_pt : idp = q)
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: refl (p ⬝ q) ⬝ whisker_left p q_pt⁻¹ ⬝ p_pt⁻¹ = p_pt⁻¹ ◾ q_pt⁻¹ :=
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by induction p_pt; induction q_pt; reflexivity
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protected definition ppi_mul_loop.lemma2 {X : Type} {x : X} (p q : x = x) (p_pt : p = idp) (q_pt : q = idp)
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: refl (p ⬝ q) ⬝ whisker_left p q_pt ⬝ p_pt = p_pt ◾ q_pt :=
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by rewrite [-(inv_inv p_pt),-(inv_inv q_pt)]; exact ppi_mul_loop.lemma1 p q p_pt⁻¹ q_pt⁻¹
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definition ppi_mul_loop {h : Πa, B a} (f g : ppi_gen.mk h idp ~~* ppi_gen.mk h idp) : f ⬝*' g = ppi_mul f g :=
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begin
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apply ap (ppi_gen.mk (λa, f a ⬝ g a)),
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apply ppi_gen.rec_on f, intros f' f_pt, apply ppi_gen.rec_on g, intros g' g_pt,
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clear f g, esimp at *, exact ppi_mul_loop.lemma2 (f' pt) (g' pt) f_pt g_pt
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end
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variable (k)
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definition trunc_ppi_loop_isomorphism_lemma
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: isomorphism.{(max u v) (max u v)}
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(Group.mk (trunc 0 (k = k)) (@trunc_group (k = k) !inf_group_loop))
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(Group.mk (trunc 0 (Π*(a : A), Ω (pType.mk (B a) (k a)))) !trunc_group) :=
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begin
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apply @trunc_isomorphism_of_equiv _ _ !inf_group_loop !inf_group_ppi (ppi_loop_equiv k),
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intro f g, induction k with k p, induction p,
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apply trans (ppi_homotopy_of_eq_homomorphism f g),
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exact ppi_mul_loop (ppi_homotopy_of_eq f) (ppi_homotopy_of_eq g)
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end
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end
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definition trunc_ppi_loop_isomorphism {A : Type*} (B : A → Type*)
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: Group.mk (trunc 0 (Ω (Π*(a : A), B a))) !trunc_group
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≃g Group.mk (trunc 0 (Π*(a : A), Ω (B a))) !trunc_group :=
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trunc_ppi_loop_isomorphism_lemma (ppi_const B)
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end group
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@ -1,7 +1,7 @@
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/-
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Copyright (c) 2016 Floris van Doorn. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Floris van Doorn
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Authors: Floris van Doorn, Ulrik Buchholtz
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Reduced cohomology of spectra and cohomology theories
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-/
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@ -80,7 +80,13 @@ sorry /- TODO FOR SSS -/
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definition parametrized_cohomology_isomorphism_shomotopy_group_spi {X : Type*} (Y : X → spectrum)
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{n m : ℤ} (p : -m = n) : pH^n[(x : X), Y x] ≃g πₛ[m] (spi X Y) :=
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sorry /- TODO FOR SSS -/
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begin
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apply isomorphism.trans (trunc_ppi_loop_isomorphism (λx, Ω (Y x (n + 2))))⁻¹ᵍ,
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apply homotopy_group_isomorphism_of_pequiv 0, esimp,
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have q : sub 2 m = n + 2,
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from (int.add_comm (of_nat 2) (-m) ⬝ ap (λk, k + of_nat 2) p),
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rewrite q, symmetry, apply ppi_loop_pequiv
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end
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definition unreduced_parametrized_cohomology_isomorphism_shomotopy_group_supi {X : Type}
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(Y : X → spectrum) {n m : ℤ} (p : -m = n) : upH^n[(x : X), Y x] ≃g πₛ[m] (supi X Y) :=
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@ -1,19 +1,44 @@
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/-
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Copyright (c) 2017 Floris van Doorn and Ulrik Buchholtz. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Floris van Doorn, Ulrik Buchholtz
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Truncatedness and truncation of spectra
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-/
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import .spectrum .EM
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namespace int
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-- TODO move this
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open trunc_index nat algebra
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open nat algebra
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section
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private definition maxm2_le.lemma₁ {n k : ℕ} : n+(1:int) + -[1+ k] ≤ n :=
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le.intro (
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calc n + 1 + -[1+ k] + k = n + 1 - (k + 1) + k : by reflexivity
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... = n : sorry) /- TODO FOR SSS -/
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calc n + 1 + -[1+ k] + k
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= n + 1 + (-(k + 1)) + k : by reflexivity
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... = n + 1 + (-1 - k) + k : by krewrite (neg_add_rev k 1)
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... = n + 1 + (-1 - k + k) : add.assoc
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... = n + 1 + (-1 + -k + k) : by reflexivity
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... = n + 1 + (-1 + (-k + k)) : add.assoc
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... = n + 1 + (-1 + 0) : add.left_inv
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... = n + (1 + (-1 + 0)) : add.assoc
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... = n : int.add_zero)
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private definition maxm2_le.lemma₂ {n : ℕ} {k : ℤ} : -[1+ n] + 1 + k ≤ k :=
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le.intro (
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calc -[1+ n] + 1 + k + n = - (n + 1) + 1 + k + n : by reflexivity
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... = k : sorry) /- TODO FOR SSS -/
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calc -[1+ n] + 1 + k + n
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= - (n + 1) + 1 + k + n : by reflexivity
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... = -n - 1 + 1 + k + n : by rewrite (neg_add n 1)
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... = -n + (-1 + 1) + k + n : by krewrite (int.add_assoc (-n) (-1) 1)
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... = -n + 0 + k + n : add.left_inv 1
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... = -n + k + n : int.add_zero
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... = k + -n + n : int.add_comm
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... = k + (-n + n) : int.add_assoc
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... = k + 0 : add.left_inv n
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... = k : int.add_zero)
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open trunc_index
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definition maxm2_le (n k : ℤ) : maxm2 (n+1+k) ≤ (maxm1m1 n).+1+2+(maxm1m1 k) :=
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begin
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@ -439,6 +439,13 @@ namespace group
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definition isomorphism_of_is_contr {G H : Group} (hG : is_contr G) (hH : is_contr H) : G ≃g H :=
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trivial_group_of_is_contr G ⬝g (trivial_group_of_is_contr H)⁻¹ᵍ
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definition trunc_isomorphism_of_equiv {A B : Type} [inf_group A] [inf_group B] (f : A ≃ B)
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(h : is_mul_hom f) : Group.mk (trunc 0 A) (trunc_group A) ≃g Group.mk (trunc 0 B) (trunc_group B) :=
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begin
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apply isomorphism_of_equiv (equiv.mk (trunc_functor 0 f) (is_equiv_trunc_functor 0 f)), intros x x',
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induction x with a, induction x' with a', apply ap tr, exact h a a'
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end
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end group open group
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namespace fiber
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@ -190,27 +190,13 @@ namespace pointed
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induction t, exact eq_ppi_homotopy_refl_ppi_homotopy_of_eq_refl k ▸ q,
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end
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definition ppi_loop_equiv_lemma (p : k ~ k)
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: (p pt ⬝ ppi_gen.resp_pt k = ppi_gen.resp_pt k) ≃ (p pt = idp) :=
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calc (p pt ⬝ ppi_gen.resp_pt k = ppi_gen.resp_pt k)
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≃ (p pt ⬝ ppi_gen.resp_pt k = idp ⬝ ppi_gen.resp_pt k)
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: equiv_eq_closed_right (p pt ⬝ ppi_gen.resp_pt k) (inverse (idp_con (ppi_gen.resp_pt k)))
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... ≃ (p pt = idp)
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: eq_equiv_con_eq_con_right
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variables (k l)
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definition ppi_loop_equiv : (k = k) ≃ Π*(a : A), Ω (pType.mk (B a) (k a)) :=
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calc (k = k) ≃ (k ~~* k)
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: ppi_eq_equiv
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... ≃ Σ(p : k ~ k), p pt ⬝ ppi_gen.resp_pt k = ppi_gen.resp_pt k
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: ppi_homotopy.sigma_char k k
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... ≃ Σ(p : k ~ k), p pt = idp
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: sigma_equiv_sigma_right
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(λ p, ppi_loop_equiv_lemma p)
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... ≃ Π*(a : A), pType.mk (k a = k a) idp
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: ppi.sigma_char
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... ≃ Π*(a : A), Ω (pType.mk (B a) (k a))
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: erfl
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begin
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induction k with k p, induction p,
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exact ppi_eq_equiv (ppi_gen.mk k idp) (ppi_gen.mk k idp)
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end
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variables {k l}
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-- definition eq_of_ppi_homotopy (h : k ~~* l) : k = l :=
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