/-
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

Reduced cohomology
-/

import algebra.arrow_group .spectrum homotopy.EM

open eq spectrum int trunc pointed EM group algebra circle sphere nat EM.ops equiv susp

namespace cohomology

definition EM_spectrum /-[constructor]-/ (G : AbGroup) : spectrum :=
spectrum.Mk (K G) (λn, (loop_EM G n)⁻¹ᵉ*)

definition cohomology (X : Type*) (Y : spectrum) (n : ℤ) : AbGroup :=
AbGroup_pmap X (πag[2] (Y (2+n)))

definition ordinary_cohomology [reducible] (X : Type*) (G : AbGroup) (n : ℤ) : AbGroup :=
cohomology X (EM_spectrum G) n

definition ordinary_cohomology_Z [reducible] (X : Type*) (n : ℤ) : AbGroup :=
ordinary_cohomology X agℤ n

notation `H^` n `[`:0 X:0 `, ` Y:0 `]`:0 := cohomology X Y n
notation `H^` n `[`:0 X:0 `]`:0 := ordinary_cohomology_Z X n

-- check H^3[S¹*,EM_spectrum agℤ]
-- check H^3[S¹*]

definition unpointed_cohomology (X : Type) (Y : spectrum) (n : ℤ) : AbGroup :=
cohomology X₊ Y n

definition cohomology_homomorphism [constructor] {X X' : Type*} (f : X' →* X) (Y : spectrum)
  (n : ℤ) : cohomology X Y n →g cohomology X' Y n :=
Group_pmap_homomorphism f (πag[2] (Y (2+n)))

definition cohomology_homomorphism_id (X : Type*) (Y : spectrum) (n : ℤ) (f : H^n[X, Y]) :
  cohomology_homomorphism (pid X) Y n f ~* f :=
!pcompose_pid

definition cohomology_homomorphism_compose {X X' X'' : Type*} (g : X'' →* X') (f : X' →* X)
  (Y : spectrum) (n : ℤ) (h : H^n[X, Y]) : cohomology_homomorphism (f ∘* g) Y n h ~*
  cohomology_homomorphism g Y n (cohomology_homomorphism f Y n h) :=
!passoc⁻¹*

end cohomology

exit
definition cohomology_psusp (X : Type*) (Y : spectrum) (n : ℤ) :
  H^n+1[psusp X, Y] ≃ H^n[X, Y] :=
calc
  H^n+1[psusp X, Y] ≃ psusp X →* πg[2] (Y (2+(n+1))) : by reflexivity
  ... ≃ X →* Ω (πg[2] (Y (2+(n+1)))) : psusp_adjoint_loop_unpointed
--  ... ≃ X →* πg[3] (Y (2+(n+1))) : _
--... ≃ X →* πag[3] (Y ((2+n)+1)) : _
  ... ≃ X →* πg[2] (Y (2+n)) :
    begin
      refine equiv_of_pequiv (pequiv_ppcompose_left _),
      refine !homotopy_group_succ_o ⬝ _,
      exact sorry --refine _ ⬝e* _ ⬝e* _
    end
  ... ≃ H^n[X, Y] : by reflexivity