import ..homotopy.spectrum ..homotopy.EM ..algebra.arrow_group ..algebra.direct_sum ..homotopy.fwedge ..choice ..homotopy.pushout ..move_to_lib

open eq spectrum int trunc pointed EM group algebra circle sphere nat EM.ops equiv susp is_trunc
     function fwedge cofiber bool lift sigma is_equiv choice pushout algebra unit pi smash

namespace homology

  /- homology theory -/

  structure homology_theory.{u} : Type.{u+1} :=
    (HH : ℤ → pType.{u} → AbGroup.{u})
    (Hh : Π(n : ℤ) {X Y : Type*} (f : X →* Y), HH n X →g HH n Y)
    (Hid : Π(n : ℤ) {X : Type*} (x : HH n X), Hh n (pid X) x = x)
    (Hcompose : Π(n : ℤ) {X Y Z : Type*} (g : Y →* Z) (f : X →* Y) (x : HH n X),
      Hh n (g ∘* f) x = Hh n g (Hh n f x))
    (Hsusp : Π(n : ℤ) (X : Type*), HH (succ n) (psusp X) ≃g HH n X)
    (Hsusp_natural : Π(n : ℤ) {X Y : Type*} (f : X →* Y),
      Hsusp n Y ∘ Hh (succ n) (psusp_functor f) ~ Hh n f ∘ Hsusp n X)
    (Hexact : Π(n : ℤ) {X Y : Type*} (f : X →* Y), is_exact_g (Hh n f) (Hh n (pcod f)))
    (Hadditive : Π(n : ℤ) {I : Set.{u}} (X : I → Type*), is_equiv (
      dirsum_elim (λi, Hh n (pinl i)) : dirsum (λi, HH n (X i)) → HH n (⋁ X))
  )

  section
    parameter (theory : homology_theory)
    open homology_theory

    definition HH_base_indep (n : ℤ) {A : Type} (a b : A)
    : HH theory n (pType.mk A a) ≃g HH theory n (pType.mk A b) :=
      calc HH theory n (pType.mk A a) ≃g HH theory (int.succ n) (psusp A) : by exact (Hsusp theory n (pType.mk A a)) ⁻¹ᵍ
                                  ... ≃g HH theory n (pType.mk A b)       : by exact Hsusp theory n (pType.mk A b)

  end

end homology