23 lines
1.1 KiB
Text
23 lines
1.1 KiB
Text
|
import .spectrum .EM ..algebra.arrow_group ..algebra.direct_sum .fwedge ..choice .pushout ..move_to_lib ..algebra.product_group
|
|||
|
|
|
|||
|
|
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
|
|||
|
|
|
|||
|
|
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))
|
|||
|
|
)
|
|||
|
|
end homology
|