diff --git a/homotopy/homology.hlean b/homotopy/homology.hlean
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+++ b/homotopy/homology.hlean
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+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