feat(cohomology): start on cohomology file

homotopy/cohomology.hlean

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+/-
+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 .EM
+
+open spectrum int trunc pointed EM group algebra circle sphere
+
+definition cohomology [reducible] (X : Type*) (Y : spectrum) (n : ℤ) : Set :=
+ttrunc 0 (X →* Ω[2] (Y (n+2)))
+
+definition ordinary_cohomology [reducible] (X : Type*) (G : CommGroup) (n : ℤ) : Set :=
+cohomology X (EM_spectrum G) n
+
+definition ordinary_cohomology_Z [reducible] (X : Type*) (n : ℤ) : Set :=
+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.]