is a natural embedding of the category of pointed CW complexes into this category: it takes to the suspension spectrum in which the nth complex is . The smash product of a spectrum and a pointed complex is a spectrum given by (associativity of the smash product yields immediately that this is indeed a spectrum). A homotopy of maps between spectra corresponds to a map , where is the disjoint union with taken to be the basepoint. The stable homotopy category, or homotopy category of (CW) spectra is defined to be the category whose objects are spectra and whose morphisms are homotopy classes of maps between spectra. Many other definitions of spectrum, some appearing very different, lead to equivalent stable homotopy categories. Finally, we can define the suspension of a spectrum by . This translation suspension is invertible, as we can desuspend too, by setting . The triangulated homotopy category of spectra The stable homotopy category is additive: maps can be added by using a variant of the track addition used to define homotopy groups. Thus homotopy classes from one spectrum to another form an abelian group. Furthermore the stable homotopy category is triangulated (Vogt (1970)), the shift being given by suspension and the distinguished triangles by the mapping cone sequences of spectra . Smash products of spectra The smash product of spectra extends the smash product of CW complexes. It makes the stable homotopy category into a monoidal category; in other words it behaves like the (derived) tensor product of abelian groups. A major problem with the smash product is that obvious ways of defining it make it associative and commutative only up to homotopy. Some more recent definitions of spectra, such as symmetric spectra, eliminate this problem, and give a symmetric monoidal structure at the level of maps, before passing to homotopy classes. The smash product is compatible with the triangulated category structure. In particular the smash product of a distinguished triangle with a spectrum is a distinguished triangle. Generalized homology and cohomology of spectra We can define the (stable) homotopy groups of a spectrum to be those given by , where is the sphere spectrum and is the set of homotopy classes of maps from to . We define the generalized homology theory of a spectrum E by and define its generalized cohomology theory by Here can be a spectrum or (by using its suspension spectrum) a