Spectrum (homotopy Theory) - Smash Products of Spectra

Smash Products of Spectra

The smash product of spectra extends the smash product of CW complexes. It is somewhat cumbersome to define. It makes the stable homotopy category into a monoidal category; in other words it behaves like the 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 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.