As A Symmetric Monoidal Product
For any pointed spaces X, Y, and Z in an appropriate "convenient" category (e.g. that of compactly generated spaces) there are natural (basepoint preserving) homeomorphisms
However, for the naive category of pointed spaces, this fails. See the following discussion on MathOverflow.
These isomorphisms make the appropriate category of pointed spaces into a symmetric monoidal category with the smash product as the monoidal product and the pointed 0-sphere (a two-point discrete space) as the unit object. One can therefore think of the smash product as a kind of tensor product in an appropriate category of pointed spaces.
Read more about this topic: Smash Product
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