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
Famous quotes containing the word product:
“In fast-moving, progress-conscious America, the consumer expects to be dizzied by progress. If he could completely understand advertising jargon he would be badly disappointed. The half-intelligibility which we expect, or even hope, to find in the latest product language personally reassures each of us that progress is being made: that the pace exceeds our ability to follow.”
—Daniel J. Boorstin (b. 1914)