Smash Product - As A Symmetric Monoidal Product

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

\begin{align} X \wedge Y &\cong Y\wedge X, \\ (X\wedge Y)\wedge Z &\cong X \wedge (Y\wedge Z). \end{align}

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.

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