Why do we use the smash product in the category of based topological spaces? I was telling someone about the smash product and he asked whether it was the categorical product in the category of based spaces and I immediately said yes, but after a moment we realized that that wasn't right.  Rather, the categorical product of $(X,x_0)$ and $(Y,y_0)$ is just $(X\times Y,(x_0,y_0))$.  (It seems like in any concrete category $(\mathcal{C},U)$, if we have a product (does a concrete category always have products?) then it must be that $U(X\times Y)=U(X)\times U(Y)$.  But I couldn't prove it.  I should learn category theory.  Maybe functors commute with products or something.)  Anyways, here's what I'm wondering: is the main reason that we like the smash product just that it gives the right exponential law?  It's easy to see that the product $\times$ I gave above has $F(X\times Y,Z)\not\cong F(X,F(Y,Z))$ just by taking e.g. $X=Y=Z=S^0$.

From nLab:

The smash product is the tensor product in the closed monoidal category of pointed sets. That is,
  $$\operatorname{Fun}_*(A\wedge B,C)\cong \operatorname{Fun}_*(A,\operatorname{Fun}_*(B,C))$$
  Here, $\operatorname{Fun}_*(A,B)$ is the set of basepoint-preserving functions from $A$ to $B$, itself made into a pointed set by taking as basepoint the constant function from all of $A$ to the basepoint in $B$.

There's more at the link.  I must admit that I know nothing about this, but I recommend nLab as a good place to look for the categorical place of mathematical constructions.
