Q:

Universal Property of the Smash Product (of pointed spaces)

Hey
Is there a universal property for the smash product (of pointed spaces or pointed CW-complexes or something of that ilk)?   I've seen the smash product of spectra defined with a universal property in terms of the smash product of pointed-spaces, but I was wondering if there was just some simple universal property you could put on these somewhat mysterious (to me) space-level operations. 
EDIT: I was hoping for something more satisfying than the internal Hom adjoint.   The tensor product can be defined similarly, but I find the universal product in terms of bilinear maps more intuitive (although, when unraveled, they are the same thing).    I was hoping for something similar for smash.   
Thanks :)

A:

$X \wedge Y$ represents maps from $X \times Y$ that are base-point-preserving separately in each variable, just as the tensor product represents maps that are linear separately in each variable. 

A:

There is a universal property of pointed spaces and the smash product, in an $\infty$-categorical sense.  Let $S_*$ be the $\infty$-category of pointed spaces.  Then the functor $S_\ast \times S_\ast \to S_\ast$ is the unique  colimit-preserving functor which satisfies $S^0 \wedge S^0 = S^0$. Since you mention spectra, this property is analogous to the characterization of the smash product of spectra as the unique functor which preserves colimits (in each variable) and such that $S^0 \wedge S^0 = S^0$ (for $S^0$ here the sphere spectrum).  
One reason you should expect such a functor to a) exist and b) give an interesting symmetric monoidal structure is the following.  The $\infty$-category $S_\ast$ is the free pointed  $\infty$-category on an object: that is, given a pointed $\infty$-category $\mathcal{C}$ with all colimits, there is an equivalence $\mathrm{Fun}^L(S_\ast, \mathcal{C}) \simeq \mathcal{C}$ between colimit-preserving functors $S_* \to \mathcal{C}$ and objects of $\mathcal{C}$ (given by evaluation on $S^0$).  This is a toy analog of the fact that $\mathrm{Fun}^L(Sp, \mathcal{C}) \simeq \mathcal{C}$ for a stable $\infty$-category $\mathcal{C}$ with all colimits: that is, spectra are the free stable $\infty$-category on one object.  
In general such "free" objects tend to admit symmetric monoidal structures.  Here's one way to get the monoidal structure: by the above, pointed spaces are precisely 
the same thing as colimit-preserving functors $S_\ast \to S_\ast$ (i.e., any such is given by smashing with a pointed space).  So the smash product of spaces comes from composing functors; in other words, the monoidal structure comes from composition on $\mathrm{Fun}^L(S_\ast, S_\ast)$. Another approach, which gives the symmetric monoidal structure, is to use Lurie's "tensor product" of presentable $\infty$-categories.  I don't understand this very well, but I think the idea is that tensoring a presentable $\infty$-category with $S_*$ corresponds to taking the "pointed envelope," and so tensoring with $S_\ast$ is actually an idempotent operation on presentable $\infty$-categories.  
This point of view is  useful with spectra; there the point is that spectra are the free stable presentable $\infty$-category on one object.  There you have to replace "pointed" with "stable" throughout.  
Most of this is in Lurie's papers and his book "Higher Algebra."  One upshot of this is that you can use it define the smash product of spectra, in a manner analogous to the tensor product of abelian groups. 

A:

This is NOT an answer.  I would like to answer Akhil's last question to me in a comment, but I just don't see any place to comment: no place to click in sight anywhere.   I'm not technologically adept.  That
is the only reason I've given past "answers'' where comments would be appropriate.   The definition of "the'' smash product of spectra (admittedly off topic: I agree that Rune gave the best answer to the original space level question) is subtle.  There is a beautiful model theoretic characterization in
Brooke Shipley's paper "Monoidal uniqueness of stable homotopy theory".  Advances in Math.  160 (2001), 
217--240.   For the construction, done in terms of diagram categories (symmetric or orthogonal spectra), there is information packed into the domain category $\mathcal{D}$ that has to be accounted for and is more that can reasonably be described in terms of stable cells and gluing.   The first such constructions were two step, first a Kan extension to build a smash product of $\mathcal{D}$-spaces, which are in no sense spectra, and then a coequalizer "tensor product'' construction to (loosely) build in the spheres.   By enlarging $\mathcal{D}$ so as to pack more information into it, one can get a one-step construction,
as explained in Mandell, May, Schwede, and Shipley "Model categories of diagram spectra".  Proc.  London Math.  Soc. ( 3) 82(2001), 441--512.   Still, the universal property that is the characterization is not helpfully thought of in terms of gluing cells.   Rather, there is a simple naive construction of an external smash product that goes from $\mathcal{D}$-spectra to $(\mathcal{D}\times\mathcal{D})$-spectra, and then the role of Kan extension is to internalize it, using a choice of $\mathcal{D}$ that is itself symmetric monoidal.    The whole problem is to build the symmetric monoidal structure.  That, on the space level, is already characterized by the function space adjunction.   On the spectrum level, defining function spectra is just as subtle as defining smash products.   Internalizing an obvious external smash product is also the theme in the $S$-module construction.  This is intrinsic to the mathematics, at least on the level of actual categories.