Universal property of the smash product (of pointed spaces)

Question

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.

Answer 1

$X \wedge Y$ 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.

Answer 2

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.

Answer 3

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.

Answer 4

Akhil, as one of those who struggled, I'd like to point out that nobody in a million years would have first come up with such an esoteric construction (if it is one) of the smash product of spectra, and nobody who actually understands their calculational role would dream of thinking that to be the primary version of the construction, or its most important property, or something that one would actually use as "the'' smash product. It may arguably be a useful point of view, even a very useful one, but not for algebraic topology as a calculationally intensive subject. It sheds no light on many of the calculationally central features of the smash product. There are different categories of spectra with different good constructions of "the" smash product, and it doesn't help to give the idea that the notion is solely the $\infty$-category version. It is not. And it really is too bad to try to tell people that "stability is an idea that you need $\infty$-categories to make sense of''. That idea has been very well understood since before I started out 50 years ago. You are referring narrowly to stability of $\infty$-categories, so you only ``need'' it when that is all that you are referring to. Not everything in life (I mean mathematics, no, I mean algebraic topology) is $\infty$-categories, not by a long shot. End of lecture.

Answer 5

I have found in my work that it is often useful to generalise from pointed spaces to pairs of spaces. This is what led me to the gluing theorem for homotopy equivalences published in my 1968 book "Elements of Modern topology", now "Topology and Groupoids".

Instead of pairs one can also do $M$-ads, where $M$ is an indexing set; there is a product, and a smash product, and an exponential law using the latter, see section 4 of my 1964 paper "Function spaces and product topologies" Quart. J. Math. (1964). As explained there: " However, the smashed product as defined here of spaces with base point $ X, Y$ is a 2-ad but is not a space with base point."

There should be some relation with the notion of Carrier, see Spanier and Whitehead, "Carriers and S-theory" so maybe there is a stable theory of carriers?

Also I do think the exponential law which is really about monoidal closed categories, not known, at least to me, in 1964, should be seen as implying a notion of bimorphism.