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In addition to the property with respect to the internal hom that Qiaochu mentions, there

There is a universal property of pointed spaces and the smash product, in an $\infty$-categorical sense. (I should add the disclaimer that I'm a beginner in all this, and I hope nothing I say is too far wrong!)

Namely, let 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. (This is one of the refrains in Lurie's work, as I understand it, and it's a really powerful fact. It means that it's very easy to give a functor out of spectra into a stable $\infty$-category! By contrast saying these things with model categories is likely to be much more awkward.)

In general such "free" things objects tend to admit symmetric monoidal structures. Here's one way to get the monoidal structure: by the above, pointed spaces are precisely

This point of view is a lot more useful with spectra--- ; there the point is that spectra are the free stable presentable $\infty$-category on one object. (Note that this is really an $\infty$-categorical phenomenon: stability is an idea that you need $\infty$-categories to make sense of.) There you have to replace "pointed" with "stable" throughout. Most of this is in Lurie's papers and his book "Higher Algebra." Anyway, the point One upshot of this is that you can get use it define the smash product of spectrapurely formally, while in previous years people had to struggle a manner analogous to define it (though the smash tensor product of spaces is not hard to define...)abelian groups.

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In addition to the property with respect to the internal hom that Qiaochu mentions, there is a universal property of pointed spaces and the smash product, in an $\infty$-categorical sense. (I should add the disclaimer that I'm a beginner in all this, and I hope nothing I say is too far wrong!)

Namely, 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. (This is one of the refrains in Lurie's work, as I understand it, and it's a really powerful fact. It means that it's very easy to give a functor out of spectra into a stable $\infty$-category! By contrast saying these things with model categories is likely to be much more awkward.)

In general such "free" things 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 a lot more useful with spectra --- there the point is that spectra are the free stable presentable $\infty$-category on one object. (Note that this is really an $\infty$-categorical phenomenon: stability is an idea that you need $\infty$-categories to make sense of.) There you have to replace "pointed" with "stable" throughout. Most of this is in Lurie's papers and his book "Higher Algebra." Anyway, the point of this is that you can get the smash product of spectra purely formally, while in previous years people had to struggle to define it (though the smash product of spaces is not hard to define...).