This is the canonical tensor product in the symmetric monoidal (infinity,1)-category of spectra. There are various model category presentations for which which are symmetric monoidal model categories (such as the highly structured spectra: S-modules, symmetric spectra and orthogonal spectra, but also for instance excisive functors). See at symmetric monoidal category of spectra for more on this. Passing to the stable homotopy category, it become a symmetric monoidal category under the smash product. Under mild assumptions, this is the essentially unique symmetric tensor product with unit the sphere spectrum (Shipley 01).
Historically the discussion proceeded in the opposite direction: available variants of the construction of smash products on sequential spectra (the “handicrafted” or “naive” smash products Boardman 65, Adams 74, part III, section 4) were found to yield a symmetric monoidal category structure only after passage to the stable homotopy category. Then models via non-sequential highly structured spectra were discovered which do admit a symmetric smash product of spectra in the sense of 1-category theory, as do excisive functors, see at model structure on excisive functors.
The symmetric smash product of spectra on, in particular, symmetric spectra and orthogonal spectra is the Day convolution product for Top-enriched functors on monoidal categories of symmetric groups of orthogonal groups, respectively (MMSS 00, theorem 1.7 and section 21.).
Similarly the symmetric smash product of spectra on the model structure for excisive functors is Day convolution for sSet-enriched functors on the plain smash product of finite pointed simplicial sets (Lydakis 98).
See also at functor with smash products.
The smash product of spectra exhibits a certain graded commutativity akin to the graded commutativity in the tensor product of chain complexes (in fact, under the stable Dold-Kan correspondence the latter maps to the former).
is given by permuting the coordinates:
This has degree .
Here has objects and as hom-spaces it has , identified as the adjunct of the canonical isomorphism .
Hence identifies the th component space of a sequential spectrum with the value of an excisive functor on the -sphere, and it identifies the structure map with part of the enriched functoriality of the excisive functor.
Now in the model category of excisive functors, the correct smash product of spectra is the Day convolution over the symmetric monoidal category . The issue then is that the restricted hom-spaces of do not see the non-trivial braiding of spheres in prop. 1 anymore.
More concretely, the enriched category does not inherit monoidal structure: defining the smash product on hom spaces requires permuting smash copies of spheres, which is not available. Thus there is no Day convolution product on sequential spectra at all.
One could further restrict along and use the monoidal structure to define at least a smash product on sequences of pointed spaces by Day convolution over as in (MMSS 00, example 4.1, Hovey-Shipley-Smith 00, below prop. 2.3.4). But then in addition to the above problem that this does not give a functorial smash product on spectra (it will not respect the structure maps), moreover is trivially braided and so, again, under restriction of excisive functors to there is no way to recover the information in the smash product of spectra that is encoded in the non-trivial braiding of the smash product of spheres.
The original “handicrafted” constructions of the smash product on the stable homotopy category are due to
Michael Boardman, Stable homotopy theory, mimeographed notes, University of Warwick, 1965 onwards
Rainer Vogt, Boardman’s stable homotopy category, lectures, spring 1969
for symmetric spectra:
and for excisive functors in
The uniqueness of the smash product on spectra is discussed in