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In 1995, Robert Thomason published “Symmetric monoidal categories model all connective spectra” in TAC. On page 2, he argues that symmetric monoidal categories are more convenient than “May’s coordinate-free spectra” for various reasons, one of which is that a symmetric monoidal structure is easier to obtain (this was before EKMM or the Hovey-Shipley-Smith paper on symmetric spectra). Thomason writes:

As convincing evidence for this claim, I refer to my talk at the Colloque en l’honneur de Michel Zisman at l’Universite Paris VII in June 1993. There I used this alternate model of stable homotopy to give the first known construction of a smash product which is associative and commutative up to coherent natural isomorphism in the model category. This will be the subject of an article to appear.

Unfortunately, he died later that year. Hence my question:

Did anyone ever work out the construction of the smash product he had in mind?

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I think that the best thing in this direction is the paper "Permutative categories, multicategories and algebraic K-theory" by Elmendorff and Mandell.

I have only skimmed this so I may well not be understanding it correctly.

Anyway, we want to consider symmetric monoidal categories, whose monoidal structure should be thought of as "addition", so I will call them "plus-monoidal categories". We want to define a kind of tensor product of plus-monoidal categories. Elmendorff and Mandell say that we should first generalise and consider plus-multicategories. We can then define multilinear functors between plus-multicategories (as at https://ncatlab.org/nlab/show/multicategory), making the class of plus-multicategories into a tensor-multicategory. One can then show that there are universal multilinear functors in a suitable sense, so we can form tensor products of plus-multicategories as desired.

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  • $\begingroup$ Hi Neil. Thanks for the answer. I guess the obvious follow-up question would be: does this monoidal product on plus-multicategories line up with the monoidal product on connective spectra, under the equivalence that Thomason provides? $\endgroup$ – David White Jan 16 '18 at 13:37
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    $\begingroup$ @DavidWhite I think that's supposed to be section 6 in the paper I mentioned, although there is a rather large network of definitions and references to unwind. $\endgroup$ – Neil Strickland Jan 16 '18 at 14:10
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As for connective spectra, a good smash product was first worked out by Manos Lydakis in the context of $\Gamma$-spaces. Thomason came to Bielefeld in the mid 1990s and spoke with Waldhausen's group about this matter. Lydakis carried it through.

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    $\begingroup$ Thanks for sharing! I knew about Lydakis, but didn't know about the connection to Thomason. $\endgroup$ – David White Jan 16 '18 at 14:25

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