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The title of this paper is a bit tricky, because (it's been known for a long time) permutative categories do *not* have a symmetric monoidal tensor product. Not symmetric monoidal *as a category*. But, what we show, is that it is symmetric monoidal *as a bicategory*. And, we include enough details so you can see just where the difference occurs.

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The paper has a lot more. It includes many of the basic things that go along with tensor products, and the details of how they work in this setting. If you're really into that kind of thing, I think you'll find it useful!

Here, I just want to add a couple of weird/neat things: (1) coproducts via the Gray tensor product; (2) subtleties about the monoidal unit.

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1. The direct sum (coproduct) of permutative categories comes from the Gray tensor product! For permutative categories A and B, you can form one-object 2-categories ΣA and ΣB. Then their Gray tensor product, ΣA ⊠ ΣB, is another 1-object 2-category, and it has a permutative category C of 1- and 2-cells.

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That permutative category C is the direct sum A ⊕ B. We also give a straightforward objects-and-morphisms definition, but I think the connection with the Gray tensor product is neat. This direct sum is equivalent (not isomorphic) to the cartesian product!

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2. The monoidal unit, S, is also known as the natural number category. It's objects are natural numbers, and it's morphisms are given by permutations of n-elements sets. A subtle but apparently important fact is that the 2-unitors are not strict.

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This *seems* related to corresponding monoidal unit subtleties for (a) EKMM S-modules, where S is the sphere spectrum, and (b) Elmendorf-Mandell M1-modules (in pointed multicategories). We don't have a mathematical statement of this relationship, just a feeling we can't put into words.

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@johncarlosbaez oh, yeah it's a coproduct in a *bi*categorical sense; adjoint equivalence instead of isomorphism. I'm sure there's a more precise term I should have used, but here's the precise statement.

Wierdly, if one restricts to *strict* symmetric monoidal functors for the hom sets, then that equivalence is an isomorphism, so it's a 1-categorical coproduct. (That's on the next page, but I don't know what to make of it.)

@nilesjohnson - Thanks! Nice!

Btw you may not have noticed it since it showed up in a funny context, but I did a bunch of 1-categorical work on PROPs here:

https://arxiv.org/abs/1707.08321

especially in the appendix, where we take advantage of a certain multisorted Lawvere theory whose algebras are PROPs.

@johncarlosbaez oh, I did not know about that; thanks! Donald and I have another thing coming out soon with more about multilinearity and connections to K-theory, so I should think about how this connects to that too.

@upennig Enjoy! I think this stuff is super cool, but I'm nervous about how long it's taken us just to pin down all the details for the ungraded tensor product.