Reference for an unbiased definition of a symmetric monoidal category

Question

In the references I know, a symmetric monoidal category is defined as a monoidal category equipped with some additional data, so in particular the monoidal product is a functor $$ \mathcal{C}\times \mathcal{C}\to \mathcal{C}. $$ The word unbiased is sometimes used to mean that there are products $$ \mathcal{C}^n\to \mathcal{C} $$ but that is not quite what I mean here. Rather, I want a monoidal category to give me an assignment for every finite set $S$ $$ \mathcal{C}^S\to \mathcal{C} $$ and for everything to be described in terms of functors involving the category of (unordered) finite sets, maybe some weak kind of functor since associativity isn't strict.

1. Does this definition exist? That is, is there some obstruction that is not obvious to me that prevents it from working?
2. Is it written down clearly and cleanly somewhere? This seems like a natural alternative to the standard definition that could sometimes be useful to eliminate certain bookkeeping.
3. Why isn't this viewpoint more common? Or is it common and I've just been reading the wrong books?

Answer 1

This isn't quite an answer to your question either, but Appendix A of my 2003 book Higher Operads, Higher Categories contains something close.

First it defines commutative monoids in the style you describe: as a set $A$ together with a function $A^S \to A$ for each finite set $S$, satisfying some axioms.

Then it defines symmetric multicategories in the same style: as a set $A_0$ (to be thought of as the set of objects) together with, for each set-indexed family of objects $(a_s)_{s \in S}$ and each object $b$, a set $Hom(a_\cdot; b)$, together with suitable composition and identities.

Although symmetric monoidal categories aren't treated there, playing the same game a third time shouldn't be hard. (Alternatively, you could describe symmetric monoidal categories as symmetric multicategories with certain special properties.)

I first learned of this style of definition from Beilinson and Drinfeld's Chiral Algebras (then a set of notes, now a book, I think).

Answer 2

This is not quite what you mean, but relevant. Remember that a strictly associative and unital symmetric monoidal category is called a permutative category. I observed ages ago (http://www.math.uchicago.edu/~may/PAPERS/13.pdf) that the unbiased version of that is the same thing as an algebra over the categorical Barratt-Eccles operad. Recently Corner and Gurski, https://arxiv.org/abs/1312.5910, defined pseudoalgebras over operads in $Cat$. A pseudoalgebra over the categorical Barratt-Eccles operad is the unbiased version of a symmetric monoidal category that you are looking for, except that you ask for functors defined on all unordered finite sets. But, as usual, it is standard and reasonable when considering diagram categories, to pass to a skeleton so as to replace essentially small domain categories with actual small ones, and then one arrives at the definition just cited.

Incidentally, this unbiased variant is essential in equivariant theory in progress where we define genuine symmetric monoidal $G$-categories and show how to construct genuine $G$-spectra from them, where $G$ is a finite group (joint work with Guillou, Merling, and Osorno, to be posted soon; see http://www.math.uchicago.edu/~may/TALKS/Chicago2015.pdf).