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Does the following mathematical gadget have a standard name? Let $R$ be an $\mathbb{N}$-graded ring together with an $S_n$ action on each $R_n$ which are compatible in the following sense. Let $i:S_a \times S_b \rightarrow S_{a+b}$ be the standard inclusion. If $x$ is in grade $a$ with $\sigma \in S_a$ and $y$ is in grade $b$ with $\tau$ in $S_b$, then $$\sigma(a) \cdot \tau(b) = i(\sigma,\tau)(a\cdot b).$$

The main example I have in mind is the ring of G-invariant tensors.

The point is to have the right language to say that such a gadget has certain generators and relations, where I think of the symmetric group actions as unary operations.

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  • $\begingroup$ Now that Nicholas has posted the answer, searching for that answer says that this question is roughly a duplicate of another question. $\endgroup$
    – Noah Snyder
    Commented Oct 29, 2017 at 19:25

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Steven Sam and Andrew Snowden and their other collaborators call these `twisted commutative algebras' and have been having fun writing papers about properties of generators and similar.

But predating this, any topologist of a certain sort would call this a monoid in the category of symmetric abelian groups. If you do the same construction in simplicial sets (or topological spaces), an example is the sphere spectrum, with nth space $S^n$. Modules over this object are what are called symmetric spectra, and serve as one of the main models of modern day stable homotopy theory. (This was an observation by Jeff Smith in the mid 1990s.)

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  • $\begingroup$ Thanks, the Sam-Snowden papers were exactly what I was looking for. The comment about spectra is interesting, though I’m not sure yet how it relates to what I was thinking about. $\endgroup$
    – Noah Snyder
    Commented Oct 29, 2017 at 12:24
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As a warmup, an $\mathbb{N}$-graded ring is a monoid object in the symmetric monoidal category of $\mathbb{N}$-graded abelian groups under the convolution tensor product, which you can think of as Day convolution from the usual addition on $\mathbb{N}$.

Similarly, this thing is a monoid object in the symmetric monoidal category of species in abelian groups (presheaves on the category $S$ of finite sets and bijections valued in abelian groups) under the convolution tensor product, which you can again think of as Day convolution from disjoint union on $S$.

So I might be inclined to call such a thing an $S$-graded ring, and you can feel free to replace $S$ with your favorite name for $S$, maybe $\text{FinSet}^{\times}$ or something.

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  • $\begingroup$ Nifty! I was thinking “this is kind of like species” but didn’t exactly understand how. $\endgroup$
    – Noah Snyder
    Commented Oct 28, 2017 at 22:29
  • $\begingroup$ @NoahSnyder Maybe something to note is that precisely this definition has appeared on the literature, but equivalently phrased as a lax symmetric monoidal functor $(\mathsf{FinSets}^{\simeq},\coprod,\emptyset)\to(\mathsf{Ab},\otimes_{\mathbb{Z}},\otimes_{\mathbb{Z}})$. See Section 2 of Bunke–Nikolaus's Twisted differential cohomology. $\endgroup$
    – Emily
    Commented Sep 6, 2021 at 5:36

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