I thought of that, but then Owen also said that he's using a fancy version of operads, so I decided to focus on the details. And in the details, the set of -ary operations is the set of polynomials in variables, and is equal to in the set of polynomials in (say) variables.

Maybe the problem is with the composition of operators. If I compose the -variable polynomial with the -variable polynomial (which is the identity) and the -variable polynomial , then I get the -variable polynomial . But if I compose the -variable polynomial with the -variable polynomial twice, then I get the -variable polynomial . And I could apply a symmetry to this to turn it into , but nothing will turn it into , which is an operation with a different arity. So in an algebra of the operad, applying to the algebra elements , , and , will typically give a different result from applying to the elements , , , and , even though we'd be tempted to write both of them as .

So it's not that we don't have ; we should be so lucky! No, instead, we have two different interpretations of , one of which equals , and one of which doesn't. Or rather, we have infinitely many interpretations of , one of which equals one of the infinitely many interpretations of , and another of which doesn't equal any of them. In fact, every expression has infinitely many distinct interpretations, even integer constants!

So the algebras are actually extremely complicated.

]]>There can’t possibly be an operad whose algebras in Set are commutative rings, since operads can only describe sets equipped with operations obeying equations with no deletion or duplication of variables. This is why there’s an operad for monoids (that is, an operad whose algebras in Set are monoids) but no operad for groups: the axiom

duplicates a variable at left and deletes it at right! There’s also no operad for rings, or commutative rings, due to the axiom

There’s an adjunction between Lawvere theories and operads, so you can take the Lawvere theory for groups or rings or commutative rings and then take its underlying operad—but the resulting operad has something *else* as its algebras in Set.

I believe this is what Owen is implicitly doing here: taking the Lawvere theory for commutative rings, whose n-ary operations are integer-coefficient polynomials in n variables, and then taking the underlying operad.

I’m not sure exactly what the algebras of this operad are like: one possibility is that they’re things like rings but not obeying the axioms where variables are duplicated or deleted, like

and

]]>I give up. I've looked at this back and forth, and as long as Owen means what I mean by ‘polynomial’ (an element of the free commutative ring on the set of variables), then it seems to me that the algebras are commutative rings. Although I should look at what the algebra homomorphisms are (which was not defined here); maybe the true answer is Morita equivalence classes of rings or something like that.

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