**Definitions.**

By an $\mathbb{N}$-

*graded set*, I mean a set $X$ together with a function $|\Box|_X:\mathbb{N} \leftarrow X,$ called the*grading.*These will simply be called*graded sets*hereafter.If $Y$ and $X$ are graded sets, then a function $f : Y \leftarrow X$ is said to be a

*morphism of graded sets*iff it preserves the grading up to equality. Meaning that: $$\left(\mathop{\forall}_{x:X}\right)\;|f(x)|_Y = | x|_X$$(There's a weaker version in which we only require $\leq$ in the above condition, but we won't be using that here.)

Write $\mathbf{Set}^\mathbb{N}$ for the category of graded sets (this is equivalent to the functor category $\mathbf{Set}^\mathbb{N}$).

If $Y$ and $X$ are graded sets, define $Y \oplus X$ as follows.

Its underlying set is $Y \times X$.

The new grading is defined additively: $$\left(\mathop{\forall}_{y,x:X}\right)\; |(y,x)|_{Y \oplus X} = |y|_Y+|x|_X$$

If $Y$ and $X$ are graded sets, define $Y \otimes X$ as follows.

$$Y \otimes X = \bigsqcup_{y:Y} X^{\oplus |y|}$$

Define a function $\mathbf{Set}^\mathbb{N} \leftarrow \mathbb{N}$ by assigning to each natural number $n$ a graded set $\underline{n}$ with precisely one element, whose grade equals $n.$

We have: $$\left(\mathop{\forall}_{a,b:\mathbb{N}}\right)\; \underline{a+b} \cong \underline{a} \oplus \underline{b}, \qquad \left(\mathop{\forall}_{a,b:\mathbb{N}}\right)\;\underline{ab} \cong \underline{a} \otimes \underline{b}$$

I haven't checked the details, but this seems to make $\mathbf{Set}^\mathbb{N}$ into a kind of categorified rig; a "rigoidal category," if you will, with identity elements $\underline{0}$ and $\underline{1}$ respectively. My main interest in this structure is to give a description of operads; it seems to be the case that an *operad* can be defined as a monoid object in the monoidal category $(\mathbf{Set}^\mathbb{N}, \otimes, \underline{1}).$ Anyway, I'd like to get more information.

Questions.

Q0.Does this description of operads work? If so, can we describe symmetric operads in a similar way? What about cartesian operads? Where can I learn more?

Q1.Does this "rigoidal category" misbehave in any unexpected ways? Failure of distributivity, etc? Further to that, supposing that we want $\mathbf{Set}^\mathbb{N}$ to form a "rigoidal category," what are the appropriate axioms of a "rigoidal category"? Is there somewhere I can learn more about such structures?