**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?
> 
> 
> 



Seeking more information regarding the "rigoidal category" of $\mathbb{N}$-graded sets
If $M$ is any monoidal category, the presheaf category $[M^{op}, \text{Set}]$ inherits a monoidal structure given by [Day convolution](https://ncatlab.org/nlab/show/Day+convolution). It is uniquely determined by the condition that it restricts to the given monoidal structure on $M$ and that it preserves colimits in both variables. Taking $M = \mathbb{N}$ (as a discrete monoidal category) we recover the monoidal category of $\mathbb{N}$-graded sets (with the first monoidal structure you describe). 


$\mathbb{N}$ is the free monoidal category on a point. Consequently, by the universal property of the Yoneda embedding, $[\mathbb{N}^{op}, \text{Set}]$ is the free monoidal cocomplete (this includes the condition that the monoidal structure distributes over colimits) category on a point. If you think of cocomplete categories as analogous to abelian groups, then monoidal cocomplete categories are analogous to rings, so this free guy is analogous to a polynomial ring. 


One way to describe the universal property above is that $[\mathbb{N}^{op}, \text{Set}]$ represents the forgetful functor from monoidal cocomplete categories to categories. Accordingly, it is also equivalent to the category of natural transformations from this forgetful functor to itself. This means that $\mathbb{N}$-graded sets are "universal endomorphisms" of monoidal cocomplete categories: explicitly, if $X\_n$ is an $\mathbb{N}$-graded set, the corresponding endomorphism of monoidal cocomplete categories acts on an object $V$ via the "ordinary generating function"


$$V \mapsto \coprod\_n X\_n \otimes V^{\otimes n}.$$


This gives $\mathbb{N}$-graded sets a new monoidal structure, the composition product (analogous to composition rather than multiplication of polynomials; this is the second monoidal structure you describe). This means that a monoid in $\mathbb{N}$-graded sets (with respect to the composition product) is a "universal monad" acting on monoidal cocomplete categories, and if you work out what that means in terms of the action above it is precisely a nonsymmetric operad.


To get symmetric operads replace "monoidal" above everywhere with "symmetric monoidal." This replaces $\mathbb{N}$ with the category $S$ of finite sets and bijections, which is the free symmetric monoidal category on a point. Consequently, the category $[S^{op}, \text{Set}]$ of [species](https://ncatlab.org/nlab/show/species) is the free symmetric monoidal cocomplete category on a point. It acts as "universal endomorphisms" of symmetric monoidal cocomplete categories as follows: if $X\_n$ is a species, the corresponding endomorphism acts on an object $V$ via the "exponential generating function"


$$V \mapsto \coprod\_n X\_n \otimes\_{S\_n} V^{\otimes n}.$$


Again we get a composition product. This means that a monoid in species (with respect to the composition product) is a "universal monad" acting on symmetric monoidal cocomplete categories, and if you work out what that means in terms of the action above it is precisely a symmetric operad. 


I haven't thought about how to get cartesian operads (these are Lawvere theories, right?) into the game, but plausibly you can do it by replacing "symmetric monoidal" with "cartesian monoidal." 


