Seeking more information regarding the "rigoidal category" of $\mathbb{N}$-graded sets

Question

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?

Answer 1

If $M$ is any monoidal category, the presheaf category $[M^{op}, \text{Set}]$ inherits a monoidal structure given by 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 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."