The category theoretic origin of arithmetic product

Question

$\newcommand\Bij{\mathrm{Bij}}\newcommand\Set{\mathrm{Set}}\newcommand\Species{\mathrm{Species}}$The paper "On the arithmetic product of combinatorial species" by Maia and Méndez introduces a combinatorial interpretation for the arithmetic product of two combinatorial species $M$, $N$.

In short, let $F:\Bij\to \Set$ be a (finite) combinatorial species and define its (modified) Dirichlet series to be $$ D_F(s) = \sum_{n=1}^\infty\frac{|F[n]|}{n!n^s}$$ where $|F[n]|$ is the cardinality of the set $F[n]$. Then, the arithmetic product $\boxdot$ is a monoidal operation on the category of species such that $D_{M\boxdot N}(s)=D_M(s)D_N(s)$.

Explicitly, one takes an (exponential) power series $\sum \frac{a_n}{n!}X^n$ for the species $M$, and one $\sum \frac{b_n}{n!}X^n$ for $N$, and their arithmetic product is $$\newcommand\mbinom{\genfrac\{\}{0pt}{}} \sum \frac{c_n}{n!}X^n \qquad c_n := \sum_{d\mid n} \mbinom n d a_db_{n/d}$$ (there is an unfortunate clash of notation between this notation and Stirling numbers of the 2nd kind here, but $\mbinom n d$ is just the "modified binomial" $\frac{n!}{d!(n/d)!}$).

With sparse remarks here and there, the paper proves what boils down to the fact that the category of species has a symmetric monoidal structure given by $\boxdot$. The only thing is, usually Dirichlet series and their convolution are not defined in this way, but instead without the $\frac 1{n!}$. At any rate, it seems to me that with the appropriate substitution/transport of structure the monoidal category $(\Species,+,\boxdot)$ can be regarded as a categorified version of the semiring of Dirichlet series with nonnegative integer coefficients.

My question is: is there a description of this monoidal structure that looks concise to a category theorist, or at least as concise as all other monoidal structures (Cartesian, Day convolution, substitution…) do? What is the universal property, if any, of the $\boxdot$ structure? (Compare this question with: the substitution product of species is the unique monoidal structure making $\Species$ monoidally equivalent to the category of analytic functors + composition, and it's called "substitution product" because it coincides with the composition of formal power series $(\sum a_n t^n)\circ g = \sum a_n g(t)^n$).

My understanding is that this structure is related to the substitution product (obtained as iterated convolution) which is what the authors (and other people studying species with more combinatorial jargon) call "assemblies": indeed, they say:

The most interesting combinatorial construction associated to the arithmetic product is the assembly of cloned structures. Informally, an assembly of cloned $G$-structures is an assembly of $G$-structures in the above sense, where all structures in the assembly are isomorphic replicas of the same structure. Moreover, information about ‘homologous vertices’ or ‘genetic similarity’ between each pair in the assembly is also provided. The structures of $F \boxdot G$ have some resemblance with the structures of the substitution $F(G)$. An element of $F \boxdot G$ can be represented as a cloned assembly of $G$-structures together with an external $F$-structure (an $F$-assembly of cloned $G$-structures). Because of the symmetry $F \boxdot G = G \boxdot F$ this structures can also be represented as $G$-assemblies of cloned $F$-structures. For example, if $L_+$ denotes the species of non-empty lists, the structures of $F \boxdot L_+$ could be thought of either as $F$-assemblies of cloned lists, or as lists of cloned $F$-structures.

But I'm having a really hard time even understanding what is the notion of a "rectangle" that the authors employ to give to $\boxdot$ a combinatorial interpretation (for example: the authors employ notation from combinatorics that is not introduced, because I believe it's standard, but very far from my knowledge). I reason way better with coends and Kan extensions….

Answer 1

The arithmetic product is indeed obtained by Day convolution – it is induced by the product of natural numbers – and is studied as the matrix monoidal structure of Dwyer–Hess's The Boardman-Vogt tensor product of operadic bimodules. (The connection between the arithmetic product and matrix product is observed in Bremner–Dotsenko's Boardman–Vogt tensor products of absolutely free operads.)

That the arithmetic product interacts nicely with the substitution monoidal structure to form a normal duoidal structure was proven by Garner and López Franco in §6 of Commutativity.

Providing a more abstract understanding of the arithmetic product is the motivation for the recent preprint Monoidal Kleisli Bicategories and the Arithmetic Product of Coloured Symmetric Sequences by Gambino, Garner, and Vasilakopoulou, in which the authors derive an arithmetic product structure on categorical symmetric sequences / generalised species, which restricts to the arithmetic product of Maia and Méndez (see §10 in particular). The introduction of this paper also provides a good survey on the topic.