It's been a long time since I posted the following question on stackexchange.

Now I think it's better to adress it to you, in the hope I will reach the right audience: Martin's comment, albeit useful, didn't help me to satisfy my curiosity.

===

A functor $F\colon \bf Sets\to Sets$ is said to be *analytic* if it results from the left Kan extension of a functor $f\colon \mathbf{Bij}(\mathbb N)\to \bf Sets$ (the "species" of the functors $F$) along the natural inclusion $\mathbf{Bij}(\mathbb N)\to \bf Sets$, where $\mathbf{Bij}(\mathbb N)$ is the category having objects natural numbers and where $\mathbf{Bij}(\mathbb N)(m,n)$ are the bijective functions $\{1,\dots,m\}\to \{1,\dots,n\}$ (empty if $n\neq m$). Representing a left Kan extension as a coend it means that
$$
F(T)\cong \int^n T^n\times f(n)
$$
(the most of you will recognize the fact that a functor is "anaytic" if it can be written in Taylor form, and the coend is in a suitable sense exactly that Taylor series) This can be expressed replacing $\bf Sets$ with any symmetric monoidally cocomplete category: there is a functor $\mathbf{Bij}(\mathbb N)^\text{op}\times \mathcal V\to \mathcal V\colon (n,V)\mapsto V\otimes\dots\otimes V=V^{\otimes n}$, which allows to define
$$
\int^n V^{\otimes n}\otimes f(n)
$$
for any "species" $f\colon \mathbf{Bij}(\mathbb N)\to \mathcal V$.

In the case of $\bf Sets$, it seems to be possible to extend the definition of an analytic functor to the case of an arbitrary cardinal $\kappa$:

- the category $\mathbf{Bij}(\kappa)$ is defined to be the category having objects cardinals $<\kappa$ and $\mathbf{Bij}(\kappa)(\mu,\nu)$ bijections between those cardinals (hence empty if $\mu\neq\nu$).
- I say that a functor $F\colon\bf Sets\to Sets$ is $\kappa$-analytic if it results from $\text{Lan}_jf$ in the diagram $$ \begin{array}{ccc} \mathbf{Bij}(\kappa) &\to& \bf Sets \\ \downarrow&\nearrow_F&\\ \bf Sets & \end{array} $$ $F(T)\cong \int^{\mu}T^\mu\times f(\mu)$ where $T^\mu$ is the product of $\mu$ copies of $T$, in the obvious sense.

Now my question: can I define something analogous in the case of a generic monoidal(ly cocomplete) category, maybe imposing some additional property? In the end all boils down to the possibility of defining a $\kappa$-fold tensor product $\otimes\colon \mathcal V^\kappa\to \mathcal V$, for any cardinal $\kappa$. Modules over a ring $R$ seem to have this "extension property", together with sets.

Being these two very explicit and natural examples of a monoidal category where "tensoring has an arbitrarily large arity", I think the problem has its own interest, and I would be surprised if the topic was new.

===

As I pointed out in the comments on math.SE, I was motivated by Martin's topic about k-ary tensors in $Mod_R$ to pose the question in the form of a "k-ary tensor product". I think there is no hope to recover "usual properties"; it's better to think about an axiomatization of the structure I need, but for the moment I don't have the slightest idea. I repeat myself: on the one side I find quite astounding the lack of a theory of those monoidal categories where tensoring has an arbitrarily large arity, and on the other, it's obviously due to the absence of "good" properties for such a structure.