I have attempted to find a definition of a monoidal category which incorporates $n$-fold tensor products instead of just binary tensor products.


**Definition.** A "multi-monoidal category" consists of


* a category $\mathcal{C}$,
* for every $n \geq 0$ a functor $T\_n : \mathcal{C}^n \to \mathcal{C}$, denoted by $(A\_1,\dotsc,A\_n) \mapsto A\_1 \otimes \dotsc \otimes A\_n$,
* an isomorphism $\eta : T\_1 \cong \mathrm{id}\_{\mathcal{C}}$,
* for all $n\_1,\dotsc,n\_k \geq 0$ an isomorphism
$$\mu\_{n\_1,\dotsc,n\_k} : T\_k \circ (T\_{n\_1} \times \dotsc \times T\_{n\_k}) \cong T\_{n\_1+\dotsc+n\_k}.$$


The following coherence conditions should hold:


* Coherence of $\eta$ with $\mu$: We have $\mu\_{n\_1} = \eta \circ T\_{n\_1} : T\_1 \circ T\_{n\_1} \to T\_{n\_1}$.
* Coherence of $\mu$ with $\mu$: The square


$$\begin{array}{cc} T\_k \circ \bigl(T\_{n\_1} \circ (T\_{m\_{11}} \times \dotsc \times T\_{m\_{1n\_1}}) \times \dotsc \times T\_{n\_k} \circ (T\_{m\_{k1}} \times \dotsc \times T\_{m\_{k n\_k}})\bigr) & \rightarrow & T\_k \circ (T\_{m\_{11}+\dotsc+m\_{1 n\_1}} \times \dotsc \times T\_{m\_{k1}+\dotsc+m\_{k n\_k}}) \\ \downarrow && \downarrow \\ T\_{n\_1+\dotsc+n\_k} \circ (T\_{m\_{11}} \times \dotsc \times T\_{m\_{k n\_k}}) & \rightarrow & T\_{m\_{11}+\dotsc+m\_{k n\_k}} \end{array}$$
commutes.


Notice that when $\mathcal{C}$ is discrete, this is the monadic definition of a monoid (as compared to the usual definition).


**Questions.** (1) Did I forget some coherence condition?
(2) Is this concept already known? Does it have a name? It really looks like the most natural thing in the world, especially when you think "operadic" or "monadic". (3) Most important for me: Is this concept equivalent to the definition of a monoidal category? If yes, what is a reference for this? The idea for the equivalence is straight forward (the $n$-fold tensor product is an iteration of binary tensor products etc.), but I believe that it will probably require much work to check the coherence conditions in both directions.



Reference for "multi-monoidal categories"
Look at Section 3 of Leinster's *[Higher Operads, Higher Categories](http://arxiv.org/abs/math/0305049)*, where the term used is "unbiased monoidal category."


