Is there any notion in the literature which captures the idea of an $\infty$-ary tensor product on a category $C$? This should include tensor products of $\alpha$-indexed families of objects in $C$ for every ordinal $\alpha$, as well as some coherence isomorphisms and compatiblity properties. Here are three motivating examples:

1. The tensor product of modules

2. A category with choosen arbitrary (co)products

3. Let $C$ be a cocomplete monoidal category such that every object $x$ is equipped with a natural morphism $1 \to x$. If this data isn't available, just pass to this slice category over $C$. For an ordinal $\alpha$ and a familiy $(x_{\beta})_{\beta < \alpha}$ in $C$ define $\bigotimes_{\beta < \alpha} x_{\beta}$ by recursion on $\alpha$. It's clear what to do for $\alpha=0$ and when $\alpha$ is a successor. When $\alpha$ is a limit ordinal, take the colimit of all the preceding tensor products. The transition maps are induced by the data above.

There are some applications for the construction in 3), for example the largest Hausdorff quotient, the associated sheaf and the coproduct of algebras. In the context of orthogonal classes in presentable categories it is also called "transfinite composition".

If there is no such notion yet, what axioms should we choose?