It is a well-known fact that on categories admitting all finite coproducts one can define a monoidal structure where the monoidal product is exactly the coproduct and the monoidal unit is the initial object.
For coproducts there is nothing that prohibits me from working with a countable (or even uncountable) number of objects. Now I was wondering if there exists any general theory on how this can be generalized to the monoidal context? (In general I have barely found anything on the extension of monoidal products to an infinite number of objects.)
This question popped up while I was studying monoidal posets (see Seven Sketches in Compositionality by Spivak & Fong: http://math.mit.edu/~dspivak/teaching/sp18/7Sketches.pdf). On one hand I am interested in this question from an educational point of view because I would in general like to know how the concepts from monoidal category theory generalize to ''infinite products''. But on the other hand I am also interested from a professional point of view because not all operations I consider come from coproducts (e.g. the addition on $\mathbb{R}^+$) and hence I cannot resort to a description in those terms.
