What kind of additional properties and/or structures one needs to impose on the category
of (commutative or noncommutative) monoids of some monoidal category
so that one can recover the original monoidal category from this data?


What kind of additional properties and/or structures one needs to impose on a category
to ensure that it is the category of monoids of some monoidal category?


The example I have in mind is the category of (commutative or noncommutative) C\*-algebras (or von Neumann algebras).
Can we obtain one of these categories as the category of monoids of some monoidal category?



Recovering a monoidal category from its category of monoids
Here is a characterization of categories of commutative monoids. I don't know the answer in the non-commutative case.


Let *C* be a category. Then *C* is the category of commutative monoids in some symmetric monoidal category if and only if *C* has finite coproducts.


For suppose that *C = CMon(M)* for some symmetrical monoidal category *M = (M, @, I)*. Then one can show that the tensor product @ of M also defines a tensor product on *C* --- and that this is, in fact, binary coproduct in *C*. (Example: if *M* is the category of abelian groups then *C* is the category of commutative rings, and the tensor product of commutative rings is the coproduct.) Similarly, the unit object *I* of *M* is a commutative monoid in a unique way, and is in fact the initial object of *C*. So *C* has finite coproducts.


Conversely, suppose that *C* has finite coproducts. Then (+, 0) defines a symmetric monoidal structure on *C*, and with respect to this structure, every object of *C* is a commutative monoid in a unique way. Thus, *C = CMon(C)*.


