# Recovering a monoidal category from its category of monoids

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?

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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).

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Notice that these constructions are not inverse. I don't think you can recover M from CMon(M), can you? –  Theo Johnson-Freyd Oct 30 '09 at 1:43
Certainly not without some extra structure, e.g., CMon(Set) is equivalent to CMon(CMon(Set)) (where I give everything the cartesian monoidal strcture). –  Reid Barton Oct 30 '09 at 5:15
These constructions aren't inverse, no, but one is adjoint to the other. –  Tom Leinster Oct 30 '09 at 12:57