The category of commutative monoid objects in a symmetric monoidal category is cocartesian, with their tensor product serving as their coproduct. This sort of result seems to date back to here:
- Thomas Fox, Coalgebras and Cartesian categories, Commun. Algebra 4 (1976), 665–667.
I'm working on a paper with Todd Trimble and Joe Moeller, and right now we need something similar one level up — that is, for symmetric pseudomonoids. (For example, a symmetric pseudomonoid in Cat is a symmetric monoidal category.)
The 2-category of symmetric pseudomonoids in a symmetric monoidal 2-category should be cocartesian, with their tensor product serving as their coproduct. I imagine the coproduct universal property will hold only up to 2-iso.
Has someone proved this already? This paper:
- Brendan Fong and David I, Spivak, Supplying bells and whistles in symmetric monoidal categories.
proves the result in the special case where the symmetric monoidal 2-category is Cat. In fact they do more, in this special case:
Theorem 2.3. The 2-category SMC of symmetric monoidal categories, strong monoidal functors, and monoidal natural transformations has 2-categorical biproducts.
Unfortunately their proof is not purely 'formal', so it doesn't instantly generalize to other symmetric monoidal 2-categories. And I believe the fact that the coproducts in SMC are biproducts must rely on the fact that Cat is a cartesian 2-category.