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:

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.

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    $\begingroup$ Two ideas: (1) It's surely been shown in general that if $C$ is a symmetric monoidal $\infty$-category, then in $Alg_{E_\infty}(C)$ the coproduct and tensor product coincide. This should specialize to what you want by taking $C$ to be the (2,1)-category of categories (maybe the appropriate comparisons (still!) haven't been done). (2) Perhaps this can be deduced representably from Fong-Spivak: a symm.pseudomonoid in $C$ should be a certain kind of lift of a representable 2-functor $C \to Cat$ through the forgetful 2-functor $SymPsMon \to Cat$, with coproduct and tensor product defined levelwise $\endgroup$ – Tim Campion Sep 15 '20 at 22:25
  • $\begingroup$ @TimCampion : in (1), don't you mean a general (2,1)-symmetric monoidal category ? Which would not be enough if John is interested in general 2-categories with $\endgroup$ – Maxime Ramzi Sep 16 '20 at 9:26
  • $\begingroup$ @MaximeRamzi Ah, good point. Although typically I should expect that coproducts in a 2-category can be detected at the (2,1) level, and similarly with the tensor product. $\endgroup$ – Tim Campion Sep 16 '20 at 14:27
  • $\begingroup$ @TimCampion : Yes, I agree, I expect so too; but I definitely don't have enough experience in the $(2,2)$-world to be sure $\endgroup$ – Maxime Ramzi Sep 16 '20 at 14:31
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    $\begingroup$ @JohnBaez This isn't true. The universal property of a coproduct of objects in a bicategory involves an equivalence of categories (not just groupoids): $Hom(\sum_i A_i,B) \simeq \prod_i Hom(A_i,B)$. However, if the bicategory admits cotensors by the walking arrow, then coproducts are detected at the level of the underlying (2,1)-category. $\endgroup$ – Alexander Campbell Sep 18 '20 at 21:54

The result I wanted is Theorem 5.2 here:

Daniel Schäppi, Ind-abelian categories and quasi-coherent sheaves, Mathematical Proceedings of the Cambridge Philosophical Society, 157 (2014), 391–423. doi:10.1017/S0305004114000401

The proof appears in Appendix A. He proves the result for symmetric pseudomonoids in a symmetric monoidal bicategory.


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