### Symmetric Pseudomonoids

#### Posted by John Baez

The category of cocommutative comonoid objects in a symmetric monoidal category is cartesian, with their tensor product serving as their product. This result seems to date back to here:

- Thomas Fox, Coalgebras and Cartesian categories,
*Comm. Alg.***4**(1976), 665–667.

Dually, the category of commutative monoid objects in a symmetric monoidal category is cocartesian. This was proved in Fox’s suspiciously similar paper in *Cocomm. Coalg.*

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, so we don’t need to?

Hmm, I just noticed that this paper:

- Brendan Fong and David I, Spivak, Supplying bells and whistles in symmetric monoidal categories.

proves the result I want in the special case where the symmetric monoidal 2-category is Cat. Namely:

**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 surely the fact that the coproducts in SMC are *biproducts* must rely on the fact that Cat is a *cartesian* 2-category; this must fail for symmetric pseudomonoids in a general symmetric monoidal 2-category.

They do more:

**Theorem 2.2.** The 2-category SMC has all small products and coproducts, and products are strict.

## Re: Symmetric Pseudomonoids

It looks to me like the part of their proof dealing with the coproduct structure is purely formal, if interpreted in the internal logic of a symmetric monoidal 2-category.