I think I have an idea of what a double multicategory should be like, and I have a live example or two, which are mentioned in arXiv:1705.04814 [math.OC]. But maybe I am wrong.

]]>Symmetric monoidal categories give rise to representable multicategories. What do monoidal double categories give rise to?

I don’t know! This concept bubbles to the surface of my mind:

• nLab, Virtual double categories.

I’m not claiming this concept is the answer to your question – it’s certainly not.

But it’s still an interesting concept. The basic sort of 2-cell in a virtual double category looks like this:

so virtual double categories are a common generalization of monoidal category, bicategory, double category, and multicategory!

Also, algebras over multicategories give semantics (I think). What do algebras over monoidal double categories give you?

More semantics. ‘Semantics’ is a very general term for ‘mapping syntactic expressions to their meanings’. Lawvere’s thesis *Functorial Semantics* showed how to do semantics using maps between categories with finite products. Later people generalized this to all kinds of categories, and in our paper we’re doing it with symmetric monoidal double categories.

Also, algebras over multicategories give semantics (I think).

What do algebras over monoidal double categories give you?

Thanks.

]]>• Eugene Lerman and David Spivak, An algebra of open continuous time dynamical systems and networks.

open electrical circuits and chemical reaction networks:

• Kenny Courser, A bicategory of decorated cospans, *Theory and Applications of Categories* **32** (2017), 995–1027.

open discrete-time Markov chains:

• Florence Clerc, Harrison Humphrey and P. Panangaden, Bicategories of Markov processes, in *Models, Algorithms, Logics and Tools*, Lecture Notes in Computer Science **10460**, Springer, Berlin, 2017, pp. 112–124.

and coarse-graining for open continuous-time Markov chains:

• John Baez and Kenny Courser, Coarse-graining open Markov processes. (Blog article here.)

As noted by Shulman, the easiest way to get a symmetric monoidal bicategory is often to first construct a symmetric monoidal double category:

• Mike Shulman, Constructing symmetric monoidal bicategories.

]]>• Fabrizio Romano Genovese and Jelle Herold, Executions in (semi-)integer Petri nets are compact closed categories.

]]>