You can read a lot more about this example here:

• John Baez, Open Petri nets, *Azimuth*, 15 August 2018.

• John Baez and Jade Master, Open Petri nets.

Last time I explained, in a sketchy way, the double category of open Petri nets. This time I’d like to describe a ‘semantics’ for open Petri nets.

]]>This is why I said Mike Shulman’s technique can be applied to get a symmetric monoidal bicategory from the symmetric monoidal double category of open Petri nets. His technique applies to fibrant double categories—or more generally, ‘isofibrant’ ones, where each *invertible* vertical 1-morphism has a conjoint and a companion.

Jade and I don’t actually discuss this isofibrancy, or how to get the symmetric monoidal bicategory. Kenny Courser and I will talk about these things more generally in our paper on structured cospans. The double category is an example of a ‘structured cospan double category’. The idea is that an open Petri net is a cospan of sets where the apex is equipped with extra structure: it’s made into the set of places of a Petri net. Structured cospan double categories are isofibrant under very general conditions!

]]>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?