### Structured vs Decorated Cospans

#### Posted by John Baez

Some of us just finished a paper clarifying the connection between two approaches to describing open systems—that is, systems that can interact with their environment, and can be composed to form larger open systems:

• John Baez, Kenny Courser and Christina Vasilakopolou, Structured versus decorated cospans.

And, next week I’m giving a talk about it at YAMCaTS! This is not a conference for felines who like sweet potatoes: it’s the Yorkshire and Midlands Category Seminar, organized by Simona Paoli, Nicola Gambino and Steve Vickers.

In my talk, I’ll start by sketching some ideas behind Halter and Patterson’s software for quickly assembling larger models of COVID-19 from smaller models. Then, I’ll dig deeper into the underlying math, where we use ‘structured’ or ‘decorated’ cospans to model open systems.

This quickly gets into some serious category theory, like symmetric monoidal double categories and the symmetric monoidal Grothendieck construction — and since YAMCaTS is a category theory seminar, I won’t shy away from that. Here are my slides:

• John Baez, Structured vs decorated cospans, YAMCaTS, Friday 5 February 2021, 17:00 UTC. Zoom link here, meeting ID 810 4239 7132; to get in use 68302x where x is a one-digit perfect number.

Abstract.One goal of applied category theory is to understand open systems: that is, systems that can interact with the external world. We compare two approaches to describing open systems as cospans equipped with extra data: structured and decorated cospans. Each approach provides a symmetric monoidal double category, and we prove that under certain conditions these symmetric monoidal double categories are isomorphic. We illustrate these ideas with applications to dynamical systems and epidemiological modeling. This is joint work with Kenny Courser and Christina Vasilakopoulou.

I don’t know if my talk will be recorded, but it will be on Zoom so recording it would be easy, and I’ll try to get the organizers to do that.

For videos and slides of two related talks go here:

For more, read these:

Evan Patterson and Micah Halter, Compositional epidemiological modeling using structured cospans.

John Baez and Kenny Courser, Structured cospans.

Kenny Courser,

*Open Systems: a Double Categorical Perspective*. (Blog articles here.)Michael Shulman, Framed bicategories and monoidal fibrations.

Joe Moeller and Christina Vasilakopolou, Monoidal Grothendieck construction.

To read more about the network theory project, go here:

## Re: Structured vs Decorated Cospans

Here’s the schedule of the Friday February 5th 2021 meeting of the Yorkshire and Midlands Category Theory Seminar. There will be three talks.

YaMCATS - Friday 5th February - University of Leeds (via Zoom)All times are UK (GMT = UTC+00:00).

14:30-15:30 Martin Escardo (University of Birmingham), Equality of mathematical structures

15:30-16:30 Sina Hazratpour (University of Leeds), Kripke-Joyal semantics for dependent type theory

16:30-17:00 Break

17:00-18:00 John Baez, Structured versus decorated cospans

Zoom linksNicola Gambino is inviting you to a scheduled Zoom meeting.

Topic: YaMCATS 23

Time: Feb 5, 2021 02:30 PM London

Join Zoom Meeting https://universityofleeds.zoom.us/j/81042397132?pwd=RTg3MFV1TUt2YzJXZVZJSkhoOEQwQT09

Meeting ID: 810 4239 7132

Passcode: 68302x where x is a one-digit perfect number

AbstractsMartin EscardoTitle:Equality of mathematical structuresAbstract.Two groups are regarded to be the same if they are isomorphic, two topological spaces are regarded to be the same if they are homeomorphic, two metric spaces are regarded to be the same if they are isometric, two categories are regarded to be the same if they are equivalent, etc. In Voevodsky’s Univalent Foundations (HoTT/UF), the above become theorems: we can replace “are regarded to be the same” by “are the same”. I will explain how this works. I will not assume previous knowledge of HoTT/UF or type theory.Sina Hazratpur(University of Leeds)Title:Kripke-Joyal semantics for dependent type theoryAbstract.Every topos has an internal higher-order intuitionistic logic. The so-called Kripke–Joyal semantics of a topos gives an interpretation to formulas written in this language used to express ordinary mathematics in that topos. The Kripke–Joyal semantics is in fact a higher order generalization of the well-known Kripke semantic for intuitionistic propositional logic. In this talk I shall report on joint work with Steve Awodey and Nicola Gambino on extending the Kripke–Joyal semantics to dependent type theories, including homotopy type theory.John Baez(University of California at Riverside)Title:Structured versus decorated cospansAbstract.One goal of applied category theory is to understand open systems: that is, systems that can interact with the external world. We compare two approaches to describing open systems as cospans equipped with extra data: structured and decorated cospans. Each approach provides a symmetric monoidal double category, and we prove that under certain conditions these symmetric monoidal double categories are equivalent. We illustrate these ideas with applications to dynamical systems and epidemiological modeling. This is joint work with Kenny Courser and Christina Vasilakopoulou.