The n-Category Café

Socrates: Synthesi! What a long time it is since you have put in at the port of the nCafé!

Synthesi: Indeed, Socrates. The journey of seeking composition in the natural realm has led me afar, those first patterns in the flow of electrons spurred me ahead. And how fruitful it has been! Now, a new road of application, of scientific modelling with deeper clarity lies ahead.

Socrates: Not only such news has reached my ears, I hear it is also a computational tool to which you aspire?

Synthesi: Your ears serve you well as ever. The turmoil of our time lent itself well to demonstrate the strength of our approach - a computational language with composition at its very core. Phenomena such as the current pandemic call for simulations designed in their very essence as being composed of subsystems which, upon connection, give rise to the dynamics we observe in the larger picture.

Socrates: So not only understanding how scientific models compose but facilitating their rapid implementation is your aim? From paper to screen, from thought to test, guided by categorical clarity?

Synthesi: You speak my heart, Socrates.

Socrates: And it is the name of AlgebraicJulia by which your language goes?

Synthesi: Indeed, a daughter to which Julia gave birth.

Socrates: So, what is it then, this structure which you clarify and encode?

Synthesi: If you ask in broad terms, Socrates, “networks” would be my answer.

Socrates: And for this broad phenomenon, a well-investigated category exists, am I correct?

Synthesi: Undoubtedly, it is the class of hypergraph categories which you imply, Socrates. The one with a Frobenius monoid structure on each object and which thus brings merging and splitting, inputs and outputs to the fore.

Socrates: And yet it is not just any hypergraph category but certain instances of cospan flavour in which you have invested your efforts?

Synthesi: Indeed, the decorated and the structured cospan have become my staple.

Socrates: And it is their composition rule, provided by the pushout, which has so roused your appetite?

Synthesi: Socrates, once you witness its powers of connection, you become convinced of its magic. Applied to the realm of electricity, the pushout perfectly encaptured that moment of union when two simple circuits become entwined. Yet ‘union’ does not do justice, it is more… it is the identification of a physical property, namely that of connection by ideal wires, that is expressed in this operation.

Socrates: It is even more that you claim, is it not? This operation may be applied beyond? It is the horizons of systems biology, fluid dynamics, ecology and more which you hope to reach?

Synthesi: I like your way of leading us on, Socrates, and I will endeavour to reveal to you the whole nature of our program so that you may fully appreciate the power of our framework.

Socrates: And yet before you lead into application, let us linger just a moment more on the foundation that has been laid. It is the category of decorated cospans with which you began?

Synthesi: It is so, Socrates. Decorations were our early tool in modelling the systems we see in the world around. Confronted with complexity, we begin to understand a smaller part of this cosmos, begin to carve out its internal functioning. And yet to give this isolated subsystem an interface to the external world, we place the decoration on the apex of a cospan. Thus, we arrive at the larger picture.

Socrates: Any yet I hear your computational tool makes use of a formalization even more refined?

Synthesi: Yes, it is the category of structured cospans. Shall we leave its discussion for a further encounter at the café?

Socrates: Yes, it is my turn to agree.

Synthesi: Then let us take a stroll over to our friends at the computer science department and listen to their thoughts on decorated cospans in the network security realm….