## April 27, 2018

### Props in Network Theory

#### Posted by John Baez

Long before the invention of Feynman diagrams, engineers were using similar diagrams to reason about electrical circuits and more general networks containing mechanical, hydraulic, thermodynamic and chemical components. We can formalize this reasoning using ‘props’: that is, strict symmetric monoidal categories where the objects are natural numbers, with the tensor product of objects given by addition. In this approach, each kind of network corresponds to a prop, and each network of this kind is a morphism in that prop. A network with $m$ inputs and $n$ outputs is a morphism from $m$ to $n.$ Putting networks together in series is composition, and setting them side by side is tensoring.

In this paper, we study the props for various kinds of electrical circuits:

- John Baez, Brandon Coya and Franciscus Rebro, Props in network theory.

We start with circuits made solely of ideal perfectly conductive wires. Then we consider circuits with passive linear components like resistors, capacitors and inductors. Finally we turn on the power and consider circuits that also have voltage and current sources.

And here’s the cool part: each kind of circuit corresponds to a prop that pure mathematicians would eventually invent on their own! So, what’s good for engineers is often mathematically natural too.

We describe the ‘behavior’ of these various kinds of circuits using morphisms between props. In particular, we give a new proof of the black-boxing theorem proved earlier with Brendan Fong. Unlike the original proof, this new one easily generalizes to circuits with nonlinear components. We also use a morphism of props to clarify the relation between circuit diagrams and the signal-flow diagrams in control theory.

Here’s a quick sketch of the main ideas.

## April 26, 2018

### What Is an n-Theory?

#### Posted by Mike Shulman

A few weeks ago at the homotopy type theory electronic seminar I spoke about my joint work with Dan Licata and Mitchell Riley on a “generic” framework for type theories. I briefly mentioned how this fits into a hierarchy of “$n$-theories” for $n=0,1,2,3$, but I didn’t have the time in the talk to develop that idea in detail. However, since that short remark has generated some good discussion and questions at the $n$Forum, and also the question of “$n$-theories” has a substantial history here at the $n$-Category Cafe (e.g. this post from 10 years ago), I thought I would expand on it somewhat here.

Let me clarify at the outset, though, that there may be many possible notions of “$n$-theory”. This is just one such notion, which gives a convenient way to describe what I see happening in our general framework for type theories. Notable features of this notion of $n$-theory include:

- It identifies 2-theories with a certain kind of doctrine, for those who know that word. (If that’s not you, don’t worry: I’ll explain it).
- It definitively expects the models of an $n$-theory to form an $n$-category, and not just an $n$-groupoid.
- It helps clarify the somewhat confusing question of “what is a type theory?”, by pointing out that the phrase “type theory” (and also “internal language”) is actually used in two distinct ways: sometimes it refers to a 1-theory and sometimes to a 2-theory.

## April 11, 2018

### Torsion: Graph Magnitude Homology Meets Combinatorial Topology

#### Posted by Simon Willerton

As I explained in my previous post, magnitude homology categorifies the magnitude of graphs. There are two questions that will jump out to seasoned students of homology.

- Are there two graphs which have the same magnitude but different homology groups?
- Is there a graph with torsion in its homology groups?

Both of these were stated as open questions by Richard Hepworth and me in our paper as we were unable to answer them, despite thinking about them a fair bit. However, recently both of these have been answered positively!

The first question has been answered positively by Yuzhou Gu in a reply to my post. Well, this is essentially answered, in the sense that he has given two graphs both of which we know (provably) the magnitude of, one of which we know (provably) the magnitude homology groups of and the other of which we can compute the magnitude homology of using mine and James Cranch’s SageMath software. So this just requires verification that the program result is correct! I have no doubt that it is correct though.

The second question on the existence of torsion is what I want to concentrate on in this post. This question has been answered positively by Ryuki Kaneta and Masahiko Yoshinaga in their paper

It is a consequence of what they prove in their paper that the graph below has $2$-torsion in its magnitude homology; SageMath has drawn it as a directed graph, but you can ignore the arrows. (Click on it to see a bigger version.)

In their paper they prove that if you have a finite triangulation $T$ of an $m$-dimensional manifold $M$ then you can construct a graph $G((T)$ so that the reduced homology groups of $M$ embed in the magnitude homology groups of $G((T)$:

$\widetilde{\mathrm{H}}_i(M)\hookrightarrow MH_{i+2, m+2}( G(T)) \,\,\,\, \text{for }\,\,0\le i \le m.$

Following the suggestion in their paper, I’ve taken a minimal triangulation $T_0$ of the real projective plane $\mathbb{R} P^2$ and used that to construct the above graph. As we know $\mathrm{H}_1(\mathbb{R} P^2)=\mathbb{Z}/2\mathbb{Z}$, we know that there is $2$-torsion in $MH_{3,4}(G({T_0}))$.

In the rest of this post I’ll explain the construction of the graph and show explicitly how to give a $2$-torsion class in $MH_{3,4}(G({T_0}))$. I’ll illustrate the theory of Kaneta and Yoshinaga by working through a specific example. Barycentric subdivision plays a key role!

## April 5, 2018

### Magnitude Homology Reading Seminar, II

#### Posted by Simon Willerton

*guest post by Scott Balchin*

Following on from Simon’s introductory post, this is the second installment regarding the reading group at Sheffield on magnitude homology, and the first installment which looks at the paper of Leinster and Shulman. In this post, we will be discussing the concept of magnitude for enriched categories.

The idea of magnitude is to capture the essence of *size* of a (finite) enriched category. By changing the ambient enrichment, this magnitude carries different meanings. For example, when we enrich over the monoidal category $[0,\infty ]$ we capture metric space invariants, while changing the enrichment to $\{ \text {true},\text {false}\}$ we capture poset invariants.

We will introduce the concept of magnitude via the use of zeta functions of enriched categories, which depend on the choice of a size function for the underlying enriching category. Then, we describe magnitude in a more general way using the theory of weightings. The latter will have the advantage that it is invariant under equivalence of categories, a highly desirable property.

What is presented here is taken almost verbatim from Section 2 of Leinster and Shulman’s Magnitude homology of enriched categories and metric spaces. It is, however, enhanced using comments from various other papers and, of course, multiple $n$-Café posts.

## April 2, 2018

### Dynamical Systems and Their Steady States

#### Posted by John Baez

*guest post by Maru Sarazola*

Now that we know how to use decorated cospans to represent open networks, the Applied Category Theory Seminar has turned its attention to open reaction networks (aka Petri nets) and the dynamical systems associated to them.

In A Compositional Framework for Reaction Networks (summarized in this very blog by John Baez not too long ago), authors John Baez and Blake Pollard put Fong’s results to good use and define cospan categories $\mathbf{RxNet}$ and $\mathbf{Dynam}$ of (open) reaction networks and (open) dynamical systems. Once this is done, the main goal of the paper is to show that the mapping that associates to an open reaction network its corresponding dynamical system is *compositional*, as is the mapping that takes an open dynamical system to the relation that holds between its constituents in steady state. In other words, they show that *the study of the whole can be done through the study of the parts*.

I would like to place the focus on dynamical systems and the study of their steady states, taking a closer look at this correspondence called “black-boxing”, and comparing it to previous related work done by David Spivak.