The n-Category Café

Talking to my student Joe Moeller today, I bumped into a little question that seems fun. If I’ve got a monoidal category $A$, is there some bigger monoidal category $\hat{A}$ such that lax monoidal functors out of $A$ are the same as strict monoidal functors out of $\hat{A}$?

Someone should know the answer already, but I’ll expound on it a little…

Simon Willerton pointed out a wonderful workshop, which unfortunately neither he nor I can attend… nor Jamie Vicary, who is usually at Birmingham these days:

• Tropical Mathematics & Optimisation for Railways, University of Birmingham, School of Engineering, Monday 18 June 2018.

If you can go, please do — and report back!

Let me explain why it’s so cool…

The Centre of Australian Category Theory is advertising for a postdoc. The position is for 2 years and the ad is here.

Applications close on 15 June. Most questions about the position would be best directed to Richard Garner or Steve Lack. You can also find out more about CoACT here.

This is a great opportunity to join a fantastic research group. Please help spread the word to those who might be interested!

My student Brandon Coya finished his thesis, and successfully defended it last Tuesday!

• Brandon Coya, Circuits, Bond Graphs, and Signal-Flow Diagrams: A Categorical Perspective, Ph.D. thesis, U. C. Riverside, 2018.

It’s about networks in engineering. He uses category theory to study the diagrams engineers like to draw, and functors to understand how these diagrams are interpreted.

His thesis raises some really interesting pure mathematical questions about the category of corelations and a ‘weak bimonoid’ that can be found in this category. Weak bimonoids were invented by Pastro and Street in their study of ‘quantum categories’, a generalization of quantum groups. So, it’s fascinating to see a weak bimonoid that plays an important role in electrical engineering!

However, in what follows I’ll stick to less fancy stuff: I’ll just explain the basic idea of Brandon’s thesis, say a bit about circuits and ‘bond graphs’, and outline his main results. What follows is heavily based on the introduction of his thesis, but I’ve baezified it a little.

Intuitionistic logic, i.e. logic without the principle of excluded middle ($P \vee \neg P$), is important for many reasons. One is that it arises naturally as the internal logic of toposes and more general categories. Another is that it is the logic traditionally used by constructive mathematicians — mathematicians following Brouwer, Heyting, and Bishop, who want all proofs to have “computational” or “algorithmic” content. Brouwer observed that excluded middle is the primary origin of nonconstructive proofs; thus using intuitionistic logic yields a mathematics in which all proofs are constructive.

However, there are other logics that constructivists might have chosen for this purpose instead of intuitionistic logic. In particular, Girard’s (classical) linear logic was explicitly introduced as a “constructive” logic that nevertheless retains a form of the law of excluded middle. But so far, essentially no constructive mathematicians have seriously considered replacing intuitionistic logic with any such alternative. I will refrain from speculating on why not. Instead, in a paper appearing on the arXiv today:

• Linear logic for constructive mathematics, arxiv:1805.07518

I argue that in fact, constructive mathematicians (going all the way back to Brouwer) have already been using linear logic without realizing it!

Let me explain what I mean by this and how it comes about — starting with an explanation, for a category theorist, of what linear logic is in the first place.

João Faria Martins and Paul Martin at the University of Leeds are advertising a 2-year research fellowship in geometric topology, topological quantum field theory and applications to quantum computing. This is part of a Leverhulme funded project.

• Further details

The deadline is Tuesday 29th May. Contact João or Paul with any informal inquiries.

A new journal! We’ve been working on it for a long time, but we finished sorting out some details at Applied Category Theory 2018, and now we’re ready to tell the world!

The pillars of society are those who cannot be bribed or bought, the upright citizens of integrity, the incorruptibles. Throw at them what you will, they never bend.

In the mathematical world, the Fisher metric is one such upstanding figure.

What I mean is this. The Fisher metric can be derived from the concept of relative entropy. But relative entropy can be deformed in various ways, and you might imagine that when you deform it, the Fisher metric gets deformed too. Nope. Bastion of integrity that it is, it remains unmoved.

You don’t need to know what the Fisher metric is in order to get the point: the Fisher metric is a highly canonical concept.

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.

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:

1. 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).
2. It definitively expects the models of an $n$-theory to form an $n$-category, and not just an $n$-groupoid.
3. 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.

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

• Magnitude homology of metric spaces and order complexes.

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!

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.

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.

joint post with Heiko Gimperlein and Magnus Goffeng.

In the previous post, On the Magnitude Function of Domains in Euclidean Space, I, Heiko and Magnus explained the main theorem in their paper

• On the magnitude function of domains in Euclidean space

(Remember that here a domain $X$ in $R^n$ means a subset equal to the closure of its interior.)

The main theorem involves the asymptoic behaviour of the magnitude function $\mathcal{M}_X(R)$ as $R\to\infty$ and also the continuation of the magnitude function to a meromorphic function on the complex numbers.

In this post we have tried to tease out some of the analytical ideas that Heiko and Magnus use in the proof of their main theorem.

Heiko and Magnus build on the work of Mark Meckes, Juan Antonio Barceló and Tony Carbery and give a recipe of calculating the magnitude function of a compact domain $X\subset \mathbb{R}^n$ (for $n=2m-1$ an odd integer) by finding a solution to a differential equation subject to boundary conditions which involve certain derivatives of the function at the boundary $\partial X$ and then integrating over the boundary certain other derivatives of the solution.

In this context, switching from one set of derivatives at the boundary to another set of derivatives involves what analysts call a Dirichlet to Neumann operator. In order to understand the magnitude function it turns out that it suffices to consider this Dirichlet to Neumann operator (which is actually parametrized by the scale factor in the magnitude function). Heavy machinary of semiclassical analysis can then be employed to prove properties of this parameter-dependent operator and hence of the magntiude function.

We hope that some of this is explained below!

In Sheffield we have started a reading seminar on the recent paper of Tom Leinster and Mike Shulman Magnitude homology of enriched categories and metric spaces. The plan was to write the talks up as blog posts. Various things, including the massive strike that has been going on in universities in the UK, have meant that I’m somewhat behind with putting the first talk up. The strike also means that we haven’t had many seminars yet!

I gave the first talk which is the one here. It is an introductory talk which just describes the idea of categorification and the paper I wrote with Richard Hepworth on categorifying the magnitude of finite graphs, this is the idea which was generalized by Tom and Mike.