## July 7, 2018

### Beyond Classical Bayesian Networks

#### Posted by John Baez

guest post by Pablo Andres-Martinez and Sophie Raynor

In the final installment of the Applied Category Theory seminar, we discussed the 2014 paper “Theory-independent limits on correlations from generalized Bayesian networks” by Henson, Lal and Pusey.

In this post, we’ll give a short introduction to Bayesian networks, explain why quantum mechanics means that one may want to generalise them, and present the main results of the paper. That’s a lot to cover, and there won’t be a huge amount of category theory, but we hope to give the reader some intuition about the issues involved, and another example of monoidal categories used in causal theory.

## July 4, 2018

### Symposium on Compositional Structures

#### Posted by John Baez

There’s a new conference series, whose acronym is pronounced “psycho”. It’s part of the new trend toward the study of “compositionality” in many branches of thought, often but not always using category theory:

• First Symposium on Compositional Structures (SYCO1), School of Computer Science, University of Birmingham, 20-21 September, 2018. Organized by Ross Duncan, Chris Heunen, Aleks Kissinger, Samuel Mimram, Simona Paoli, Mehrnoosh Sadrzadeh, Pawel Sobocinski and Jamie Vicary.

The Symposium on Compositional Structures is a new interdisciplinary series of meetings aiming to support the growing community of researchers interested in the phenomenon of compositionality, from both applied and abstract perspectives, and in particular where category theory serves as a unifying common language. We welcome submissions from researchers across computer science, mathematics, physics, philosophy, and beyond, with the aim of fostering friendly discussion, disseminating new ideas, and spreading knowledge between fields. Submission is encouraged for both mature research and work in progress, and by both established academics and junior researchers, including students.

More details below! Our very own David Corfield is one of the invited speakers.

Posted at 6:40 PM UTC | Permalink | Followups (1)

## June 27, 2018

### Elmendorf’s Theorem

#### Posted by John Huerta

I want to tell you about Elmendorf’s theorem on equivariant homotopy theory. This theorem played a key role in a recent preprint I wrote with Hisham Sati and Urs Schreiber:

We figured out how to apply this theorem in mathematical physics. But Elmendorf’s theorem by itself is a gem of homotopy theory and deserves to be better known. Here’s what it says, roughly: given any $G$-space $X$, the equivariant homotopy type of $X$ is determined by the ordinary homotopy types of the fixed point subspaces $X^H$, where $H$ runs over all subgroups of $G$. I don’t know how to intuitively motivate this fact; I would like to know, and if any of you have ideas, please comment. Below the fold, I will spell out the precise theorem, and show you how it gives us a way to define a $G$-equivariant version of any homotopy theory.

Posted at 7:03 PM UTC | Permalink | Followups (23)

## June 15, 2018

### ∞-Atomic Geometric Morphisms

#### Posted by Mike Shulman

Today’s installment in the ongoing project to sketch the $\infty$-elephant: atomic geometric morphisms.

Chapter C3 of Sketches of an Elephant studies various classes of geometric morphisms between toposes. Pretty much all of this chapter has been categorified, except for section C3.5 about atomic geometric morphisms. To briefly summarize the picture:

• Sections C3.1 (open geometric morphisms) and C3.3 (locally connected geometric morphisms) are steps $n=-1$ and $n=0$ on an infinite ladder of locally n-connected geometric morphisms, for $-1 \le n \le \infty$. A geometric morphism between $(n+1,1)$-toposes is locally $n$-connected if its inverse image functor is locally cartesian closed and has a left adjoint. More generally, a geometric morphism between $(m,1)$-toposes is locally $n$-connected, for $n\lt m$, if it is “locally” locally $n$-connected on $n$-truncated maps.

• Sections C3.2 (proper geometric morphisms) and C3.4 (tidy geometric morphisms) are likewise steps $n=-1$ and $n=0$ on an infinite ladder of n-proper geometric morphisms.

• Section C3.6 (local geometric morphisms) is also step $n=0$ on an infinite ladder: a geometric morphism between $(n+1,1)$-toposes is $n$-local if its direct image functor has an indexed right adjoint. Cohesive toposes, which have attracted a lot of attention around here, are both locally $\infty$-connected and $\infty$-local. (Curiously, the $n=-1$ case of locality doesn’t seem to be mentioned in the 1-Elephant; has anyone seen it before?)

So what about C3.5? An atomic geometric morphism between elementary 1-toposes is usually defined as one whose inverse image functor is logical. This is an intriguing prospect to categorify, because it appears to mix the “elementary” and “Grothendieck” aspects of topos theory: a geometric morphisms are arguably the natural morphisms between Grothendieck toposes, while logical functors are more natural for the elementary sort (where “natural” means “preserves all the structure in the definition”). So now that we’re starting to see some progress on elementary higher toposes (my post last year has now been followed by a preprint by Rasekh), we might hope be able to make some progress on it.

Posted at 4:16 PM UTC | Permalink | Followups (31)

### The Behavioral Approach to Systems Theory

#### Posted by John Baez

guest post by Eliana Lorch and Joshua Tan

As part of the Applied Category Theory seminar, we discussed an article commonly cited as an inspiration by many papers1 taking a categorical approach to systems theory, The Behavioral Approach to Open and Interconnected Systems. In this sprawling monograph for the IEEE Control Systems Magazine, legendary control theorist Jan Willems poses and answers foundational questions like how to define the very concept of mathematical model, gives fully-worked examples of his approach to modeling from physical first principles, provides various arguments in favor of his framework versus others, and finally proves several theorems about the special case of linear time-invariant differential systems.

In this post, we’ll summarize the behavioral approach, Willems’ core definitions, and his “systematic procedure” for creating behavioral models; we’ll also examine the limitations of Willems’ framework, and conclude with a partial reference list of Willems-inspired categorical approaches to understanding systems.

Posted at 3:05 AM UTC | Permalink | Followups (6)

## June 13, 2018

### Fun for Everyone

#### Posted by John Baez

There’s a been a lot of progress on the ‘field with one element’ since I discussed it back in “week259”. I’ve been starting to learn more about it, and especially its possible connections to the Riemann Hypothesis. This is a great place to start:

Abstract. This text serves as an introduction to $\mathbb{F}_1$-geometry for the general mathematician. We explain the initial motivations for $\mathbb{F}_1$-geometry in detail, provide an overview of the different approaches to $\mathbb{F}_1$ and describe the main achievements of the field.

Posted at 6:17 AM UTC | Permalink | Followups (14)

## June 9, 2018

### Sets of Sets of Sets of Sets of Sets of Sets

#### Posted by John Baez

The covariant power set functor $P : Set \to Set$ can be made into a monad whose multiplication $m_X: P(P(X)) \to P(X)$ turns a subset of the set of subsets of $X$ into a subset of $X$ by taking their union. Algebras of this monad are complete semilattices.

But what about powers of the power set functor? Yesterday Jules Hedges pointed out this paper:

The authors prove that $P^n$ cannot be made into a monad for $n \ge 2$.

Posted at 7:05 PM UTC | Permalink | Followups (28)

## June 4, 2018

### Applied Category Theory: Resource Theories

#### Posted by John Baez

My course on applied category theory is continuing! After a two-week break where the students did exercises, I went back to lecturing about Fong and Spivak’s book Seven Sketches. The second chapter is about ‘resource theories’.

## May 28, 2018

### Applied Category Theory: Ordered Sets

#### Posted by John Baez

My applied category theory course based on Fong and Spivak’s book Seven Sketches is going well. Over 300 people have registered for the course, which allows them to ask question and discuss things. But even if you don’t register you can read my “lectures”.

Posted at 7:39 PM UTC | Permalink | Followups (5)

## May 25, 2018

### Laxification

#### Posted by John Baez

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…

Posted at 9:40 PM UTC | Permalink | Followups (13)

## May 24, 2018

### Tropical Algebra and Railway Optimization

#### Posted by John Baez

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:

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

Let me explain why it’s so cool…

Posted at 10:39 PM UTC | Permalink | Followups (5)

## May 20, 2018

### Postdoc at the Centre of Australian Category Theory

#### Posted by Emily Riehl

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!

## May 19, 2018

### Circuits, Bond Graphs, and Signal-Flow Diagrams

#### Posted by John Baez

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.

Posted at 11:07 PM UTC | Permalink | Followups (4)

### Linear Logic for Constructive Mathematics

#### Posted by Mike Shulman

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:

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.

Posted at 4:53 AM UTC | Permalink | Followups (27)

## May 14, 2018

### Research Fellowship at the University of Leeds

#### Posted by Simon Willerton

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.

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

Posted at 10:01 AM UTC | Permalink | Followups (1)