## 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.

## 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.

### 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 papers^{1} 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.

## 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:

- Oliver Lorscheid, $\mathbb{F}_1$ for everyone.

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.

## 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:

- Bartek Klin and Julian Salamanca, Iterated covariant powerset is not a monad.

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

## 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’.