## December 16, 2017

### Entropy Modulo a Prime (Continued)

#### Posted by Tom Leinster

In the comments last time, a conversation got going about *$p$-adic*
entropy. But here I’ll return to the original subject: entropy *modulo
$p$*. I’ll answer the question:

Given a “probability distribution” mod $p$, that is, a tuple $\pi = (\pi_1, \ldots, \pi_n) \in (\mathbb{Z}/p\mathbb{Z})^n$ summing to $1$, what is the right definition of its entropy $H_p(\pi) \in \mathbb{Z}/p\mathbb{Z}?$

## December 14, 2017

### Entropy Modulo a Prime

#### Posted by Tom Leinster

In 1995, the German geometer Friedrich Hirzebruch retired, and a private booklet was put together to mark the occasion. That booklet included a short note by Maxim Kontsevich entitled “The $1\tfrac{1}{2}$-logarithm”.

Kontsevich’s note didn’t become publicly available until five years later, when it was included as an appendix to a paper on polylogarithms by Philippe Elbaz-Vincent and Herbert Gangl. Towards the end, it contains the following provocative words:

Conclusion:If we have a random variable $\xi$ which takes finitely many values with all probabilities in $\mathbb{Q}$ then we can define not only the transcendental number $H(\xi)$ but also its “residues modulo $p$” for almost all primes $p$ !

Kontsevich’s note was very short and omitted many details. I’ll put some flesh on those bones, showing how to make sense of the sentence above, and much more.

## December 11, 2017

### The Icosahedron and E8

#### Posted by John Baez

Here’s a draft of a little thing I’m writing for the *Newsletter of the London Mathematical Society*. The regular icosahedron is connected to many ‘exceptional objects’ in mathematics, and here I describe two ways of using it to construct $\mathrm{E}_8$. One uses a subring of the quaternions called the ‘icosians’, while the other uses Du Val’s work on the resolution of Kleinian singularities. I leave it as a challenge to find the connection between these two constructions!

(Dedicated readers of this blog may recall that I was struggling with the second construction in July. David Speyer helped me a lot, but I got distracted by other work and the discussion fizzled. Now I’ve made more progress… but I’ve realized that the details would never fit in the *Newsletter*, so I’m afraid anyone interested will have to wait a bit longer.)

You can get a PDF version here:

• From the icosahedron to E_{8}.

But blogs are more fun.

## December 4, 2017

### The 2-Dialectica Construction: A Definition in Search of Examples

#### Posted by Mike Shulman

An adjunction is a pair of functors $f:A\to B$ and $g:B\to A$ along with a natural *isomorphism*

$A(a,g b) \cong B(f a,b).$

**Question 1:** Do we get any interesting things if we replace “isomorphism” in this definition by something else?

- If we replace it by “function”, then the Yoneda lemma tells us we get just a natural transformation $f g \to 1_B$.
- If we replace it by “retraction” then we get a unit and counit, as in an adjunction, satisfying one triangle identity but not the other.
- If $A$ and $B$ are 2-categories and we replace it by “equivalence”, we get a biadjunction.
- If $A$ and $B$ are 2-categories and we replace it by “adjunction”, we get a sort of lax 2-adjunction (a.k.a. “local adjunction”)

Are there other examples?

**Question 2:** What if we do the same thing for multivariable adjunctions?

A two-variable adjunction is a triple of functors $f:A\times B\to C$ and $g:A^{op}\times C\to B$ and $h:B^{op}\times C\to A$ along with natural isomorphisms

$C(f(a,b),c) \cong B(b,g(a,c)) \cong A(a,h(b,c)).$

What does it mean to “replace ‘isomorphism’ by something else” here? It could mean different things, but one thing it might mean is to ask instead for a *function*

$A(a,h(b,c)) \times B(b,g(a,c)) \to C(f(a,b),c).$

Even more intriguingly, if $A,B,C$ are 2-categories, we could ask for an ordinary *two-variable adjunction* between these three hom-categories; this would give a certain notion of “lax two-variable 2-adjunction”. Question 2 is, are notions like this good for anything? Are there any natural examples?

Now, you may, instead, be wondering about

**Question 3:** In what sense is a function $A(a,h(b,c)) \times B(b,g(a,c)) \to C(f(a,b),c)$ a “replacement” for isomorphisms $C(f(a,b),c) \cong B(b,g(a,c)) \cong A(a,h(b,c))$?

But that question, I can answer; it has to do with comparing the Chu construction and the Dialectica construction.

## November 23, 2017

### Real Sets

#### Posted by John Baez

Good news! Janelidze and Street have tackled some puzzles that are perennial favorites here on the $n$-Café:

- George Janelidze and Ross Street, Real sets,
*Tbilisi Mathematical Journal*,**10**(2017), 23–49.

Abstract.After reviewing a universal characterization of the extended positive real numbers published by Denis Higgs in 1978, we define a category which provides an answer to the questions:• what is a set with half an element?

• what is a set with π elements?

The category of these extended positive real sets is equipped with a countable tensor product. We develop somewhat the theory of categories with countable tensors; we call the commutative such categories

series monoidaland conclude by only briefly mentioning the non-commutative possibility calledω-monoidal. We include some remarks on sets having cardinalities in $[-\infty,\infty]$.

## November 22, 2017

### Internal Languages of Higher Categories II

#### Posted by Emily Riehl

*Guest post by Chris Kapulkin*

Two years ago, I wrote a post for the n-Cafe, in which I sketched how to make precise the claim that intensional type theory (and ultimately HoTT) is the internal language of higher category theory. I wrote it in response to a heated discussion on Michael Harris’ blog Mathematics Without Apologies about what (if anything) HoTT is good for, with the goal of making certain arguments mathematically precise, rather than taking a side in the discussion. Back then, I had very little hope that we will be able to carry out the work necessary to prove these conjectures in any foreseeable future, but things have changed and the purpose of this post is to report on the recent progress.

First off, a tl;dr version.

- Shortly after my post, Mike Shulman posted a paper describing a new class of models of the Univalence Axiom, this time in categories of simplicial presheaves over what he calls EI-categories.
- Peter Lumsdaine and I figured out how to equip the category of type theories with a left semi-model structure and were able to give a precise statement of the internal language conjectures.
- Mike Shulman proposed a tentative definition of an elementary $\infty$-topos, which, conjecturally, gives the higher-categorical counterpart of HoTT.
- A few weeks ago, Karol Szumiło and I proved a version of the first of the conjectures, relating type theories with $\mathrm{Id}$- and $\Sigma$-types with finitely cocomplete $\infty$-categories.
- And maybe the most surprising of all: Michael Harris and I are organizing a conference together (and you should attend it!).

## November 17, 2017

### Star-autonomous Categories are Pseudo Frobenius Algebras

#### Posted by Mike Shulman

A little while ago I talked about how multivariable adjunctions naturally form a polycategory: a structure like a multicategory, but in which codomains as well as domains can involve multiple objects. Now I want to talk about some structures we can define *inside* this polycategory $MVar$.

What can you define inside a polycategory? Well, to start with, a polycategory has an underlying multicategory, consisting of the arrows with unary target; so anything we can define in a multicategory we can define in a polycategory. And the most basic thing we can define in a multicategory is a monoid object — in fact, there are some senses in which this is the *canonical* thing we can define in a multicategory.

So what is a monoid object in $MVar$?

## November 13, 2017

### HoTT at JMM

#### Posted by Mike Shulman

At the 2018 U.S. Joint Mathematics Meetings in San Diego, there will be an AMS special session about homotopy type theory. It’s a continuation of the HoTT MRC that took place this summer, organized by some of the participants to especially showcase the work done during and after the MRC workshop. Following is the announcement from the organizers.

## November 11, 2017

### Topology Puzzles

#### Posted by John Baez

Let’s say the closed unit interval $[0,1]$ **maps onto** a metric space $X$ if there is a continuous map from $[0,1]$ onto $X$. Similarly for the Cantor set.

**Puzzle 0.** Does the Cantor set map onto the closed unit interval, and/or vice versa?

**Puzzle 1.** Which metric spaces does the closed unit interval map onto?

**Puzzle 2.** Which metric spaces does the Cantor set map onto?

The first one is easy; the second two are well-known… but still, perhaps, not well-known enough!

## November 9, 2017

### The 2-Chu Construction

#### Posted by Mike Shulman

Last time I told you that multivariable adjunctions (“polyvariable adjunctions”?) form a polycategory $MVar$, a structure like a multicategory but in which codomains as well as domains can involve multiple objects. This time I want to convince you that $MVar$ is actually (a subcategory of) an instance of an exceedingly general notion, called the *Chu construction*.

As I remarked last time, in defining multivariable adjunctions we used opposite categories. However, we didn’t need to know very much about the opposite of a category $A$; essentially all we needed is the existence of a hom-functor $hom_A : A^{op}\times A \to Set$. This enabled us to define the representable functors corresponding to multivariable morphisms, so that we could then ask them to be isomorphic to obtain a multivariable adjunction. We didn’t need any special properties of the category $Set$ or the hom-functor $hom_A$, only that each $A$ comes equipped with a map $hom_A : A^{op}\times A \to Set$. (Note that this is sort of “half” of a counit for the hoped-for dual pair $(A,A^{op})$, or it would be if $Set$ were the unit object; the other half doesn’t exist in $Cat$, but it does once we pass to $MVar$.)

Furthermore, we didn’t need any cartesian properties of the product $\times$; it could just as well have been any monoidal structure, or even any *multicategory* structure! Finally, if we’re willing to end up with a somewhat larger category, we can give up the idea that each $A$ should be equipped with $A^{op}$ and $hom_A$, and instead allow each objects of our “generalized $MVar$” to make a free choice of its “opposite” and “hom-functor”.

## November 7, 2017

### The Polycategory of Multivariable Adjunctions

#### Posted by Mike Shulman

Adjunctions are well-known and fundamental in category theory. Somewhat less well-known are *two-variable adjunctions*, consisting of functors $f:A\times B\to C$, $g:A^{op}\times C\to B$, and $h:B^{op}\times C\to A$ and natural isomorphisms

$C(f(a,b),c) \cong B(b,g(a,c)) \cong A(a,h(b,c)).$

These are also ubiquitous in mathematics, for instance in the notion of closed monoidal category, or in the hom-power-copower situation of an enriched category. But it seems that only fairly recently has there been a wider appreciation that it is worth defining and studying them in their own right (rather than simply as a pair of parametrized adjunctions $f(a,-)\dashv g(a,-)$ and $f(-,b) \dashv h(b,-)$).

Now, ordinary adjunctions are the morphisms of a 2-category $Adj$ (with an arbitrary choice of direction, say pointing in the direction of the left adjoint), whose 2-cells are compatible pairs of natural transformations (a fundamental result being that either uniquely determines the other). It’s obvious to guess that two-variable adjunctions should be the binary morphisms in a multicategory of “$n$-ary adjunctions”, and this is indeed the case. In fact, Eugenia, Nick, and Emily showed that multivariable adjunctions form a *cyclic* multicategory, and indeed even a cyclic *double* multicategory.

In this post, however, I want to argue that it’s even better to regard multivariable adjunctions as forming a slightly different structure called a polycategory.

## November 3, 2017

### Applied Category Theory Papers

#### Posted by John Baez

In preparation for the Applied Category Theory special session at U.C. Riverside this weekend, my crew dropped three papers on the arXiv.

### Magnitude Homology is Hochschild Homology

#### Posted by Mike Shulman

Magnitude homology, like magnitude, was born on this blog. Now there is a paper about it on the arXiv:

- Tom Leinster and Mike Shulman,
*Magnitude homology of enriched categories and metric spaces*, arXiv:1711.00802

I’m also giving a talk about magnitude homology this Saturday at the AMS sectional meeting at UC Riverside (this is the same meeting where John is running a session about applied category theory, but my talk will be in the Homotopy Theory session, 3 pm on Saturday afternoon). Here are my slides.

This paper contains basically everything that’s been said about magnitude homology so far on the blog (somewhat cleaned up), plus several new things. Below the fold I’ll briefly summarize what’s new, for the benefit of a (hypothetical?) reader who remembers all the previous posts. But if you don’t remember the old posts at all, then I suggest just starting directly with the preprint (or the slides for my talk).

I also have a request for help with terminology at the end.

## October 28, 2017

### The Adjoint School

#### Posted by John Baez

The deadline for applying to this ‘school’ on applied category theory is Wednesday November 1st. I hear they are still looking for a few really good applicants. Hurry up—this is a great opportunity!

- Applied Category Theory: Adjoint School: online sessions starting in January 2018, followed by a meeting 23–27 April 2018 at the Lorentz Center in Leiden, the Netherlands. Organized by Bob Coecke (Oxford), Brendan Fong (MIT), Aleks Kissinger (Nijmegen), Martha Lewis (Amsterdam), and Joshua Tan (Oxford).

## October 26, 2017

### Categorification and the Cosmic Cube

#### Posted by David Corfield

I see that Tobias Dyckerhoff’s A categorified Dold-Kan correspondence has just appeared, looking, as its title suggests, to categorify the nLab: Dold-Kan correspondence. As it says there, the latter

interpolates between homological algebra and general simplicial homotopy theory.

So with Dyckerhoff’s paper we seem to be dipping down to the lower layer of the flamboyantly named ‘cosmic cube’, see slide 10 of these notes by John, and discussed at nLab: cosmic cube. Via chain complexes of stable $(\infty, 1)$-categories Dyckerhoff speaks of a ‘categorified homological algebra’, and also through a categorified Eilenberg-Mac Lane spectrum, of a ‘categorified cohomology’.

For old time’s sake, let’s see if anyone is up for the kind of grand vision thing we used to talk about. For one thing we might wonder what plays the role of a categorified homotopy theory, the kind of world where Mike’s suggestions on directed homotopy type theory might find a home.

I see I was raising stratified spaces as relevant back there. In the meantime we now have useful models from A stratified homotopy hypothesis. It turns out that $(\infty, 1)$-categories are equivalent to ‘striation sheaves’, a certain kind of sheaf on ‘conically smooth’ stratified spaces. The relevant fundamental $(\infty, 1)$-category is the exit-path $(\infty, 1)$-category, rather than the entry and exit paths of our older discussions which brought in duals.

In line with Urs’s claim that cohomology concerns mapping spaces in $(\infty, 1)$-toposes (nLab: cohomology), perhaps for categorified cohomology we should be looking for parallels in $(\infty, 2)$-toposes, an important one of which will be that containing all $(\infty, 1)$-categories, or equivalently, all striation sheaves.