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July 28, 2015

Internal Languages of Higher Categories

Posted by Mike Shulman

(guest post by Chris Kapulkin)

I recently posted the following preprint to the arXiv:

Btw if you’re not the kind of person who likes to read mathematical papers, I also gave a talk about the above mentioned work in Oxford, so you may prefer to watch it instead. (-:

I see this work as contributing to the idea/program of HoTT as the internal language of higher categories. In the last few weeks, there has been a lot of talk about it, prompted by somewhat provocative posts on Michael Harris’ blog.

My goal in this post is to survey the state of the art in the area, as I know it. In particular, I am not going to argue that internal languages are a solution to many of the problems of higher category theory or that they are not. Instead, I just want to explain the basic idea of internal languages and what we know about them as far as HoTT and higher category theory are concerned.

Disclaimer. The syntactic rules of dependent type theory look a lot like a multi-sorted essentially algebraic theory. If you think of sorts called types and terms then you can think of rules like Σ\Sigma-types and Π\Pi-types as algebraic operations defined on these sorts. Although the syntactic presentation of type theory does not quite give an algebraic theory (because of complexities such as variable binding), it is possible to formulate dependent type theory as an essentially algebraic theory. However, actually showing that these two presentations are equivalent has proven complicated and it’s a subject of ongoing work. Thus, for the purpose of this post, I will take dependent type theories to be defined in terms of contextual categories (a.k.a. C-systems), which are the models for this algebraic theory (thus leaving aside the Initiality Conjecture). Ultimately, we would certainly like to know that these statements hold for syntactically-presented type theories; but that is a very different question from the \infty-categorical aspects I will discuss here.

A final comment before we begin: this post derives greatly from my (many) conversations with Peter Lumsdaine. In particular, the two of us together went through the existing literature to understand precisely what’s known and what’s not. So big thanks to Peter for all his help!

Posted at 3:30 AM UTC | Permalink | Followups (1)

July 19, 2015

Category Theory 2015

Posted by John Baez

Just a quick note: you can see lots of talk slides here:

Category Theory 2015, Aveiro, Portugal, June 14-19, 2015.

The Giry monad, tangent categories, Hopf monoids in duoidal categories, model categories, topoi… and much more!

Posted at 9:19 AM UTC | Permalink | Followups (1)

July 8, 2015

Mary Shelley on Invention

Posted by Tom Leinster

From the 1831 introduction to Frankenstein:

Invention, it must be humbly admitted, does not consist in creating out of void, but out of chaos; the materials must, in the first place, be afforded: it can give form to dark, shapeless substances, but cannot bring into being the substance itself. […] Invention consists in the capacity of seizing on the capabilities of a subject, and in the power of moulding and fashioning ideas suggested to it.

She’s talking about literary invention, but it immediately struck me that her words are true for mathematical invention too.

Except that I can’t think of a part of mathematics I’d call “dark” or “shapeless”.

Posted at 12:54 PM UTC | Permalink | Followups (4)

June 29, 2015

What is a Reedy Category?

Posted by Mike Shulman

I’ve just posted the following preprint, which has apparently quite little to do with homotopy type theory.

The notion of Reedy category is common and useful in homotopy theory; but from a category-theoretic point of view it is odd-looking. This paper suggests a category-theoretic understanding of Reedy categories, which I find more satisfying than any other I’ve seen.

Posted at 6:33 PM UTC | Permalink | Followups (20)

Feynman’s Fabulous Formula

Posted by Simon Willerton

Guest post by Bruce Bartlett.

There is a beautiful formula at the heart of the Ising model; a formula emblematic of all of quantum field theory. Feynman, the king of diagrammatic expansions, recognized its importance, and distilled it down to the following combinatorial-geometric statement. He didn’t prove it though — but Sherman did.

Feynman’s formula. Let GG be a planar finite graph, with each edge ee regarded as a formal variable denoted x ex_e. Then the following two polynomials are equal:

H evenGx(H)= [γ]P(G)(1(1) w[γ]x[γ])\displaystyle \sum_{H \subseteq_{even} G} x(H) = \prod_{[\vec{\gamma}] \in P(G)} \left(1 - (-1)^{w[\vec{\gamma}]} x[\vec{\gamma}]\right)


I will explain this formula and its history below. Then I’ll explain a beautiful generalization of it to arbitrary finite graphs, expressed in a form given by Cimasoni.

Posted at 7:47 AM UTC | Permalink | Followups (17)

June 21, 2015

What’s so HoTT about Formalization?

Posted by Mike Shulman

In my last post I promised to follow up by explaining something about the relationship between homotopy type theory (HoTT) and computer formalization. (I’m getting tired of writing “publicity”, so this will probably be my last post for a while in this vein — for which I expect that some readers will be as grateful as I).

As a potential foundation for mathematics, HoTT/UF is a formal system existing at the same level as set theory (ZFC) and first-order logic: it’s a collection of rules for manipulating syntax, into which we can encode most or all of mathematics. No such formal system requires computer formalization, and conversely any such system can be used for computer formalization. For example, the HoTT Book was intentionally written to make the point that HoTT can be done without a computer, while the Mizar project has formalized huge amounts of mathematics in a ZFC-like system.

Why, then, does HoTT/UF seem so closely connected to computer formalization? Why do the overwhelming majority of publications in HoTT/UF come with computer formalizations, when such is still the exception rather than the rule in mathematics as a whole? And why are so many of the people working on HoTT/UF computer scientists or advocates of computer formalization?

Posted at 6:47 AM UTC | Permalink | Followups (10)

June 12, 2015

Carnap and the Invariance of Logical Truth

Posted by David Corfield

I see Steve Awodey has a paper just out Carnap and the invariance of logical truth. We briefy discussed this idea in the context of Mautner’s 1946 article back here.

Steve ends the article by portraying homotopy type theory as following in the same tradition, but now where invariance is under homotopy equivalence. I wonder if we’ll see some variant of the model/theory duality he and Forssell found in the case of HoTT.

Posted at 12:09 PM UTC | Permalink | Followups (2)

June 9, 2015

Semigroup Puzzles

Posted by John Baez

Suppose you have a semigroup: that is, a set with an associative product. Also suppose that

xyx=x x y x = x

for all xx and all yy.

Puzzle 1. Prove that

xyz=xz x y z = x z

for all x,yx,y and zz.

Puzzle 2. Prove that

xx=x x x = x

for all xx.

Posted at 2:30 PM UTC | Permalink | Followups (41)

May 26, 2015

A 2-Categorical Approach to the Pi Calculus

Posted by John Baez

guest post by Mike Stay

Greg Meredith and I have a short paper that’s been accepted for Higher-Dimensional Rewriting and Applications (HDRA) 2015 on modeling the asynchronous polyadic pi calculus with 2-categories. We avoid domain theory entirely and model the operational semantics directly; full abstraction is almost trivial. As a nice side-effect, we get a new tool for reasoning about consumption of resources during a computation.

It’s a small piece of a much larger project, which I’d like to describe here in a series of posts. This post will talk about lambda calculus for a few reasons. First, lambda calculus is simpler, but complex enough to illustrate one of our fundamental insights. Lambda calculus is to serial computation what pi calculus is to concurrent computation; lambda calculus talks about a single machine doing a computation, while pi calculus talks about a network of machines communicating over a network with potentially random delays. There is at most one possible outcome for a computation in the lambda calculus, while there are many possible outcomes in a computation in the pi calculus. Both the lazy lambda calculus and the pi calculus, however, have as an integral part of their semantics the notion of waiting for a sub-computation to complete before moving onto another one. Second, the denotational semantics of lambda calculus in Set is well understood, as is its generalization to cartesian closed categories; this semantics is far simpler than the denotational semantics of pi calculus and serves as a good introduction. The operational semantics of lambda calculus is also simpler than that of pi calculus and there is previous work on modeling it using higher categories.

Posted at 9:28 PM UTC | Permalink | Followups (36)

SoTFoM III and The Hyperuniverse Programme

Posted by David Corfield

Following SoTFom II, which managed to feature three talks on Homotopy Type Theory, there is now a call for papers announced for SoTFoM III and The Hyperuniverse Programme, to be held in Vienna, September 21-23, 2015.

Here are the details:

The Hyperuniverse Programme, launched in 2012, and currently pursued within a Templeton-funded research project at the Kurt Gödel Research Center in Vienna, aims to identify and philosophically motivate the adoption of new set-theoretic axioms.

The programme intersects several topics in the philosophy of set theory and of mathematics, such as the nature of mathematical (set-theoretic) truth, the universe/multiverse dichotomy, the alternative conceptions of the set-theoretic multiverse, the conceptual and epistemological status of new axioms and their alternative justificatory frameworks.

The aim of SotFoM III+The Hyperuniverse Programme Joint Conference is to bring together scholars who, over the last years, have contributed mathematically and philosophically to the ongoing work and debate on the foundations and the philosophy of set theory, in particular, to the understanding and the elucidation of the aforementioned topics. The three-day conference, taking place September 21-23 at the KGRC in Vienna, will feature invited and contributed speakers.

I wonder if anyone will bring some category theory along to the meeting. Perhaps they can answer my question here.

Further details:

Posted at 2:50 PM UTC | Permalink | Followups (5)

May 22, 2015

PROPs for Linear Systems

Posted by John Baez

PROPs were developed in topology, along with operads, to describe spaces with lots of operations on them. But now some of us are using them to think about ‘signal-flow diagrams’ in control theory—an important branch of engineering. I talked about that here on the n-Café a while ago, but it’s time for an update.

Posted at 8:43 PM UTC | Permalink | Followups (3)

How to Acknowledge Your Funder

Posted by Tom Leinster

A comment today by Stefan Forcey points out ways in which US citizens can try to place legal limits on the surveillance powers of the National Security Agency, which we were discussing in the context of its links with the American Mathematical Society. If you want to act, time is of the essence!

But Stefan also tells us how he resolved a dilemma. Back here, he asked Café patrons what he should do about the fact that the NSA was offering him a grant (for non-classified work). Take their money and contribute to the normalization of the NSA’s presence within the math community, or refuse it and cause less mathematics to be created?

What he decided was to accept the funding and — in this paper at least — include a kind of protesting acknowledgement, citing his previous article for the Notices of the AMS.

I admire Stefan for openly discussing his dilemma, and I think there’s a lot to be said for how he’s handled it.

Posted at 2:28 PM UTC | Permalink | Followups (2)

May 21, 2015

The Origin of the Word “Quandle”

Posted by John Baez

A quandle is a set equipped with a binary operation with number of properties, the most important being that it distributes over itself:

a(bc)=(ab)(ac) a \triangleright (b \triangleright c) = (a \triangleright b)\triangleright (a \triangleright c)

They show up in knot theory, where the operation \triangleright describes what happens when one strand crosses over another… and the laws are chosen so that the quandle gives an invariant of a knot, independent of how you draw it. Even better, the quandle is a complete invariant of knots: if two knots have isomorphic quandles, there’s a diffeomorphism of 3\mathbb{R}^3 mapping one knot to the other.

I’ve always wondered where the name ‘quandle’ came from. So I decided to ask their inventor, David Joyce—who also proved the theorem I just mentioned.

Posted at 7:41 AM UTC | Permalink | Followups (33)

May 18, 2015

The Revolution Will Not Be Formalized

Posted by Mike Shulman

After a discussion with Michael Harris over at the blog about his book Mathematics without apologies, I realized that there is a lot of confusion surrounding the relationship between homotopy type theory and computer formalization — and that moreover, this confusion may be causing people to react negatively to one or the other due to incorrect associations. There are good reasons to be confused, because the relationship is complicated, and various statements by prominent members of both communities about a “revolution” haven’t helped matters. This post and its sequel(s) are my attempt to clear things up.

Posted at 8:46 PM UTC | Permalink | Followups (8)

May 13, 2015

Categorifying the Magnitude of a Graph

Posted by Simon Willerton

Tom Leinster introduced the idea of the magnitude of graphs (first at the Café and then in a paper). I’ve been working with my mathematical brother Richard Hepworth on categorifying this and our paper has just appeared on the arXiv.

Categorifying the magnitude of a graph, Richard Hepworth and Simon Willerton.

The magnitude of a graph can be thought of as an integer power series. For example, consider the Petersen graph.

Petersen graph

Its magnitude starts in the following way. #P =1030q+30q 2+90q 3450q 4 +810q 5+270q 65670q 7+. \begin{aligned} \#P&=10-30q+30q^{2}+90q ^{3}-450q^{4}\\ &\quad\quad+810q^{5} + 270 q^{6} - 5670 q^{7} +\dots. \end{aligned}

Richard observed that associated to each graph GG there is a bigraded group MH *,*(G)\mathrm{MH}_{\ast ,\ast }(G), the graph magnitude homology of GG, that has the graph magnitude #G# G as its graded Euler characteristic. #G = k,l0(1) krank(MH k,l(G))q l = l0χ(MH *,l(G))q l. \begin{aligned} #G &= \sum _{k,l\geqslant 0} (-1)^{k}\cdot \mathrm{rank}\bigl (\mathrm{MH}_{k,l}(G)\bigr )\cdot q^{l}\\ &= \sum _{l\geqslant 0} \chi \bigl (\mathrm{MH}_{\ast ,l}(G)\bigr )\cdot q^{l}. \end{aligned} So graph magnitude homology categorifies graph magnitude in the same sense that Khovanov homology categorifies the Jones polynomial.

For instance, for the Petersen graph, the ranks of MH k,l(P)\mathrm{MH}_{k,l}(P) are given in the following table. You can check that the alternating sum of each row gives a coefficient in the above power series.

k 0 1 2 3 4 5 6 7 0 10 1 30 2 30 3 120 30 l 4 480 30 5 840 30 6 1440 1200 30 7 7200 1560 30 \begin{array}{rrrrrrrrrr} &&&&&&k\\ &&0&1&2&3&4&5&6&7 \\ &0 & 10\\ & 1 & & 30 \\ &2 & && 30 \\ &3 &&& 120 & 30 \\ l &4 &&&& 480 & 30 \\ &5 &&&&& 840 & 30 \\ &6 &&&&& 1440 & 1200 & 30 \\ &7 &&&&&& 7200 & 1560 & 30 \\ \\ \end{array}

Many of the properties that Tom proved for the magnitude are shadows of properties of magnitude homology and I’ll describe them here.

Posted at 4:18 PM UTC | Permalink | Followups (19)