The n-Category Café

We’ve discussed the prospects for adding modalities to type theory for many a year, e.g., here at the Café back at Modal Types, and frequently at the nLab. So now I’ve written up some thoughts on what philosophy might make of modal types in this preprint. My debt to the people who helped work out these ideas will be acknowledged when I publish the book.

This is to be the fourth chapter of a book which provides reasons for philosophy to embrace modal homotopy type theory. The book takes in order the components: types, dependency, homotopy, and finally modality.

The chapter ends all too briefly with mention of Mike Shulman et al.’s project, which he described in his post – What Is an n-Theory?. I’m convinced this is the way to go.

PS. I already know of the typo on line 8 of page 4.

I’ve just done something I’ve been meaning to do for ages: compiled a bibliography of all the publications on magnitude that I know about. More people have written about it than I’d realized!

This isn’t an exercise in citation-gathering; I’ve only included a paper if magnitude is the central subject or a major theme.

I’ve included works on magnitude of ordinary, un-enriched, categories, in which context magnitude is usually called Euler characteristic. But I haven’t included works on the diversity measures that are closely related to magnitude.

Enjoy! And let me know in the comments if I’ve missed anything.

Well orderings have slightly perplexed me for a long time, so every now and then I have a go at seeing if I can understand them better. The insight I’m about to explain doesn’t resolve my perplexity, it’s pretty trivial, and I’m sure it’s well known to lots of people. But it does provide a fresh perspective on well orderings, and no one ever taught me it, so I thought I’d jot it down here.

In short: the axiom of choice allows you to choose one element from each nonempty subset of any given set. A well ordering on a set is a way of making such a choice in a consistent way.

I’ve been distracted by other things for the last few months, but in that time several interesting-looking papers on magnitude (co)homology have appeared on the arXiv. I will just list them here with some vague comments. If anyone (including the author!) would like to write a guest post on any of them then do email me.

• Magnitude meets persistence. Homology theories for filtered simplicial sets by Nina Otter

For years a standing question was whether magnitude was connected with persistent homology, as both had a similar feel to them. Here Nina relates magnitude homology with persistent homology.

• Magnitude cohomology by Richard Hepworth

In both mine and Richard’s paper on graphs and Tom Leinster and Mike Shulman’s paper on general enriched categories, it was magnitude homology that was considered. Here Richard introduces the dual theory which he shows has the structure of a non-commutative ring.

• Smoothness filtration of the magnitude complex by Kiyonori Gomi

I haven’t looked at this yet as I only discovered it last night. However, when I used to think a lot about gerbes and Deligne cohomology I was a fan of Kiyonori Gomi’s work with Yuji Terashima on higher dimensional parallel transport.

• Graph magnitude homology via algebraic Morse theory by Yuzhou Gu

This is the write-up of some results he announced in a discussion here at the Café. These results answered questions asked by me and Richard in our original magnitude homology for graphs paper, for instance proving the expression for magnitude homology of cyclic graphs that we’d conjectured and giving pairs of graphs with the same magnitude but different magnitude homology.

This blog was born in 2006 when a philosopher, a physicist and a mathematician found they shared an interest in categorification — and in particular, categorical groups, also known as 2-groups. So it’s great to see 2-groups showing up in theoretical condensed matter physics. From today’s arXiv papers:

• J.P. Ang and Abhishodh Prakash, Higher categorical groups and the classification of topological defects and textures.

Abstract. Sigma models effectively describe ordered phases of systems with spontaneously broken symmetries. At low energies, field configurations fall into solitonic sectors, which are homotopically distinct classes of maps. Depending on context, these solitons are known as textures or defect sectors. In this paper, we address the problem of enumerating and describing the solitonic sectors of sigma models. We approach this problem via an algebraic topological method – combinatorial homotopy, in which one models both spacetime and the target space with algebraic objects which are higher categorical generalizations of fundamental groups, and then counts the homomorphisms between them. We give a self-contained discussion with plenty of examples and a discussion on how our work fits in with the existing literature on higher groups in physics.

The fun will really start when people actually synthesize materials described by these materials! Condensed matter physicists are doing pretty well at realizing theoretically possible phenomena in the lab, so I’m optimistic. But I don’t think it’s happened yet.

Samuel Vidal has kindly LaTeXed some notes by Todd Trimble:

• Todd Trimble, Bar constructions and combinatorics of polyhedra for $n$-categories.

Todd wrote these around 1999, as far as I know. I’ve always enjoyed them; they give a clearer introduction to the bar construction than any I’ve seen, and they also suggest a number of fascinating directions for research on the relation between higher categorical structures and polyhedra.

Higher gauge theory has the potential to describe the behavior of 1-dimensional objects and higher-dimensional membranes much as ordinary gauge theory describes the behavior of point particles. But ordinary gauge theory is also a source of fascinating differential equations, which yield interesting results about topology if one uses enough analysis to prove rigorous results about their solutions. What about higher gauge theory?

Andreas Gastel has a new paper studying higher gauge theory using some techniques of analysis that are commonly used in ordinary gauge theory. He’s finding some interesting similarities but also some ways in which higher gauge theory is simpler:

• Andreas Gastel, Canonical gauges in higher gauge theory.

Abstract. We study the problem of finding good gauges for connections in higher gauge theories. We find that, for 2-connections in strict 2-gauge theory and 3-connections in 3-gauge theory, there are local “Coulomb gauges” that are more canonical than in classical gauge theory. In particular, they are essentially unique, and no smallness of curvature is needed in the critical dimensions. We give natural definitions of 2-Yang-Mills and 3-Yang-Mills theory and find that the choice of good gauges makes them essentially linear. As an application, (anti-)selfdual 2-connections over $B^6$ are always 2-Yang-Mills, and (anti-)selfdual 3-connections over $B^8$ are always 3-Yang-Mills.

I’m thinking about finite groups these days, from a Klein geometry perspective where we think of a group $G$ as a source of $G$-sets. Since the category of $G$-sets is a topos, this lets us translate concepts, facts and questions about groups into concepts, facts and questions about topoi. I’m not at all good at this, so here are a bunch of basic questions.

The major annual category theory conference will be held in Edinburgh next year:

Category Theory 2019

University of Edinburgh

7-13 July 2019

Organizing committee: Steve Awodey, Richard Garner, Chris Heunen, Tom Leinster, Christina Vasilakopoulou.

As John has just pointed out, this is followed two days later by the Applied Category Theory conference and school in Oxford, very conveniently for anyone wishing to go to both.

I’m helping organize ACT 2019, an applied category theory conference and school at Oxford, July 15-26, 2019. Here’s a ‘pre-announcement’.

More details will come later, but here’s some good news: it’s right after the big annual worldwide category theory conference, which is in Edinburgh in 2019. So, conference-hopping category theorists can attend both!

One of the main reasons I’m interested in type theory in general, and homotopy type theory (HoTT) in particular, is that it has categorical semantics. More precisely, there is a correspondence between (1) type theories and (2) classes of structured categories, such that any proof in a particular type theory can be interpreted into any category with the corresponding structure. I wrote a lot about type theory from this perspective in The Logic of Space. The basic idea is that we construct a particular structured category $Syn$ out of the syntax of the type theory, and prove that it is the initial such category. Then we can interpret any syntactic object $A$ in a structured category $C$ by regarding $A$ as living in $Syn$ and applying the unique structured functor $Syn\to C$.

Unfortunately, we don’t currently have any very general definitions of what “a type theory” is, what the “corresponding class of structured categories” is, or a very general proof of this “initiality theorem”. The idea of such proofs is easy — just induct over the construction of syntax — but its realization in practice can be long and tedious. Thus, people are understandably reluctant to take the time and space to write out such a proof explicitly, when “everyone knows” how the proof should go and probably hardly anyone would really read such a proof in detail anyway. This is especially true for dependent type theory, which is qualitatively more complicated in various ways than non-dependent type theories; to my knowledge only one person (Thomas Streicher) has ever written out anything approaching a complete proof of initiality for a dependent type theory.

My online course is now done. We finished the fourth chapter of Fong and Spivak’s book Seven Sketches—my classes at U.C.R. are starting up now, so I had to stop there.

Chapter 4 is about collaborative design: building big projects from smaller parts. This is based on work by Andrea Censi:

• Andrea Censi, A mathematical theory of co-design.

The main mathematical content of this chapter is the theory of enriched profunctors. We’ll mainly talk about enriched profunctors between categories enriched in monoidal preorders. The picture above shows what one of these looks like!

Sometimes you check just a few examples and decide something is always true. But sometimes even $1.5 \times 10^{43}$ examples is not enough.

In general, there are two kinds of bicategories: those like $Cat$ and those like $Span$. In the $Cat$-like ones, the morphisms are “categorified functions”, which generally means some kind of “functor” between some kind of “category”, consisting of functions mapping objects and arrows from domain to codomain. But in the $Span$-like ones (which includes $Mod$ and $Prof$), the morphisms are not “functors” but rather some kind of “generalized relations” (including spans, modules, profunctors, and so on) which do not map from domain to codomain but rather relate the domain and codomain in some way.

In $Span$-like bicategories there is usually a subclass of the morphisms that do behave like categorified functions, and these play an important role. Usually the morphisms in this subclass all have right adjoints; sometimes they are exactly the morphisms with right adjoints; and often one can get away with talking about “morphisms with right adjoints” rather than making this subclass explicit. However, it’s also often conceptually and technically helpful to give the subclass as extra data, and arguably the most perspicuous way to do this is to work with a double category instead. This was the point of my first published paper, though others had certainly made the same point before, and I think more and more people are coming to recognize it.

Today a new installment in this story appeared on the arXiv: Cartesian Double Categories with an Emphasis on Characterizing Spans, by Evangelia Aleiferi. This is a project that I’ve wished for a while someone would do, so I’m excited that at last someone has!

I’ve been trying to learn a bit of the theory of finite groups. As you may know, Sylow’s theorems say that if you have a finite group $G$, and $p^k$ is the largest power of a prime $p$ that divides the order of $G$, then $G$ has a subgroup of order $p^k$, which is unique up to conjugation. This is called a Sylow $p$-subgroup of $G$.

Sylow’s theorems also say a lot about how many Sylow $p$-subgroups $G$ has. They also say that any subgroup of $G$ whose order is a power of $p$ is contained in a Sylow $p$-subgroup.

I didn’t like these theorems as an undergrad. The course I took whizzed through them in a desultory way. And I didn’t go after them myself: I was into group theory for its applications to physics, and the detailed structure of finite groups doesn’t look important when you’re first learning physics: what stands out are continuous symmetries, so I was busy studying Lie groups.

Since I didn’t really master Sylow’s theorems, and had no strong motive to do so, I didn’t like them — the usual sad story of youthful mathematical distastes.

But now I’m thinking about Sylow’s theorems again, especially pleased by Robert A. Wilson’s one-paragraph proof of all three of these theorems in his book The Finite Simple Groups. And I started wondering if the importance of groups of prime power order — which we see highlighted in Sylow’s theorems and many other results — is all related to localization in algebraic topology, which is a technique to focus attention on a particular prime.