The n-Category Café

Spans are a wonderfully simple idea, and, as such, they are ubiquitous mathematics. Why? Well, for one, any span, which is a pair of arrows with common domain, from a space (set, groupoid, object, etc.) $A$ to a space $B$: $$\begin{matrix} &S&\\ &\swarrow \searrow&\\ B&&A\\ \end{matrix}$$ can be turned around without any “fuss” about injectivity or surjectivity to obtain a span from the space $B$ to the space $A$: $$\begin{matrix} &S&\\ &\swarrow \searrow&\\ A&&B\\ \end{matrix}$$ See, I just did it!

But before we get carried away, spans have an ugly, dark side as well. Composition of spans is not associative. So spans, considered as morphisms between sets, for example, do not even form a category. However, with a sunny disposition and a healthy dose of optimism, unable to have a category, we happily settle for a (weak) $2$-category, or bicategory, of spans.

In fact, Bénabou defined bicategories to handle exactly this type of situation. By defining a suitable notion of ‘maps between spans’, Benabou was able to produce, as an early example of a bicategory, a structure consisting of:

• sets as objects,
• spans of sets as $1$-morphisms, and
• maps of spans of sets as $2$-morphisms.

So how are spans composed? Given composable spans $$\begin{matrix} &S&&R&\\ &\swarrow \searrow&&\swarrow \searrow&\\ C&&B&&A\\ \end{matrix}$$ we can form a composite span $$\begin{matrix} &&S R&&\\ &&\swarrow\searrow &&\\ &S&&R&\\ &\swarrow \searrow&&\swarrow \searrow&\\ C&&B&&A\\ \end{matrix}$$

We haven’t yet defined $S R$. Let’s continue to consider the example of spans of sets a bit longer. The category of sets is complete, meaning that it has all limits. In particular, we can define $SR$ to be the pullback, sometimes called the fibered product. Pullbacks are limits of diagrams of the following shape: $$\begin{matrix} S&&&&R\\ &\searrow&&\swarrow&\\ &&B&&\\ \end{matrix}$$ called a `cospan’.

The big idea here is that we can form a bicategory $Span(\mathcal{C})$ with spans as $1$-morphisms from any category $\mathcal{C}$ with pullbacks. If $\mathcal{C}$ also has finite products (really, just adding a terminal object to a category with pullbacks is enough), then $Span(\mathcal{C})$ can also be given a monoidal structure.

The span construction is very well-known, but the seemingly minor nuisance of having non-associative composition, can be more troublesome than it might first appear.

It is quite common for mathematicians to work with spaces, which are themselves categories, or at least have, in addition to a notion of maps, a notion of maps between maps.

So, given a $2$-category $\mathcal{B}$ with pullbacks, what kind of structure is $Span(\mathcal{B})$?

The answer, which probably belongs to the realm of `folk theorems’, is a tricategory. This is the beginning of a pattern that, while nice, makes the span construction rather difficult to describe functorially. This is:

• Given a category $\mathcal{C}$ with products and pullbacks, there is a monoidal bicategory $Span(\mathcal{C})$.
• Given a bicategory $\mathcal{B}$ with products and pullbacks, there is a monoidal tricategory $Span(\mathcal{B})$.

But, $$\textstyle{What is a monoidal tricategory?}$$

I’ve just come back from the successful thesis defence of Samer Allouch, a student of Carlos Simpson in Nice. Among other things, Allouch’s thesis completely answers the question:

which finite directed graphs can be equipped with the structure of a category in at least one way?

The answer turns out to be rather satisfying: it’s neither simple enough that you’d guess it without prolonged thought, nor prohibitively complicated.

But here’s a curious thing: each of the conditions in Allouch’s theorem involves at most four vertices or objects. Let’s say that a directed graph is categorical if it can be given the structure of a category. Then for a finite directed graph $G$,

$G$ is categorical if and only if each full subgraph of $G$ with $\leq 4$ vertices is categorical.

Why?

In other words, can you prove this directly, without using Allouch’s theorem?

Peter Freyd has given that title to his 2011 Thomas and Yvonne Williams Lecture for the Advancement of Logic and Philosophy to be delivered on Monday, April 11, 4:30 - 6:00 p.m. at the Wu & Chen Auditorium, Levine Hall, 3330 Walnut Street, Philadelphia, PA. If anyone can report on the lecture, we’d love to hear about it.

It’s a dangerous business making promises about what will happen “next time” when I don’t have “next time” written yet. I said last time that I intended to talk about the univalence axiom next, but then I realized there is more I want to say about equivalences first, and perhaps functional extensionality.

But before I get into that, I’d like to mention that Steve Awodey has written a very nice post about intensional type theory and its relation to homotopy type theory from an intensional type theorists’ point of view; read it at the HoTT blog!

I’m giving two talks at Hong Kong University this week:

1. Energy, the environment, and what mathematicians can do.
   
   
2. Higher gauge theory, division algebras and superstrings.

These are roughly the first talk of my new life, and the last of my old. We’re chatting about talk 1 over on Azimuth, here and here. But the n-Café is the right place for chatting about talk 2!

First, an announcement: the homotopy type theory project now has its own web site! Follow the blog there for announcements of current developments.

Now, let’s pick up where we left off. The discussion in the comments at the last post got somewhat advanced, which is fine, but in the main posts I’m going to try to continue developing things sequentially, assuming you haven’t read anything other than the previous main posts. (I hope that after I’m done, you’ll be able to go back and read and understand all the comments.)

Last time we talked about the correspondence between the syntax of intensional type theory, and in particular of identity types, and the semantics of homotopy theory, and in particular that of a nicely-behaved weak factorization system. This time we’re going to start developing mathematics in homotopy type theory, mostly following Vladimir Voevodsky’s recent work.

Last week I was at a mini-workshop at Oberwolfach entitled “The Homotopy Interpretation of Constructive Type Theory.” Some aspects of this subject have come up here a few times before, but mostly as a digression from other topics; now I want to describe it in a more structured way.

Roughly, the goal of this project is to develop a formal language and semantics, similar to the language and semantics of set theory, but in which the fundamental objects are homotopy types (a.k.a. $\infty$-groupoids) rather than sets. There are several motivations for this, which I’ll mention below, but the most radical hope (which has been put forward most strongly by Voevodsky) is to create a new foundation for all of mathematics which is natively homotopical—that is, in which homotopy types are the basic objects, rather than being built up out of other basic objects such as sets.

I find this idea extremely exciting; at times I think it has the potential to transform the practice of everyday mathematics in a way that no foundational development has done since (probably) Cantor. But the most exciting thing of all is that the essential core of this language already exists in a completely different area of mathematics: intensional type theory. All that needs to be done is to reinterpret some words and perhaps add some additional axioms. (I especially enjoy the coincidence of terminology which allows me to say “homotopy type theory” and mean both “the homotopy version of type theory” and “the theory of homotopy types”!)

One reason this is especially exciting is that intensional type theory is (with good reason) the foundation of some of the most successful proof assistants for computer-verified mathematics, such as Coq and Agda. Thus, the $\infty$-categorical revolution, if carried out in the language of homotopy type theory, will support and be supported by the inevitable advent of better computer-aided tools for doing mathematics. I never would have guessed that the computerization of mathematics would be best carried out, not by set-theory-like reductionism, but by an enrichment of mathematics to be natively $\infty$-categorical.

My student John Huerta is looking for a job. You should hire him! And not just because he’s a great guy. He’s also done some great work.

He recently gave a talk at the School on Higher Gauge Theory, TQFT and Quantum Gravity in Lisbon:

• John Huerta, A categorified supergroup for string theory.

This has got to be the first talk that combines tricategories and the octonions in a mathematically rigorous way to shed light on the foundations of M-theory! It’s a preview of his thesis.

I have been rather remiss, I feel, in promoting some mutual scrutiny between our $n$-categorical community and that part of the metaphysics community which interests itself in structuralism. Since I heard in the mid-1990s about the metaphysical theory of structural realism put forward by various philosophers of physics, I have thought that category theory should have much to say on the issue. For one thing, a countercharge against those ontic structural realists, who believe that all that science discovers in the world are structures, maintains that the very notion of a relation within a structure involves the notion of relata, things which are being related. Structures must structure some things. Category theoretic understanding ought to have something to say on this matter.

It’s interesting then to see a recent paper by Jonathan Bain – Category-Theoretic Structure and Radical Ontic Structural Realism – which argues that the countercharge can only be made from a set-theoretic perspective.

So there’s one question: If we adopt the nPOV on physics, can we say what we are committing ourselves to the existence of?

There is to be a Symposium – Sets Within Geometry – held in Nancy, France on 26-29 July, 2011. Confirmed speakers are: FW Lawvere (Buffalo), Yuri I. Manin (Bonn and IHES), Anders Kock (Aarhus), Christian Houzel (Paris), Colin McLarty (CWRU Cleveland), Martha Bunge (Montreal), Jean-Pierre Marquis (Montreal) and Alberto Peruzzi (Florence).

Statement of aims:

Those who have come together to organise this Symposium believe that the ultimate aim of foundational efforts is to provide clarifying guidance to teaching and research in mathematics, by concentrating the essential aspects of past such endeavors. By mathematics we mean the investigation of the Relations between Space and Quantity, of the reflected relations between quantity and quantity and between space and space, and the development of our knowledge of these in other words Geometry.

Using tools developed by Cantor and his contemporaries, much more explicit forms of the relation between space and quantity were developed in the 1930s in the field of functional analysis by Stone and Gelfand, partly through the notion of Spectrum (a space corresponding to a given system of quantities). In the 1950s Grothendieck applied those same tools, around the notion of Spectrum, to algebraic geometry by using and developing the further powerful tool of category theory . Further developments have strongly suggested that it is now possible to incorporate the whole set-theoretic “foundation” of Geometry, explicitly as part of that space-quantity dialectic, in other words as a chapter in an extended Algebraic Geometry.

Liang Kong gave what was probably the first talk at the Centre for Quantum Technologies to explicitly mention tricategories:

• Liang Kong, Levin-Wen models and tensor categories. (Joint work with Alexei Kitaev.)

But as his talk shows, tricategories are a quite natural formalism for studying models of 2d condensed matter physics. Two dimensions of space, one dimension of time: a tricategory!

The relation to the work of Fjelstad–Fuchs–Runkel–Schweigert is visible, but the focus on lattice models of condensed matter physics — in particular, the so-called Levin–Wen models — gives Liang Kong’s work a somewhat different flavor.

Jim Stasheff is asking me to forward the announcement of the

• NSF/CBMS Conference on Deformation Theory of Algebras and Modules

  May 16-20, 2011

  North Carolina State University

  website

The main event is a lecture series by Martin Markl on deformation theory of ∞-algebras.

Generalized means are things like arithmetic means and geometric means. They can be ‘fair’, giving all their inputs equal status, or they can be weighted. I guess the first major result on them was the theorem that the arithmetic mean is always greater than or equal to the geometric mean.

Another, later, landmark was the 1934 book Inequalities of Hardy, Littlewood and Pólya, where they proved a characterization theorem for generalized means. It looks like this:

If you have some sort of ‘averaging operation’ with all the properties you’d expect of something called an averaging operation, then there aren’t many options: it must be of a certain prescribed form.

That’s ancient history. It could be even more ancient than Hardy, Littlewood and Pólya: I don’t know whether the characterization in their book is due to them, or whether it’s older still.

Yesterday, however, I posted about a new theorem of Guillaume Aubrun and Ion Nechita that gives a startlingly simple characterization of the $p$-norms. Since $p$-norms and generalized means are closely related, I wondered, out loud, whether it might be possible to deduce from their result a simple new characterization of generalized means. And if I’m not mistaken, the answer is yes.

This March we have the second QVEST meeting:

• Quaterly seminar on topology and geometry

  Utrecht University

  March 11, 2011 .

  (seminar website)

The speakers are

• Domenico Fiorenza (Rome, La Sapienza)

  The periods map of a complex projective manifold: an oo-category perspective

• Boris Shoikhet (MPI Bonn)

  Methods of category/homotopy theory in deformation theory

• Ulrich Penning (Münster)

  Twisted K-Theory with coefficients in a $C^*$-algebra and 2-Bimodule bundles

If you would like to attend and have any questions, please drop me a message.

The first QVEST meeeting was here.

Some mathematical objects acquire a reputation for being important. We know they’re important because our lecturers told us so when we were students, and because we’ve observed that they’re treated as important by large groups of research mathematicians. If you stood up in public and asked exactly what was so important about them, you might fear getting laughed at as an ignoramus… but perhaps no one would have a really good answer. There’s only a social proof of importance.

I have a soft spot for theorems that take a mathematical object known socially to be important and state a precise mathematical sense in which it’s important. This might, for example, be a universal property (‘it’s the universal thing with these good properties’) or a unique characterization (‘it’s the unique thing with these good properties’).

Previously I’ve enthused about theorems that do this for the category $\Delta$, the topological space $[0, 1]$, and the Banach space $L^1$. Today I’ll enthuse about a theorem that does it for the $p$-norms $\Vert\cdot\Vert_p$. The theorem is from a recent paper of Guillaume Aubrun and Ion Nechita.

The statement is beautifully simple.