The n-Category Café

The $n$-Category Café will be closed for 48 hours starting Friday, January 6th, 2012 at 6 pm CST. My calculations say this is midnight Greenwich Mean Time, the end of January 6th and beginning of January 7th.

They are temporarily shutting off all power and internet to the building at the University of Texas at which the Café secretly resides.

We all love a good universal property. Objects with a simple universal property are usually important. So you might guess (mightn’t you?) that objects with two simple universal properties are more important still.

Perhaps the most famous example is direct sum. The direct sum of $A \oplus B$ of two vector spaces, modules, etc., is both the product and the coproduct of $A$ and $B$. And the importance of direct sums is written all over homological algebra.

In that example, the two universal properties are dual to one another. This is often the way. A less obvious example, but one whose importance becomes more and more apparent the deeper you dig into category theory, is the splitting of idempotents. This process can be viewed as either a limit or a colimit.

Objects with two universal properties are extra special, then. When the universe hands you one, it’s a treat.

Now: it’s Christmas. Ordinary people give each other socks. But since you already have enough socks, I give you instead: a machine for producing objects with two dual universal properties. Merry Christmas!

I spent yesterday morning in the computer science department at Strathclyde, for the 60th birthday celebrations of Peter Hancock. Often these events are months away from the person’s actual birthday, but in this case it was only one day off. Happy birthday, Hank!

One of the talks I went to was by Alex Simpson, the title of which was too good not to use as the title of this post. Simpson’s recent work has re-examined foundational questions about probability and randomness. It’s also the most exciting use of locales I think I’ve ever encountered. I want to tell you something about it here.

This year has seen a dramatic deterioration in the relationship between mathematicians working in the UK and our main funding body, the Engineering and Physical Sciences Research Council (EPSRC). Maybe you’ve read about it on Tim Gowers’s blog. Or maybe you’ve seen Burt Totaro’s roundup of strongly worded letters to the EPSRC and the government, from probably every learned society that has anything to do with British mathematics.

It’s depressing stuff, even if you’re not in the UK. Hopefully we’ll serve as an example to others, but there’s always the suspicion that policy changes in one country are part of a worldwide trend. Indeed, there seem to be resemblances to the situation in Canada — see, for instance, these posts by Nassif Ghoussoub (and the splendidly titled “UK mathematicians unload on intransigent patronizing bureaucracy”).

Somehow 5 years have slipped by since my post Back from NIPS 2006. These NIPS (Neural Information Processing Systems) conferences bring together the machine learning community every December for a main conference in a city, followed by two days of workshops in a ski resort nearby. This time it was the turn of Granada and Sierra Nevada.

I could only manage time off for the workshops, where I was invited to participate in the Machine Learning and Philosophy session. The main thrust of my talk was to indicate that philosophy of science has shown that there’s much more to learning than what can be achieved by classifying and regression algorithms. There’s a tendency to believe that results in formal learning theory have a great deal to say about learning in general. It seems to me rather like believing that theorems in proof theory are the final word on what is philosophically interesting about mathematics. I have always believed instead that concept formation in science and mathematics is of the utmost importance.

I have just arrived in Denmark, where I am attending the

4th Odense Winter School on Geometry and Theoretical Physics (website, program ).

I’ll try to report in the comment section below on what’s going on.

Myself, I’ll be lecturing on Stacks, differential geometry and action functionals, following section 1.1 “Motivation” with technical details sprinkled in from section 1.3 “Models and applications” of the cohesive document.

I had given similar lectures (with roughly twice the time, though) last weekend in Warsaw, at the

Polish Seminar on Category Theory and its Applications (website)

with an accompanying homotopy type theoretic comment in the Warsaw category theory seminar on the geometric and the homotopical interval/circle in homotopy type theory (see here for what this is about).

I am trying to use these events to fine-tune the section 1. Introduction further (not that there is not lots of room to improve it). I am aware of a bunch of deficiencies of the exposition, that I am trying to find the time to fix. But there must be also a bunch of deficiencies that I am not yet aware of. If you tell me about those that bug you, I’d be very grateful. (Many thanks at this point to John Huerta, who has recently given me a list of useful feedback on this.)

Mathematicians should know how to count. Most of us do. But still there are some mysteries left when we try to count things in a way that gives a negative, fractional, irrational or complex answer.

Luckily, we’ve been making lots of progress. The Euler characteristic of a space can be a negative integer. The cardinality of a groupoid can be a nonnegative real number. The Euler characteristic of a category can even be a negative real number! And here’s yet another approach:

• Igor Frenkel, Catharina Stroppel, Joshua Sussan, Categorifying fractional Euler characteristics, the Jones-Wenzl projector and 3j-symbols.

The International Mathematical Union (IMU) and the International Council for Industrial and Applied Mathematics (ICIAM) are important math organizations. For example, the IMU runs the big prestigious International Congress of Mathematicians. Now they have a blog, and they want to know our ideas on what to do about high-priced journals:

• Blog on Mathematical Journals, What might be done about high prices of journals?, December 7th, 2011.

I’ll disable comments here — I want you to comment over there! But I’ll tell you what I said.

(It takes a while for comments to appear, so you may not see what I said over there yet.)

(Guest post by Emily Riehl)

The most substantial difference between Quillen’s original definition of a model category and the one in use today (which he called closed) is that, e.g., the cofibrations were defined to be any class of maps for which the usual factorizations and lifts exist — in particular, it is not necessary that all maps which lift against the trivial fibrations in the sense of the diagram $$\array{ \quad\cdot & \to & \cdot\qquad \\ {}^{\mathrm{cof}}\downarrow & {}^{\exists}\nearrow & \downarrow^{\mathrm{triv}}{}^{\mathrm{fib}} \\ \quad\cdot & \to & \cdot\qquad}$$ are cofibrations.

For instance, there is a (non-closed) model structure on spaces given by relative cell complexes, Serre fibrations, and weak homotopy equivalences. There is a model structure on chain complexes of modules over a commutative ring given by injections with free cokernel, surjections, and quasi-isomorphisms. But any retract of a map with a particular lifting property inherits that same lifting property, so for these examples to be model categories as presently understood, we must enlarge the class of cofibrations to include all retracts of relative cell complexes in the first case and all injections with projective cokernel in the second. Quillen uses this “closure” to show that the homotopy category of a model category is saturated, meaning every map that becomes an isomorphism was originally a weak equivalence.

Despite the overwhelming popularity of the closed definition of model categories, even in the modern literature, the distinction between cellular cofibrations (e.g., the relative cell complexes above) and generic ones is still maintained — at least in the case where this class is cofibrantly generated. In what follows, we’ll present several unexpectedly strong existence results for certain maps between cofibrantly generated algebraic model categories that hold precisely when certain cofibrations are cellular and not merely retracts of cellular cofibrations, providing a new justification for the classically-held distinction.

At its most basic, a discrete-time dynamical system is an object equipped with an endomorphism. The word ‘dynamical’ becomes appropriate if we plan to iterate the endomorphism. For example, the concept of image is not dynamical, because the image of an endomorphism $f$ is the set of outputs after doing $f$ just once. But the concept of eventual image — defined precisely below — is dynamical: it’s the set of outputs after iterating $f$ indefinitely.

This post is a plea for someone to do some category theory. I’ll describe three categories in which dynamical systems behave in a strikingly similar way… but the three proofs I have are all different. If someone comes along, sees what’s really going on, and unifies the three strands, that will make me happy.

The three categories are these: finite sets, finite-dimensional vector spaces, and compact metric spaces. To give a hint of what they have in common:

• Every injective endomorphism of a finite set is bijective.
• Every injective linear operator on a finite-dimensional vector space is bijective.
• Every distance-preserving endomorphism of a compact metric space is bijective.

(The first two are very well-known; the third I learned from MathOverflow.)

But the three categories have a lot more in common than that. For example, in each case, every endomorphism $f: X \to X$ gives rise canonically to an idempotent endomorphism $f^\infty: X \to X$. Why?

A while back I posted a draft of a paper on the construction of a monoidal tricategory of spans. Since then the paper has changed quite a bit, and a new version appeared on the arXiv on Monday.

The following theorem is the main result of the paper:

Given a strict $2$-category $\mathcal{C}$ with finite pseudolimits, there is a monoidal tricategory $Span(\mathcal{C})$, consisting of:

• the same objects as $\mathcal{C}$,
• spans in $\mathcal{C}$ as $1$-morphisms,
• maps of spans in $\mathcal{C}$ as $2$-morphisms,
• maps of maps of spans in $\mathcal{C}$ as $3$-morphisms.

I usually think of spans as categorified linear operators, for example, the spans of groupoids in groupoidification. Also, spans, or more specifically, correspondences, are central in convolution operations in geometric representation theory. Here on the blog we have seen this in Ben-Zvi’s notes on geometric function theories. Of course, spans generalize relations between sets, so they show up in many other contexts as well.

This theorem has not changed from the previous version, although some changes to the structure of $Span(\mathcal{C})$ have been made. I want to comment on some of the changes here. In particular, I want to highlight the inclusion of Todd Trimble’s definition of a tetracategory.

Today I spoke in our category theory seminar on

Homotopy Axiomatic Cohesion – Some basic ideas (pdf)

This is with an audience with a background in toposes and logic in mind.

For anyone following our discussions here the slides contain no news, but maybe there is some use for basic introductory and survey material.

In my last post, I described how in the search for an “internal” description of reflective subcategories of a category $H$, we are led to axiomatize instead the following notions:

1. A reflective subfibration of $H$ is a system of reflective subcategories $C_x\subseteq H/x$, such that pullback preserves the $C$’s and commutes with the reflectors. A reflective subfibration has an underlying reflective subcategory $C_1 \subseteq H$.

2. A stable factorization system is an orthogonal factorization system $(E,M)$ for which $E$-maps (and hence, the factorizations) are stable under pullback. Given such, we define $C_x = M/x$, the category of maps $y\to x$ lying in $M$. The factorizations make $C_x \subseteq H/x$ reflective, and stability of $E$ shows that pullback commutes with the reflectors; thus every stable factorization system has an underlying reflective subfibration.

   Conversely, a reflective subfibration arises in this way exactly when it has the property that if $x\to y$ is an object of $C_y$ and $y\to z$ is an object of $C_z$, then the composite $x\to y\to z$ is also an object of $C_z$. (I think this observation is due to Carboni and Janelidze and Kelly and Paré, “On localization and stabilization for factorization systems”.) The underlying reflective subcategory is $M/1$, the category of all $x$ such that $x\to 1$ is in $M$.

3. A lex-reflective subcategory is, of course, a reflective subcategory whose reflector preserves finite limits. Given such a subcategory $C\subseteq H$, we define $E$ to be the class of maps inverted by the reflector $L$, and $M$ the class of maps right orthogonal to $E$. Then $(E,M)$ is a stable factorization system whose underlying reflective subcategory is $C$. Conversely, a stable factorization system arises from a lex-reflective subcategory in this way exactly when $E$ satisfies the 2-out-of-3 property, which makes it a reflective factorization system.

In simply trying to axiomatize “a reflective subcategory”, we were essentially forced to these stronger notions, because everything in the “internal logic” automatically happens “locally”, i.e. coherently in each slice. But in this post, I’m not going to go into the internal logic at all (so you can breathe a sigh of relief if you’re getting tired of all that stuff). Rather, I want to know more about reflective subfibrations and stable factorization systems categorically, which seem like rather unfamiliar beasts compared to lex-reflective subcategories. Specifically, I want to address the question: when does a reflective subcategory underlie some reflective subfibration or stable factorization system?

There’s a plugin that lets you write TeX in your email if you bow down and accept that Google now controls the world:

• GmailTeX.

I haven’t tried it yet. Have you?

Below are various recent job offers in maths that colleagues are asking me to circulate. One in Liverpool on motives. One in Prague, on geometry and algebra. One in Erlangen, on higher categories and TQFT. One in Adelaide, on differential geometry.