The n-Category Café

What has Urs been going on about?

1. Doing higher category theory infinitesimally. We are all n-category café patrons. We are interested in higher categories, right? We’re certainly interested in higher groupoids… lest Grothendieck should turn over in his um… bed, and curse the lot of us.

How do we capture the data of a weak higher groupoid? Well, we go and listen to John’s talk, for starters. Then we gesticulate and philosophize whimsically about coherence equations, pentagonators, homotopy theory and Kan complexes.

Okay… but when we stalk talking about smooth weak higher groupoids, there’s a beautiful simplifcation. Everything becomes easy!

Why? Because, fundamentally, a higher groupoid is about symmetry. To know about the one-morphisms, all you need to know about is what they look like ‘infinitesimally’ near the identity morphism of a single object… the rest can be deduced by integration. To know about the 2-morphisms, all you need to know about is what they look like in an infinitesimal neighbourhood of the identity 2-morphism on the identity 1-cell! The rest follows by symmetry and integration. And so on for the higher morphisms.

So that entire collection of 1-morphisms, 2-morphisms, 3-morphisms, …, and all the coherence diagrams that go with them, are highly redundant , and needlessly give us headaches. We should cast the data infinitesimally!

Here’s where a miracle occurs. It turns out that all that coherence data for weak higher groupoids - the compositors and the associators and the pentagonators and the ‘thousand natural shocks that flesh is heir to’ - together with all the coherence diagrams - wait for it… can be encoded in a single equation!

The infinitesimal data of a weak $\infty$-groupoid is called an $L_\infinity$ algebra (these were invented by Jim Stasheff, a fellow café patron!) : a graded vector space $L = L_0 \oplus L_1 \oplus L_2 \oplus \cdots$ (where each $L_i$ is thought of as the ‘infinitesimal morphisms emanating from the identity $i$-morphism’), together with a whole bunch of Lie brackets. But it’s nicer to dualize, and set $L^\vee$ to be the graded vector space with $L^\vee_{n-1} = \Hom(L_{n-1}, \mathbb{R})$ in degree $n$, and then define the Chevalley-Eilenberg algebra of $L$ to be the symmetric algebra on $L^\vee$:

The transposes of all the Lie brackets assimilate into a map $d : CE(L)_\cdot \rightarrow CE(L)_{\cdot + 1}$, and all the coherence equations between the Lie brackets translate into the single equation

These kinds of structures are called differential graded commutative algebras - and this is where Urs finds his playground. The great gain is that the world of weak higher groupoids has, at a stroke, suddenly become well-defined, simple, and amenable for concrete computations.

2. Studying spaces via the algebra of differential forms. Define a smooth space $X$ as a sheaf on the site $S$ of open subsets of Euclidean space. So for each open $U \subset \mathbb{R}^n$, we have a set $X(U)$, which we think of as the smooth functions from $U$ into a mythical manifold “$X$”, natural with respect to inclusion of open sets, and gluing together appropriately.

Just as in the world of algebraic geometry there is an adjunction between the category of schemes and the category of commutative rings, in smooth geometry (the world of calculus) there appears to be an adjunction between the category of smooth spaces and the category of graded-commutative algebras equipped with a differential:

Given a smooth space $X$, we send it to its algebra $\Omega(X)$ of differential forms,

Here $\Omega$ is the ‘differential forms smooth space’ : it sends an open set $U$ of Euclidean space to the differential forms on $U$, $U \mapsto \Omega(U)$.

Given a differential graded commutative algebra $A$, we can form a smooth space $X_A$ via the formula

This technology gives us a way to think of an element of a differential graded commutative algebra as a differential form on a smooth space . That’s very powerful!

For instance, what is the smooth space corresponding to the Chevalley-Eilenberg algebra of a Lie algebra $\mathfrak{g}$? Well, it is a magical smooth space $X_{\mathfrak{g}}$ with the property that its smooth fundamental groupoid recovers the Lie group $G$!

This is the magic of smooth spaces: in one step we have integrated a Lie algebra to a Lie group!

3. Integration by the avoidance of differentiation. In the mathematical physics Urs and I play around with, there’s this important but strange concept called ‘transgression’. It’s usually formulated as a certain procedure which you apply to differential forms, involving integration and ‘the pullback of the evaluation map’.

But higher-category minded people like us always like to think globally. What is an $n$-form? It’s the infinitesimal data of an $n$-functor! What is a vector-bundle-with-connection on a manifold? It is a functor from the path groupoid to to vector spaces!

Urs and his coworker Konrad Waldorf have made this ‘$n$-functor’ viewpoint of differential forms precise. They deal with $n=1$ in this paper, and with $n=2$ in this one. This is the ‘higher gauge theory’ program which John Baez is a founder of, and it all works out beautifully!

So… we have a way of formulating differential form type data globally as $n$-functors. One of Urs’ main points is that in this global ‘functorial’ picture, transgression is simply post-composition!

That is, suppose we have a space $X$, and some kind of ‘background field’ $n$-functor $\nabla : X \rightarrow T$ where $T$ represents the higher category of ‘fibers’ for the bundle (by the way, in this formula ‘$X$’ is really a shorthand for the $n$-path groupoid of $X$). Just think of $\nabla$ as a geometric structure living on the space $X$, like a vector bundle with connection.

Now suppose we have a space $\Sigma$. Then, simply by postcomposing with $\nabla$ , we automatically get a geometric structure living on the space of maps from $\Sigma$ into $X$!

This is the global version of transgression. Urs and Konrad have shown in many speacial cases that if you differentiate this transport functor, you precisely recover the transgression formula which is formulated in terms of differential forms, and integration and evaluation and all that.

Urs calls this integration without integration… but perhaps it could also be called integration by the avoidance of differentiation!

4. Extended TQFT is about a single geometric structure. Before I met Urs, I thought quantum field theory was about the mysterious path integral,

The ingredient in this viewpoint is the action… it’s something which assigns a number to a field.

Then I read Dan Freed’s Quantum groups from path integrals and found out that in an $n$-dimensional quantum field theory, there is a space of fields for every manifold with $\dim M \leq n$. Moreover, there is an ‘action’ for every dimension… it eats up a field on $M$ and spits out a $(n-\dimM - 1)$-category! So at the top dimension, the action spits out a number, but at lower dimensions, it spits out higher and higher categories.

But then I met Urs. He explained to me (a viewpoint that Brylinski also had) that instead of thinking about a whole tower of actions, one for each space of fields on each manifold, it is better to understand all these actions as arising as the transgression of a single fundamental geometric structure .

In Chern-Simons theory, this geometric structure is a ‘2-gerbe over $B G$’. The entire classical field theory (in the sense of Freed) can be derived via ‘pulling back’ this fundamental geometric structure to the relevant mapping spaces.

So far, so good.

But Urs goes further: he believes that there is an elementary higher categorical procedure which will take this fundamental geometric structure and, in one fell swoop, produce the entire extended TQFT n-functor!

Remember, the game of ‘extended’ TQFT is to try and make precise (and find examples of) higher representations of higher cobordism categories:

The ultimate goal of Urs’ programme is to show that one can construct $Z$ from these primeval geometric structures (like the ‘2-gerbe on $BG$’) by some elementary abstract nonsense.

To do this, Urs, Jim Stasheff and Hisham Sati have developed a formalism for being able to perform concrete higher-categorical computations… solely in the language of differential graded commutative algebras! Moreover, Urs has extended this formalism to the point where he can ‘take sections’ of the resulting geometric structures, and compare the results to those obtained by other (more ad-hoc) approaches… and they agree! This has really got me excited about his research program.