## February 26, 2008

### What I learned from Urs

#### Posted by Urs Schreiber

Guest post by Bruce Bartlett

I’m sure you’ll agree with me that there is a remarkable person on this blog : Urs. The rate at which he produces new posts and deep ideas is nothing short of a phenomenon. Indeed, he is so fast that perhaps many of you are like me and have been left in the dust long ago!

If so, this post is for you! I was lucky enough to have Urs visit me recently, and after much patience on his part I think I am finally beginning to see the first glimmers of daylight. Let me mention some of the things he explained to me; perhaps it will help some of you to understand what Urs has been going on about.

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 $L^\vee$:

(1)$CE (L) = \mathbb{R} \oplus \big[L_0^\vee\big] \oplus \big[\Lambda^2 L_0^\vee \oplus L_0^\vee\big] \oplus \big[\Lambda^3 L_0^\vee\oplus (L_0^\vee \otimes L_1^\vee) \oplus L_2^\vee\big] \oplus \cdots$

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

(2)$d^2 = 0.$

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:

(3)$Smooth spaces \leftrightarrow differential graded commutative algebras$

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

(4)$\Omega(X) = \Hom_{Smooth spaces} (X, \Omega).$

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

(5)$X_A (U) = \Hom_{DGCAs} (A, \Omega(U))$

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 $G$!

(6)$G = \Pi_1 (X_{CE(\mathfrak{g})})$

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 $X$!

(7)$post(\nabla) : X^\Sigma \rightarrow T^\Sigma$

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,

(8)$Z(M) = \int_{all fields} \mathcal{D}A exp(-S[A]).$

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:

(9)$Z : nCob \rightarrow nVect$

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.

Posted at February 26, 2008 9:20 AM UTC

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### Re: What I learned from Urs

I think I can see an important step on the way to solving John’s problem.

Posted by: David Corfield on February 26, 2008 11:26 AM | Permalink | Reply to this

### Re: What I learned from Urs

Well done, Bruce!
I also like the emphasis on trangression, which is one of the (if not the main)ingredients of the Chern-Simons business (sorry for the involuntary pun ;))

Posted by: Alessandro on February 26, 2008 11:40 AM | Permalink | Reply to this

### Re: What I learned from Urs

Thanks Alessandro! Sigh. Sadly, some of us poor souls on the café, like Alessando and I, have to write up our thesis! This gives us a foul disadvantage (in terms of keeping up with Urs) compared with some of the lecturers and researchers who hang around in these parts, who have nothing to do all day but sip pina-coladas and watch the undergrads walk by! ;-)

Posted by: Bruce Bartlett on February 26, 2008 12:01 PM | Permalink | Reply to this

### Re: What I learned from Urs

Speedy Gonzales was a good choice. But I also think of the Road Runner who, just when you think you might have almost caught up to him, suddenly kicks into hyper-speed, setting aflame the roads, and leaving one in the dust, with one’s mouth dropped to the ground in complete disbelief.

I’m going to have to study it some more, but already I can see this is a terrific post, Bruce. Thanks!

Posted by: Todd Trimble on February 26, 2008 12:15 PM | Permalink | Reply to this

### Re: What I learned from Urs

Sometimes when I look at the Speedy Gonzales picture,

the thought occurs to me that instead of representing Urs, Speedy really represents those of us at the café who, on the days when our mental biorhythms are at a shamefully low ebb, take one look at Urs’ posts, feel terribly intimidated, and run like hell! Lol… it’s gonna blow!

Todd, here’s a question. It would be nice if those functors going between smooth spaces and DGCA’s really are an adjunction… can you remind me how it works? The one unit (or co-unit, can’t remember) is canonical, but it seems the other one requires a gluing argument. My understanding is that you and Urs have dealt with this question before, but I can’t seem to find that comment.

Posted by: Bruce Bartlett on February 26, 2008 1:43 PM | Permalink | Reply to this

### Re: What I learned from Urs

but it seems the other one requires a gluing argument

When you say “gluing”, it sounds like you mean that smooth spaces should be sheaves of some sort (as opposed to just presheaves) – is that what you meant?

$Set^{S^{op}} \stackrel{\leftarrow}{\to} DGCA^{op}$

where on the left is just plain old presheaves. This adjunction comes about via one of these ambimorphic thingies, which Urs calls $\Omega^\bullet$, which is a presheaf valued in DGCA’s. It just means there is a natural bijection

$\frac{A \to Hom_{Set^{S^{op}}}(X, \Omega^\bullet)}{X \to Hom_{DGCA}(A, \Omega^\bullet)}$

where the morphisms above the bar live in DGCA, and the morphisms below the bar live in $Set^{S^{op}}$.

I haven’t tracked down the post where I proved this. Basically it boiled down to the fact that every presheaf is a colimit of representables, and so it suffices to establish the bijection in the case where $X$ is a representable $hom(-, U)$. In the representable case it basically follows from the Yoneda lemma. (So that argument was soft and said nothing about gluing.) I can give more detail if you want.

Posted by: Todd Trimble on February 27, 2008 5:58 PM | Permalink | Reply to this

### Re: What I learned from Urs

I haven’t tracked down the post where I proved this

You gave the statement, and the concept of its proof, here. I had reproduced that then in the second half of the entry Transgression (to be found there when scrolling down to the boldface “Todd’s first comment”).

I can give more detail if you want.

I would volunteer to write up a nice detailed LaTeXified note on this, following your explanations.

By the way, the closely related adjunction between suitable qDGCAs and simplicial spaces is in most every book on rational homotopy theory (let me look up precise page and verse…).

Posted by: Urs Schreiber on February 27, 2008 6:30 PM | Permalink | Reply to this

### Re: What I learned from Urs

Bruce,
Many thanks! Speedy Gonzales indeed! Can you imagine what it’s liek to try to write a paper *with* Urs?
In your displayed formuala, one of the L_0 shoudl be an L_1 - I’ll let the readers find it on their own.
Living life in terms of DGCAs is also known as rational homotopy theory. ALL the differential homological algebra involved is just that - smoothness comes in only when you want to integrate’or want to feel you are doing geometry’.
Your picture of a 2-hole torus aka bretzel surface is meant only symbolically? i.e. not the integration of the formula given?
Though there is a correspondence between connection and transport, I think it is important to maitain the distinction.
You finessed nicely the issue of which way the transgression goes in terms of forms on a bundle, but then the Urs variant of the virus bit you in the context of X^\Sigma. Physics and math suffered for decades by having different language for the same objects. I hope new terminology doesn’t create a new divide.
Still many many thanks. I even gained one new insight into the Road Runners’ mind.

Posted by: jim stasheff on February 26, 2008 1:45 PM | Permalink | Reply to this

### Re: What I learned from Urs

Can you imagine what it’s like to try to write a paper with Urs?

I am hoping that one day Bruce won’t have to just imagine it but will actually experience it. I am sure if I’d be staying in Sheffield with him it would have already happened.

Well, and if there weren’t a thesis to be written on his part, demanding priority.

There is a golden bridge I am trying to build here (this has actually been one of my motivations) from Bruce’s thesis work on finite group Dijkgraaf-Witten theories to full Chern-Simons theory:

even though he didn’t quite put it that way, Simon Willerton, Bruce’s advisor, in … via gerbes and finite groupoids conceives (following of course Dijkgraaf-Witten) the Chern-Simons 3-bundle on $B G$, for $G$ a finite group, as a pseudofunctor

$\mathbf{B} G \to \mathbf{B} B^2 U(1) \,,$

(where, as now usual, I write $\mathbf{B} G$ for the one-object $n$-groupoid coming from the $n$-group $G$, and $B^2 U(1)$ for the strict 3-group of doubly shifted $U(1)$)

namely as a group 3-cocycle.

Notice that as we let $G$ become a Lie group with Lie algebra $g$ and $\mu \in CE(g)$ the corresponding Lie algebra 3-cocycle, this pseudofunctor becomes the DGCA-morphism

$CE(g) \stackrel{\mu}{\leftarrow} CE(b^2 u(1))$

which appears in our $L_\infty$-model of the Chern-Simons 3-bundle on $B G$ (lower left of p. 90):

this morphism sends the single degree 3-generator of $CE(b^2 u(1))$ to the 3-cocycle $\mu = \langle \cdot, [\cdot, \cdot]\rangle : g \otimes g \otimes g \to \mathbb{R}$ itself, regarded as a (closed) elelement of $CE(g)$.

That’s the power of arrow theory:

a “background field” ($n$-transport) is a morphism

$\nabla : \mathbf{X} \to \mathbf{T}$

satisfying this and that, and that’s true in whichever context we are.

Next, Simon Willerton gave a formula for the transgression of the group 3-cocycle to a groupoid 2-cocycle on

$\Lambda G := Funct(\mathbf{B}\mathbb{Z}, \mathbf{B} G) \,.$

I am claiming that this transgressed 2-cocycle is nothing but the image of the pseudofunctor

$\mu : \mathbf{B} G \to \mathbf{B} B^2 U(1)$

under the internal hom (what Bruce addressed as “postcomposition” in his entry above), namely the pseudofunctor

$\array{ Funct(\mathbf{B}\mathbb{Z},\mathbf{B} G) &\stackrel{Funct(\mathbf{B}\mathbb{Z},\mu )}{\to}& Funct(\mathbf{B}\mathbb{Z},\mathbf{B} B^2 U(1)) \\ \uparrow^= && \downarrow^= \\ \Lambda G && \mathbf{B} B U(1) } \,.$

It’s a little tedious to check this, though, and I admit that I only checked it explicitly (together with Bruce, on a blackboard) for the transgression of 2-cocycles to 1-cocycles. But it has to be be true.

Then go back to p. 90, now the bottom right part, and see at work precisely that transgression at work there. Same arrow theory, once again. Bruce mentioned that more details of the consequences are being worked out here.

That’s why I am eager to see Bruce’s thesis in it’s finished form: I’ll take the entire thing, internalize from finite groupoids to DGCAs and get lots of cool information about Chern-Simons theory proper, just by crank-turning. :-)

Posted by: Urs Schreiber on February 26, 2008 9:36 PM | Permalink | Reply to this

### Re: What I learned from Urs

…and how hard it is to write a paper with Urs while working on your thesis? With 24-hour communication delays (small violins playing in the background)? Luckily the paper has been accepted, and I am now actually working on getting my degree. Which means I shouldn’t be visiting the Cafe so often…

If we could get funding and a fall-guy, I mean organiser, a Cafe-patron-conference would be an interesting experience.

Posted by: David Roberts on February 27, 2008 3:03 AM | Permalink | Reply to this

### Re: What I learned from Urs

David Roberts complained (with violins in the background)

…and how hard it is to write a paper with Urs while working on your thesis?

I very much enjoyed our article. Among other things, it is a nice example for the usefulness of the Café:

I had these notes on $INN(G_2)$ sitting here when I got this email from David Roberts saying something like: hey Urs, I looked at your notes, your diagrams really suck, I have rewritten that now… And a little later we had a nice article done (going considerably beyond my original notes, of course).

Posted by: Urs Schreiber on February 27, 2008 6:18 PM | Permalink | Reply to this

### Re: What I learned from Urs

Posted by: David Roberts on February 29, 2008 12:05 AM | Permalink | Reply to this
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