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September 24, 2007

Obstructions, Tangent Categories and Lie N-tegration

Posted by Urs Schreiber

Last week I visited Yale, where I talked with Hisham Sati about supergravity, Chern-Simons 3-bundles and other higher structures in String theory, with Mikhail Kapranov about parallel surface transport, with Todd Trimble about Hecke algebras, groupoidification and the sociology of quantum gravity, and with Anton Zeitlin about BV formulation of Yang-Mills theory. With a short stop at home I went from there straight to Croatia, where I am currently attending Categories in Geometry and Physics.

The first day here was quite remarkable, with an astonishing amount of Croatian media attention for us followed by a couple of rather impressive talks. After Pavol Ševera’s talk I had the chance to chat with with him about differentiation and integration of Lie nn-algebras, much to my delight.

Last night at about 3 am I learned that I am supposed to give a public talk to high school kids and journalists next Friday. And sure enough, later towards the pm part of the day I found myself being interviewed by some TV team on how I plan to convey the ideas of Categories in Geometry and Physics to the broad public.

Trying to think of something which is both interesting and relevant, as well as easy to understand and expressible in terms of lots of pictures, I thought I’d give a talk revolving around the Eckman-Hilton argument, illustrated by two party-balloons joined at their tip and labeled “A” and “B”.

(The very moment that I am posting this, a journalist sends me the following two questions:

Q1. Your field of interest in mathematics is the relation between categories and quantum physics. During the Split categories conference you will be giving a popular lecture on this topic to the high school students, so can you briefly summarize that lecture for our readers?

Q2. You are also one of the admins at nn-category cafe blog. As it has become an interesting and influential internet based communication and discussion tool for resarchers in category theory, can you tell me a bit more about it - how did the idea start, how it developed, and what is its influence on the category maths community professional work?

)

Hence there’d be many fun things to write about here. And I should. But right now I will instead talk more about two of the issues I had mentioned recently in Obstructions for nn-bundle lifts and related.

I’ll say a couple of things that need to be said. But even more needs to be said. Some of which can already be found in the incomplete

String and Chern-Simons nn-transport.

The big obstruction diagram presented below exists as a nice big hand-drawn picture which includes a copy of the diagram that spells out the explicit meaning of each and every arrow in full concrete detail. I am hoping to be able to provide an electronic copy of that in a while.

Curvature of parallel nn-transport

For G (n)G_{(n)} a Lie nn-group, XX a smooth space and Π (n+1)(X)\Pi_{(n+1)}(X) the fundamental (n+1)(n+1)-groupoid of XX, a (general, non-fake flat) G (n)G_{(n)}-nn-bundle with connection is a pullback diagram of the suspended universal G (n)G_{(n)}-nn-bundle

TΣINN(G (n))ΣG (n) \array{ T \leftarrow \Sigma \mathrm{INN}(G_{(n)}) \leftarrow \Sigma G_{(n)} }

(where TT should have a universal realization but is for the moment part of the concrete data)

to

Π (n)(X) Π (n+1)(P) ΣG (n) (tra,curv) T ΣINN(G (n)) ΣG (n), \array{ \Pi_{(n)}(X) &\leftarrow& \Pi_{(n+1)}(P) &\leftarrow& \Sigma G_{(n)} \\ \downarrow &\Downarrow& \downarrow^{(\mathrm{tra},\mathrm{curv})} &\Downarrow& \downarrow \\ T &\leftarrow& \Sigma \mathrm{INN}(G_{(n)}) &\leftarrow& \Sigma G_{(n)} } \,,

with everything demanded to be smooth, in a straightforward generalization of the magic square for fake-flat parallel nn-transport.

Here the leftmost functor is the “classifying map”, the one in the middle is the parallel nn-transport with its (n+1)(n+1)-curvature and the rightmost one is the G (n)G_{(n)}-nn-cocycle corresponding to this differential G (n)G_{(n)}-nn-cocycle.

Mapping cones and tangent categories

Recall from the discussion at The inner automorphism group of a strict 2-group that the mapping cone of the identity 2-functor CCC \to C, in the world of strict 2-groupoids, amounts to forming what I called the “tangent categoryTCT C, which is obtained by mapping the interval \bullet \stackrel{\sim}{\to} \circ into CC and admitting only those homotopies between such maps which fix the left endpoint.

Now, you won’t be surprised to learn that the mapping cone of an arbitrary injection t:C 0C 1 t : C_0 \hookrightarrow C_1 turns out to correspond to maps of the interval into C 1C_1, whose homotopies are resticted to fix the left endpoint and to have a right endpoint transerve a 2-path in the image of tt. But I mention it nevertheless. Since I think it is true and useful.


Obstructions of gg-bundles through String-like extensions

Given a Lie n+1n+1-cocycle μ\mu on a Lie algebra gg, we obtain an extension

Lie(Σ n1U(1))g μg \mathrm{Lie}(\Sigma^{n-1}U(1)) \to g_\mu \to g

of gg by a (n1)(n-1)-fold shifted Lie(U(1))\mathrm{Lie}(U(1)) to a Lie nn-algebra of Baez-Crans type. For μ\mu the 3-cocycle on a semisimple Lie algebra, this is the extension which gives the (skeletal version of the) String Lie 2-algebra.

Whenever (see String and Chern-Simons Lie nn-algebras) the cocycle μ\mu here is related by transgression to an invariant polynomial kk, the Lie nn-algebra g μg_\mu sits in a weakly exact sequence

g μ(cs k(g)inn(g μ))ch k(g) g_\mu \to (\mathrm{cs}_k(g) \simeq \mathrm{inn}(g_\mu)) \to ch_k(g)

of Lie (n+1)(n+1)-algebras.

Noticing that for any Lie nn-algebra g (n)g_{(n)} the sequence

g (n)inn(g)(bg (n)H (BG)) g_{(n)} \to \mathrm{inn}(g) \to (b g_{(n)} \simeq H^\bullet(BG)) is the Lie version of the universal g (n)g_{(n)}-nn-bundle we obtain in fact a sequence of sequences

Lie(Σ nU(1)) inn(Lie(Σ n1U(1))) Lie(Σ n1U(1)) ch k(g) cs k(g)inn(g) g μ k bg inn(g) g \array{ \mathrm{Lie}(\Sigma^{n} U(1)) &\leftarrow& \mathrm{inn}(\mathrm{Lie}(\Sigma^{n-1} U(1))) &\leftarrow& \mathrm{Lie}(\Sigma^{n-1} U(1)) \\ \downarrow && \downarrow && \downarrow \\ \mathrm{ch}_k(g) &\leftarrow& \mathrm{cs}_k(g) \simeq \mathrm{inn}(g) &\leftarrow& g_{\mu_k} \\ \downarrow && \downarrow && \downarrow \\ b g &\leftarrow& \mathrm{inn}(g) &\leftarrow& g }

describing how the universal g μg_{\mu}-nn-bundle is the extension by the universal Σ n1U(1)\Sigma^{n-1}U(1)-bundle of the universal gg-bundle.

Any concrete gg-bundle with a classifying map

TXbg (n) T X \to b g_{(n)}

and differential parallel nn-transport

TPinn(g) T P \to \mathrm{inn}(g)

completed to a diagram

T li(|G (n)|) g TP (A,F A) inn(g) TX bg (n) \array{ T_{li} (|G_{(n)}|) &\stackrel{\simeq}{\to}& g \\ \downarrow &\Downarrow& \downarrow \\ T P &\stackrel{(A,F_A)}{\to}& \mathrm{inn}(g) \\ \downarrow &\Downarrow& \downarrow \\ T X &\to& b g_{(n)} }

(which should be a pullback, but I need to clarify this. The 2-morphisms here have to respect the sequence property of the left and right vertical composites, meaning that they need to vanish after the corresponding whiskering. The space |G (n)||G_{(n)}| is discussed below.)

We are asking under which conditions this diagram factors through the String extension

Lie(Σ n1U(1))g μg. \mathrm{Lie}(\Sigma^{n-1}U(1)) \to g_\mu \to g \,.

To check this we first puff up gg as the mapping cone

g(Lie(Σ n1U(1))g μ) g \simeq (\mathrm{Lie}(\Sigma^{n-1}U(1)) \to g_\mu)

then look at the embedding of g μg_\mu into this

i:g μ(Lie(Σ n1U(1))g μ) i : g_\mu \hookrightarrow (\mathrm{Lie}(\Sigma^{n-1}U(1)) \to g_\mu)

and consider the pushforward along the strict cokernel

g μi(Lie(Σ n1U(1))g μ)(coker(i)=Lie(Σ nU(1)). g_\mu \stackrel{i}{\hookrightarrow} (\mathrm{Lie}(\Sigma^{n-1}U(1)) \to g_\mu) \to (\mathrm{coker}(i) = \mathrm{Lie}(\Sigma^n U(1)) \,.

Doing this to the entire sequence of universal nn-bundles above

Lie(Σ nU(1)) inn(Lie(Σ n1U(1))) Lie(Σ n1U(1)) ch k(g) cs k(g)inn(g) g μ k bg inn(g) g inn(Lie(Σ n1U(1))g μ) (Lie(Σ n1U(1))g μ) Lie(Σ n+1U(1)) inn(Lie(Σ nU(1))) Lie(Σ nU(1)) \array{ \mathrm{Lie}(\Sigma^{n} U(1)) &\leftarrow& \mathrm{inn}(\mathrm{Lie}(\Sigma^{n-1} U(1))) &\leftarrow& \mathrm{Lie}(\Sigma^{n-1} U(1)) \\ \downarrow && \downarrow && \downarrow \\ \mathrm{ch}_k(g) &\leftarrow& \mathrm{cs}_k(g) \simeq \mathrm{inn}(g) &\leftarrow& g_{\mu_k} \\ \downarrow && \downarrow && \downarrow \\ b g &\leftarrow& \mathrm{inn}(g) &\leftarrow& g \\ \downarrow && \downarrow && \downarrow \\ &\leftarrow& \mathrm{inn}(\mathrm{Lie}(\Sigma^{n-1}U(1)) \to g_\mu) &\leftarrow& (\mathrm{Lie}(\Sigma^{n-1}U(1)) \to g_\mu) \\ \downarrow && \downarrow && \downarrow \\ \mathrm{Lie}(\Sigma^{n+1} U(1)) &\leftarrow& \mathrm{inn}(\mathrm{Lie}(\Sigma^{n} U(1))) &\leftarrow& \mathrm{Lie}(\Sigma^{n} U(1)) }

and then composing with the pullback

T li(|G (n)|) g TP inn(g) TX bg (n) \array{ T_{li} (|G_{(n)}|) &\to& g \\ \downarrow &\Downarrow& \downarrow \\ T P &\to& \mathrm{inn}(g) \\ \downarrow &\Downarrow& \downarrow \\ T X &\to& b g_{(n)} }

yields the obstructing Σ nU(1)\Sigma^{n} U(1)-(n+1)(n+1)-bundle

TX TP g (A,F A) bg inn(g) g inn(Lie(Σ n1U(1))g μ) (Lie(Σ n1U(1))g μ) Lie(Σ n+1U(1)) inn(Lie(Σ nU(1))) Lie(Σ nU(1)). \array{ T X &\leftarrow& T P &\leftarrow& g \\ \downarrow && \downarrow^{(A,F_A)} && \downarrow \\ b g &\leftarrow& \mathrm{inn}(g) &\leftarrow& g \\ \downarrow && \downarrow && \downarrow \\ &\leftarrow& \mathrm{inn}(\mathrm{Lie}(\Sigma^{n-1}U(1)) \to g_\mu) &\leftarrow& (\mathrm{Lie}(\Sigma^{n-1}U(1)) \to g_\mu) \\ \downarrow && \downarrow && \downarrow \\ \mathrm{Lie}(\Sigma^{n+1} U(1)) &\leftarrow& \mathrm{inn}(\mathrm{Lie}(\Sigma^{n} U(1))) &\leftarrow& \mathrm{Lie}(\Sigma^{n} U(1)) } \,.

All this can be spelled out very explicitly and one finds that it indeed correctly describes how the obstruction is the (Σ nU(1)\Sigma^n U(1)-(n+1n+1)-bundle classified by) the characteristic class k(F A)k(F_A).

Integration of Lie nn-algebras to Lie nn-groupoids

For g (n)g_{(n)} any Lie nn-algebroid, let

BG (n):=(XHom nLie(TX,g (n))) B G_{(n)} := (X \mapsto \mathrm{Hom}_{n\mathrm{Lie}}(T X , g_{(n)}))

be the corresponding presheaf on smooth manifolds.

This is not quite a Chen-smooth structure, but it is useful to think of it as a slight generalization of Chen-smooth structures. We may work entirely inside smooth spaces realized by presheaves over smooth manifolds. So for now such a presheaf is a “smooth space” and an nn-category internal to such presheaves is a smooth nn-category, an nn-group internal to these presheaves a Lie nn-group.

Using this setup, the Lie nn-group(oid) integrating g (n)g_{(n)} should be

G (n):=Π n(BG (n)), G_{(n)} := \Pi_n( B G_{(n)}) \,,

where, recall, BG (n)B G_{(n)} is the smooth space from above, and where Π n()\Pi_n(\cdot) forms the smooth internal fundamental nn-groupoid.

I am proposing that this is the right way to think of the process sketched at the beginning of Pavol Ševera’s Some title containing the words “homotopy” and “symplectic”, e.g. this one, which we had discussed here recently

The fun thing is that with this definition one easily finds that there is canonically pp-form data on BG (n)B G_{(n)} which plays the role of left-invariant pp-forms and obeys the differential graded algebra rulese that define g (n)g_{(n)} in the Koszul dual picture.

This quasi-tautological (but also quasi-tautological things need to be figured out first…) construction is pretty useful. It gives for instance a 1-line proof that String k(G)\mathrm{String}_k(G) is something like 3-connected: BG (n)B G_{(n)} tautologically carries a flat gg-valued 1-form such that A[AA]\langle A \wedge [A \wedge A]\rangle is exact. This should say that its third cohomology is trivial.

Posted at September 24, 2007 8:04 PM UTC

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8 Comments & 7 Trackbacks

Re: Obstructions, tangent categories and Lie n-tegration

Have fun in Split! I hope your high-school talk gets all those kids excited about nn-categories!

Regarding this talk: when introducing people to the Eckmann–Hilton argument, I like to emphasize that this argument is really how we prove to kids that addition is commutative. Take 3 rocks and set them on the table next to 2 rocks:

\bullet \bullet \bullet \quad \bullet \bullet

This is 3+23+2 rocks, by definition.

Now, slide the 3 rocks around the 2 rocks, so they trade places:

\bullet \bullet \quad \bullet \bullet \bullet

Now we have 2+32+3 rocks, by definition. We have the same number of rocks — more precisely, the process of sliding the rocks around is an isomorphism of sets. Therefore, we have an equation

3+2=2+33 + 2 = 2 + 3

or more precisely, a braiding isomorphism

B 3,2:3+22+3B_{3,2} : 3 + 2 \stackrel{\sim}{\to} 2 + 3

The fun thing is that with the Eckmann–Hilton argument, this proof becomes completely rigorous! The process of carrying out the proof becomes a braid. And, we see why we need at least 2 dimensions of space to carry out this argument!

If you’ve ever read Flatland, you may remember the scene in ‘Lineland’, where space is 1-dimensional and everyone is stuck talking to their two neighbors: their neighbor to the left and their neighbor to the right. There’s not enough room for them to trade places. For such people, the commutative law might seem counterintuitive — something only high-powered mathematicians could understand.

Posted by: John Baez on September 25, 2007 4:24 AM | Permalink | Reply to this

Re: Obstructions, tangent categories and Lie n-tegration

Thanks for the rock trick.

Do you have any advice for the counterexample: I was thinking that it might be fun to have some simple kind of nonabelian group and point out how that allows us to label 1-dimensional processes, but not 2-dimensional ones.

Igor cautioned me that my audience might not even know matrices. So I need to be careful.

Maybe I should take these rocks, and then color them. Then the permutation group acts on them nontrivially and non-commutatively. Nobody needs to have a general concept of group to get that.

So, ideally, I would like to motivate everything from the desire to describe physicsl processes. Here a system consisting of five rocks, ordered and aligned in a row. In each time step we permute them. Say, because we are the god of the world of rocks and that’s the way we intend to enforce dynamics on our little world.

So there should be a nice intuitive way to understand what parallel transport Π 1(S 1)ΣS n \Pi_1(S^1) \to \Sigma S_n is.

Maybe I could mumble something about “being” and “becoming” and about “state” and “process”. Or maybe that’s too much philosophical baggage.

In any case, I thought it might be good to first run into a contradiction by assuming that our permutations still work nicely when we start performing processes in two dimensions.

Then I’d say: okay, but we can fix this by uncoloring those rocks.

I am also imagining showing two rectangles, AA and BB, next to each other, and pointing out how that forbids passing them around each other. But as we shrink the sides of these rectangles to points, while keeping their interior, thus forming balloons joined at a single point (namely 2-morphisms in a one-object one-morphisms 2-category) we do obtain the means to pass AA from one side of BB to the other.

And then I end by pointing out that this is the reason why people understood the spinning particle for a long time, while the superstring requires more care. Or maybe I won’t say that… ;-)

Posted by: Urs Schreiber on September 25, 2007 9:24 AM | Permalink | Reply to this

Re: Obstructions, tangent categories and Lie n-tegration

Tom Leinster, who is here in Split, too, further helped me come up with a decent idea for what to say in that public talk.

He reminded me of the pedagogical move of choice to explain noncommutativity to the layman:

It makes a different whether you first put on your socks and then your shoes, or the other way around. It makes a difference whether you first open the window and then stick out your head, or the other way round.

(In the second case, doing it the other way round is not even invertible anymore! So maybe its even a good way to understand the difference between a monoid and a group.)

I’ll try to compile an outline of how to proceed in a talk and discuss it over breakfast here with my fellow conference participants.

So here is an idea for how to proceed:

– write to the blackboard

3+2=2+3 3 + 2 = 2 + 3

say something like “You all understand this equation. Or you think you do. Let’s look at it mode closely”.

– to wit, it ain’t always like that: think of shoes and socks and of heads and windows, etc.: these are noncommutative operations.

– noncommutative operations give rise to 1-dimensional processes:

in the morning, you first get up. Second you put your socks on. Third your shoes. Fourth you open the window, fifth you stick your head out. sixth you decide it’s a horrible day to go to work. seventh you go back to bed. Eighth you realize that you still have your shoes on. And so on.

I should draw a picture at this point

getupsocksonshoesonwindowetc. \stackrel{get up}{\to} \stackrel{socks on}{\to} \stackrel{shoes on}{\to} \stackrel{window}{\to} \stackrel{etc.}{\to}

– for process which do commute there is more of a choice of in which order to do it

makecoffee getnewspaper getnewspaper makecoffee \array{ &\stackrel{make coffee}{\to}& \\ \;\;\;\;\;\downarrow^{get newspaper} && \;\;\;\;\;\downarrow^{get newspaper} \\ &\stackrel{make coffee}{\to}& }

so we get a 2-dimensional process

– to make this 2-dimensionality more vivid, now John’s example for how commutativity of addition comes from moving rocks around in two dimensions

– at some point then I’d want to draw some bigons

+3 \array{ & \nearrow \searrow \\ \bullet & \Downarrow^{+ 3}& \bullet \\ & \searrow \nearrow }

and explain that they can be composed in two directions. And finally the Eckmann-Hilton argument.

Then the party balloons to highlight that this makes use of the special properties of disks whose boundary is constant.

Or something like that. I’ll further think about it. All comments are welcome.

Posted by: Urs Schreiber on September 25, 2007 12:28 PM | Permalink | Reply to this

n-tegration

I have time to say more about the way to look at integration of Lie nn-algebras which I mentioned. The following gives a couple of more details concerning the remark I made in the above entry and then ends with a question.

I am hoping that with the right way to think of the Lie nn-group G (n)G_{(n)} integrating a given Lie nn-algebra g (n)g_{(n)} the statement of local parallel transport which establishes an isomorphism between smooth functors

tra:Π n(X)ΣG (n) \mathrm{tra} : \Pi_n(X) \to \Sigma G_{(n)}

and certain differential form data on XX becomes a tautological statement.

So let g (n)g_{(n)} be a given Lie nn-algebra regarded as a one-object Lie nn-algebroid.

Let I nI^n be the nn-dimensional standard cube.

As I argued above, I think we should think of the smooth space of smooth nn-paths in the smooth space “BG (n)B G_{(n)}” as the presheaf on manifolds which I denote

P n(BG (n)) P_n(B G_{(n)})

and which is defined to send each smooth manifold UU to the set of smooth Lie nn-algebroid maps from T(U×I n)T (U \times I^n) to g (n)g_{(n)}:

P n(BG (n)):UHom nLie(T(U×I n),g (n)). P_n(B G_{(n)}) : U \mapsto \mathrm{Hom}_{n\mathrm{Lie}}(T (U \times I^n), g_{(n)}) \,.

By the usual Yoga, this is nothing but the collection of differential form data (A,B,C,)(A,B,C,\cdots) on U×I nU \times I^n which models the differential graded commutative algebra Koszul dual to g (n)g_{(n)}.

For instance, for g (n)=g (1)=gg_{(n)} = g_{(1)} = g an ordinary Lie algebra, we simply have

P 1(BG (1)):U{AΩ 1(U×I,g),F A=0}. P_1(B G_{(1)}) : U \mapsto \{ A \in \Omega^1(U\times I, g), F_A = 0 \} \,.

Now, a smooth local nn-transport on some smooth space XX with values in G (n)G_{(n)} is in particular a smooth map from the space of nn-paths P n(X)P_n(X) of XX to P n(BG (n)) P_n(B G_{(n)}) .

Now, the smooth space of nn-paths on XX is simply the sheaf

P n(X):UHom smooth(U×I n,X). P_n(X) : U \mapsto \mathrm{Hom}_{\mathrm{smooth}}(U \times I^n, X) \,.

Hence a smooth parallel nn-transport

tra:P n(X)P n(BG (n)) \mathrm{tra} : P_n(X) \to P_n(B G_{(n)})

is in particular a morphism of sheaves:

U(tra U:Hom smooth(U×I n,X)Hom nLie(T(U×I n),g (n))). U \mapsto ( \mathrm{tra}_U : \mathrm{Hom}_{\mathrm{smooth}}(U \times I^n, X) \to \mathrm{Hom}_{n\mathrm{Lie}}(T (U \times I^n), g_{(n)}) ) \,.

The question is hence how to characterize all such morphisms of sheaves.

It is clear that a large class of such morphisms comes from postcomposition with a fixed morphism of Lie nn-algebroids from the tangent Lie algebroid of XX

f:TXg (n). f : T X \to g_{(n)} \,.

That’s good. Such a morphism is nothing but the corresponding differential form data globally on XX. For instance for our n=1n=1 example such morphisms are in bijection with flat gg-valued 1-forms on XX.

I want to obtain a statement that this already exhausts all possible morphisms P n(X)P n(BG (n)). P_n(X) \to P_n(B G_{(n)}) \,.

Currently I am notr sure if all morphisms of the sheaves on manifolds which I have here need to compe from morphisms ff as above.

On the other hand: those coming from the maps ff as above are precisely those which fit into the philosophy of Chen-smooth structures:

morphisms of Chen-smooth spaces (which are presheaves with special properties) are required by definition to come from postcomposition with a map.

In fact, I could replace equivalently the set of smooth maps of smooth manifolds

Hom smooth(U×I n,X) \mathrm{Hom}_{\mathrm{smooth}}(U \times I^n, X)

by the canonically isomorphic set of morphisms of the corresponding tangent Lie algebroids

Hom nLie(T(U×I n),TX). \mathrm{Hom}_{n\mathrm{Lie}}(T(U \times I^n), T X) \,.

Maybe then I want to pass to the obvious modification of Chen-smooth spaces where all occurences of ordinary manifolds and their smooth maps are replaced by the notion of their tangent Lie algebroids and their morphisms.

Doing so would tend to make the statement I am after a complete tautology. Which would be good. That’s what I am hoping it will be in the end. But there seems to be room for further tuning the way to say this best.

Posted by: Urs Schreiber on September 25, 2007 12:06 PM | Permalink | Reply to this

The first interview question

Q1. Your field of interest in mathematics is the relation between categories and quantum physics. During the Split categories conference you will be giving a popular lecture on this topic to the high school students, so can you briefly summarize that lecture for our readers?

We are all familiar with processes that take place in time. A bus takes us from here to there. The earth travels around the sun over the year. An electron travels through the vacuum tube of our TV set.

Such processes take place one after each other, forming a long chain of events. For a long time, physics was the study of such chains of processes occurring in nature.

Almost 300 years ago it was discovered that the right language to describe such processes are mathematical formulas. By cleverly manipulating strings of symbols, people were able to understand the causality of fundamental processes to an unprecedented accuracy.

We can compute to many decimal places the precise position of the earth in its orbit around our sun many billions of years ahead. And all of this just by manipulating strings of symbols. This success of the description of physical laws by mathematical formulas has led to a deep interaction and interrelation between mathematics and physics.

But remarkably, in the middle of the 20th century, physicists and mathematicians independently began trying to explore the possibility that linear chains of entities may not be the end of the story.

Physicists began speculating that the elementary particles, like the electrons in our TV sets, which appear to be just points that trace out 1-dimensional curves as they zip along, might maybe turn out to look like little loops when we look very, very closely. Such loops would trace out not 1-dimensional paths in space, but 2-dimensional surfaces.

Simple as it sounds, this speculation has led theoretical physicists to discover a whole new universe of ideas. And some of these ideas turned out to have no good description in terms of the kind of mathematics that has worked so well in physics for centuries.

But by a lucky coincidence of history, mathematicians had – completely independently – began formulating a kind of mathematical language which is to the one-dimensional formulas that we are taught in school like a 2-dimensional surface is to a line. Or like a 3-dimensional space. Or like something even higher dimensional.

The name of this language is category theory. Or nn-category theory, if one wants to emphasize its nn-dimensional nature.

As with the algebra which turned out to be so very useful for describing the movement of the planets, this piece of higher dimensional mathematics is very rich and interesting in its own right. But on top of that, it miraculously turned out to be precisely the right language to naturally describe the new ideas in theoretical physics.

And turns out. The study of interrelation between categories and nn-categories with physics of point particles and their speculated loop-like generalizations is in its infancy.

As with all great and deep ideas, that of nn-categories and the physics it describes has underlying it a couple of very beautiful and very simple ideas, understandable by every layman.

In my public talk I want to highlight some of these simple beautiful ideas in a way that requires no mathematical or physical education. Using simple but careful everyday reasoning, we will try to get a good conceptual understanding of what characterizes processes in nature and in mathematics, and what happens when we start passing from 1-dimensional chains of symbols for describing these to 2-dimensional pictures.

Everybody who knows that 3+23 + 2 equals 2+32 + 3 will thereby be led to discover a simple but profound result in nn-category theory – the so-called Eckmann-Hilton argument.

If time permits, I might be tempted to end the talk with entertaining the audience by making some remarks on how this is relevant for some of the modern ideas which pervaded theoretical high energy physics in the last couple of decades.

Posted by: Urs Schreiber on September 25, 2007 1:27 PM | Permalink | Reply to this

The second interview question

Q2. You are also one of the admins at nn-category cafe blog. As it has become an interesting and influential internet based communication and discussion tool for researchers in category theory, can you tell me a bit more about it - how did the idea start, how it developed, and what is its influence on the category maths community professional work?

Before the nn-category Café came into existence, the three hosts, John Baez, David Corfield and myself, had shared the interest in nn-categories and their role in math, physics and philosophy, and the interest in sharing and discussing these issues on the internet with people from all around the world – but what we didn’t share was a single site to suit those needs.

John Baez, who is not only the urblogger of both math and physics (or was he the überblogger?), but also the one who initiated thinking about applications of higher category theory to physics, had his own web-based column This week’s finds in mathematical physics since 1993. David Corfield and myself a while ago started using two blogs to communicate on the web (now dormant: Philosophy of Real Mathematics and The String Coffee Table, respectively) when we thought that it might make all of us happier if we had a single blog unifying these activities.

Jacques Distler kindly offered to provide the required technology and since then the nn-category café is running on his equipment.

That’s how it started. More details on this can be found in Two Café Owners Interviewed.

I am rather pleased with how the nn-category café developed. For me, it has become a primary source for intellectual and scientific exchange.

I believe there is evidence that the kind of discussion we and our numerous and esteemed visitors at the nn-category Café had and have helped demonstrate what a couple of years ago was not widely considered probable or even desireable: that useful and high-quality scientific exchange in public web-based sites is possible, useful and enjoyable and not un-comparable in relevance for the participants with other scientific exchange as on conferences and in seminars.

Lately the web has seen a considerable increase in the number of mathematical blogs concerned with serious scientific discussion and exchange. Prominent examples (without the aim of completeness) are Fields medalist Terence Tao’s blog, Fields medalist Tim Gower’s blog, Fields medalist Richerd Borcherds’ blog, The secret blogging seminar, John Armstrong’s blog and a couple more of them.

I think it is clear that a new medium for serious high-quality scientific communiation is being explored here.

On the other hand, I don’t feel in the position yet to speculate about what

its influence on the category maths community professional work

is in general.

But I can make a couple of observations:

I believe we have seen some young students become aware of the topics which we are discussing here first and maybe only due to this blog. From my own time as a student I remember how very difficult up to almost impossible it was, without the presence of senior researchers on the web, to get access to insider information about truly actual research, that which has not yet made it into textbooks. It seems that bridging a certain information barrier between students (and we all remain students in most of the topics that are out there for all our lives) and experienced researchers is one of the things serious scientific blogs can be very good for.

Then I think we have seen exchange of information among experts here, which is unlikely to have occured elsewhere than on the blog, if only because the persons involved would not have come into contact at all otherwise. I consider myself as having benefited immensely from information that was provided here, as a reaction to things I said on the blog or questions that I asked, by commenters. I have found collaborators on the blog which I would hardly have come into contact with otherwise, some of whom I have still never met in person (which however needs to change).

On conferences, I see a certain benefit in the fact that people who I have never met before tend to know what I am up to from this blog. This is helpful for getting into contact in person even if there was no online contact before.

I do imagine that this kind of effect the nn-category café has on me, personally, as a member of some community, might in the long run become an effect that scientific blogging activity has on the entire community. But I don’t feel to be in the position to speculate on that far beyond what applies to me personally.

With a little luck John Baez and David Corfield may find and take the time to say more on this issue.

Posted by: Urs Schreiber on September 25, 2007 2:49 PM | Permalink | Reply to this
Read the post An Invitation to Higher Dimensional Mathematics and Physics
Weblog: The n-Category Café
Excerpt: A public talk supposed to illustrate some of the basic reasoning underlying higher dimensional algebra.
Tracked: September 25, 2007 6:00 PM

Re: Obstructions, Tangent Categories and Lie N-tegration

Last night at about 3 am I learned that I am supposed to give a public talk to high school kids and journalists next Friday.

Good luck Urs! Looks like the Croatians don’t mess about when it comes to higher categories :-)

Posted by: Bruce Bartlett on September 26, 2007 10:18 PM | Permalink | Reply to this
Read the post Detecting Higher Order Necklaces
Weblog: The n-Category Café
Excerpt: Nils Baas on higher order structures,Enrico Vitale on weak cokernels and a speculation on weak Lie n-algebras triggered by discussion with Pavol Severa.
Tracked: September 28, 2007 12:44 AM
Read the post Cohomology of the String Lie 2-Algebra
Weblog: The n-Category Café
Excerpt: On the Lie 2-algebra cohomlogy of the String Lie 2-algebra and its relation to twisted K-theory.
Tracked: October 2, 2007 10:45 PM

Re: Obstructions, Tangent Categories and Lie N-tegration

Parts of the discussion of Chern-Simons-like obstructions to lifts of GG-bundles through String-like extension can now be found in the section

String- and Chern-Simons nn-Transport

in the set of slides of (almost) the same title: On String- and Chern-Simons nn-Transport.

Posted by: Urs Schreiber on October 9, 2007 7:19 PM | Permalink | Reply to this
Read the post On Lie N-tegration and Rational Homotopy Theory
Weblog: The n-Category Café
Excerpt: On the general ideal of integrating Lie n-algebras in the context of rational homotopy theory, and about Sullivan's old article on this issue in particular.
Tracked: October 20, 2007 4:33 PM
Read the post n-Bundle Obstructions for Bruce
Weblog: The n-Category Café
Excerpt: On the global description of n-bundles obstructing lifts through shifted central extensions.
Tracked: November 4, 2007 5:55 PM
Read the post Differential Forms and Smooth Spaces
Weblog: The n-Category Café
Excerpt: On turning differential graded-commutative algebras into smooth spaces, and interpreting these as classifying spaces.
Tracked: January 30, 2008 10:16 AM
Read the post Lie oo-Connections and their Application to String- and Chern-Simons n-Transport
Weblog: The n-Category Café
Excerpt: A discussion of connections for general L-infinity algebras and their application to String- and Chern-Simons n-transport.
Tracked: January 30, 2008 11:29 AM

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