The n-Category Café

Thanks! Fixed!

I should add that while this blog entry is rather unfriendly — it basically says ‘leave the room and go read my talk!’ — Cartan geometry is very beautiful. First of all, Cartan geometry completes this commutative square:

$$\array{ Euclidean geometry & \rightarrow & Riemannian geometry \\ \downarrow & & \downarrow \\ Klein geometry & \rightarrow & Cartan geometry }$$

It’s like what category theorists call a ‘pushout’ — the ‘least common generalization’ of both Klein geometry and Riemannian geometry, which both generalize Euclidean geometry. So, if you like these other forms of geometry, you’re bound to like Cartan geometry!

Second of all, Cartan geometry is very down-to-earth: it amounts to describing a space by saying how you’d roll a Klein geometry on it. A Klein geometry is a homogeneous space: a plane, or a sphere, or a hyperbolic plane, or something fancier like that. The instructions for how to do the rolling are packed into a gadget called a ‘Cartan connection’.

Cartan geometry has been sadly neglected after his student Ehresmann came up with the more general but more abstract approach to connections on principal bundles that we use today. From what I gather, Cartan wanted Ehresmann to formalize the notion of Cartan connection, and Ehresmann did, but then he went further…

So, it’s a pleasant revenge that to understand MacDowell and Mansouri’s approach to gravity, you really need Cartan connections — as the picture in my blog entry hints!

Derek and Jim have been having lots of conversations over the last few years, only a bit of which found their way into Derek’s thesis. Derek should start explaining this stuff to the world!

However, even now that it’s summer, he’s pretty busy. He’s not roaming Europe like us lazy guys with permanent positions! He’s teaching summer school, polishing up his thesis before putting it on the arXiv, and — to make a few extra bucks — improving the format of old editions of This Week’s Finds.

The idea of a torsoroid is pretty nice, so I should explain it before I forget it.

When a space $M$ is given a Cartan geometry modelled on the Klein geometry $G/H$, it must have the same dimension as $G/H$. Even better, it comes with a principal $H$-bundle $P \to M$ where $P$ has the same dimension as $G$.

We should think of $P$ as a space that mimics $G$, in a manner that’s completely successful in the $H$ directions, but only ‘infinitesimally’ successful in the $G/H$ directions.

At one extreme, if $H = G$, $P$ is a perfect copy of $G$! More precisely, it’s a $G$-torsor.

At the other extreme, if $H$ is trivial, $P$ is a lousy copy of $G$, except infinitesimally: it’s just any manifold with the same dimension as $G$.

This leads us to call $P$ a ‘$G$-torsoroid, flat in the $H$ directions’.

Derek and Jim formalized this a bit, roughly as follows.

Suppose $G$ is a Lie group, and let $Lie(G)$ be its Lie algebra. Let $P$ be a smooth manifold.

If $G$ acts smoothly on $P$ we can differentiate this action to get a Lie algebra homomorphism:

$$\phi: Lie(G) \to Vect(P)$$

Moreover, if $P$ is a $G$-torsor, the linear map

$$\array{ Lie(G) &\to& T_x P \\ v &\mapsto& \phi(v)_x }$$

is an isomorphism for each $x \in P$.

We can also turn around these statements, at least when $P$ is compact and $G$ is simply connected, so every action of $Lie(G)$ as vector fields on $P$ generates an action of $G$ on $P$. Under these assumptions, if we have a Lie algebra homomorphism

$$\phi: Lie(G) \to Vect(P)$$

it always comes from differentiating an action of $G$ on $P$. If moreover

$$\array{ Lie(G) &\to& T_x P \\ v &\mapsto& \phi(v)_x }$$

is an isomorphism for each $x \in P$, $P$ is a $G$-torsor!

We get the concept of ‘torsoroid’ — which is secretly almost the concept of Cartan geometry! — by weakening this setup a little.

Suppose $H$ is any Lie subgroup of $G$. We define a $G$-torsoroid to be a smooth manifold $P$ equipped with a linear map

$$\phi: Lie(G) \to Vect(P)$$

such that

$$\array{ Lie(G) &\to& T_x P \\ v &\mapsto& \phi(v)_x }$$

is a vector space isomorphism for each $x \in P$.

Note, we drop the condition that $\phi$ is a Lie algebra homomorphism!

However, if $\phi$ restricted to $Lie(H) \subseteq Lie(G)$ is a Lie algebra homomorphism, we say our $G$-torsoroid is flat in the $H$ directions. If $H$ is simply-connected and $P$ is compact, we then obtain an action of $H$ on $P$, and defining

$$M = P/H$$

we get a principal $H$-bundle

$$P \to M$$

But, we get more: if $G$ is simply-connected, we get a Cartan geometry on $M$!

Even better, we can define a curvature that measures the failure of $\phi$ to be a Lie algebra homomorphism on all of $Lie(G)$:

$$curv(v,w) = \phi([v,w]) - [\phi(v),\phi(w)]$$

This vanishes for all $v, w \in Lie(H)$ if and only if our $G$-torsoroid is ‘flat in the $H$ directions’.

Let me translate this into something more resembling English. A $G$-torsoroid is a manifold such that if you’re sitting at any point and someone hands you an element of $Lie(G)$, you know which direction to march in. So, you might think you were living on a copy of $G$.

However, if you attempt to march around following these instructions, you may not come back to where you started when you expect to!

We say the $G$-torsoroid is ‘flat in the $H$ directions’ if you do come back where you started whenever you expect to, as long as you only move in directions given by the Lie subalgebra $Lie(H) \subseteq Lie(G)$. The ‘curvature’ defined above then vanishes when $v$ and $w$ are in $Lie(H)$. And under some mild conditions, your torsoroid will then be a bundle of $H$-torsors. In other words, it becomes a principal $H$-bundle over $P/H = M$.

If I had half an hour more, and could draw you pictures, you’d see the beauty of this idea. I can do it when we meet in Delphi this Thursday!

I shouldn’t say $P = \mathbb{Z}$. It’s more like a spiral staircase, extending infinitely in each direction.

Your definition of torsoroids was in terms of Lie groups and Lie algebras, which appeared to rule out the discrete setting. Does Urs’ idea about the tangent space of any category allow for smooth transfer of definition?:

Let $C$ be any category. It’s tangent space at any object $x \in \mathrm{Obj}(C)$, which I’ll write

$$T_x(C)$$

ought to be the category whose objects are morphisms in $C$ starting at $x$ and whose morphisms are commuting triangles.

I tend to be pretty open with drafts of my own papers, but a bit more protective when my students are involved.

I eventually got so impatient waiting for Derek to polish his thesis and put it on the arXiv that I put a version of it on my website. If you try to view this using your web browser, it may crash — at least, that’s what always happens with me! But this is not a fiendish trick to prevent people from stealing his results; I don’t know why it happens. If I download the PDF file and then view it using Adobe Acrobat, it works okay.

Derek says he’ll polish up his thesis and put it on the arXiv ‘soon’.