The n-Category Café

An algebraic geometer will want to distinguish “tangent space” (answers 3 and 3) and “tangent cone” (answers 2? and definitely 2).

I think of tangent vectors via physical intuition: direction and speed for a particle moving in the space in question. So I would answer 3 in both cases. I suppose the kinematic tangent space is the formalization of this for the category of Smooth Spaces?

This type of tangent space will not be a vector space at a singular point. But it is very important that the total differential of a smooth map between manifolds is linear (when restricted the fibers). So I wonder, what is the algebraic structure on more general tangent spaces, such that the differential of a map will respect this structure? At the very least, there will be an action of the positive reals, and a partially defined addition operation. And upon the extension to tangent multivectors, the exterior product of two tangent vectors will be defined precisely when addition is defined between the same. It seems to me that these structures are to vector spaces what groupoids are to groups, and that they formalize the algebraic structure of (kinematic) tangent spaces at singular points.

If I think of $[0,1]$ as a manifold with corners, then $T_0 [0,1]$ is $T_0 \mathbb{R}$, which you may identify with $\mathbb{R}$. If I think of $[0,1]$ as $S^1/(\mathbb{Z}/2)$ then $T_0 [0,1]$ is isomorphic to $\mathbb{R}/(\mathbb{Z}/2)$. Whether one should think of $\mathbb{R}/(\mathbb{Z}/2)$ as $[0,\infty)$ is another story…

$\mathbb{R}\vee \mathbb{R}$ maybe thought of as a subset of $\mathbb{R}^2$, i.e., as a subCartesian space. Or it may be thought of as a stratified space. And there choices of the stratification. So there is more than one “right” answer to your question.

My feeling is that answeres depend on what you are trying to accomplish.

Since the unit interval is the naive quotient of $\mathbb{R}$ by the natural action of the infinite dihedral group $D_\infty$, as having once been a physicist and always fond of mirrors, I cannot think of a better tangent space $T_0 [0,1]$ than $\mathbb{R}$, with exponential map the one that folds up $\mathbb{R}$ — modify this as seems sensible for other points of $[0,1]$. You can tell that I like my interval to be metrized in a sensible way.

To sing a song that others can surely sing better…

In a synthetic diff geom setting, we’ll have a “tiny” space $T$ — think of it as a real interval of infinitesimal length — with the property that for any space $X$, the tangent bundle is $X^T$. Actually, $T$ is a pointed space, and the map $1 \to T$ picking out the point induces the projection map $T X = X^T \to X$. As you’d expect, its fibre over $x \in X$ is the tangent space at $x$.

So in this setting, it all comes down to (i) choosing your category, and (ii) choosing your tiny object $T$. But if I’m correctly catching your drift, Andrew, you’re considering the possibility of multiple different $T$s. Am I understanding?

Just a quick line to suggest reading this blog post by Jeffrey Morton about Cartan geometry and spacetime. He points out (and this is not new, but nicely explained), that Cartan geometries are like manifolds where the ‘tangent space’ is a homogeneous space more general than a vector space - in fact modelled on some coset space $G/H$ for a Lie group $G$.

Robin Cockett and I just wrote a paper on this exact subject! We were looking to give an axiomatization for abstract tangent functors, based on functors that occured in Cartesian differential categories. The axiomiatization we arrived at was similar to what you mention above: an endofunctor with various natural transformations and coherence equations.

Soon after we submitted the paper, however, Anders Kock kindly pointed out to us that Rosicky had come up with almost the exact same axiomatization almost 30 years ago! He was trying to understand the similarities between the tangent functor on the category of smooth manifolds, the tangent functors in SDG, and the tangent functors in algebraic geometry. His paper can be found here.

After we recovered from our shock, we realized that we still had something to say on the subject. Namely, (1) the “tangent spaces” of any abstract tangent functor give a Cartesian differential category, and (2) we could improve Rosicky’s result about representable tangent functors. Specifically, the end result is that if you start with an abstract tangent functor which you know is representable, you recover the setting of SDG: a commutative rig R whose infinitesimals D satisfy the Kock-Lawvere axiom: R^D = R x R.

So, on the one hand, restricted to particularly simple spaces, abstract tangent functors are the same as Cartesian differential categories, while on the other hand, if the tangent functor is representable, you recover SDG.

In between these two extremes are all sorts of other examples: the canonical example of finite-dimensional smooth manifolds, the convenient manifolds of Kriegl and Michor (where the tangent bundle is the kinematic one), and various categories of algebraic geometry. Hopefully other examples of generalised smooth spaces, such as diffeology or Fermat spaces, should also satisfy the axioms.

Anyways, our updated paper can be found here.

Andrew, interesting that you posted this now, as I’m in the middle of writing a paper with Enxin Wu which includes a section about tangent spaces to diffeological spaces. It’s not ready for distribution yet, but I will try to remember to post a link here when it is.

In any case, I think that there is no one right answer, and that what you define your tangent vectors to be depends on what you’d like to do with them. If they should relate to flows, then you probably want to take something like equivalence classes of curves (and then you won’t end up with a vector space). Or you may take sums of such things to get a vector space, which is probably the same as taking the colimit of the tangent spaces of the plots. Or you might take derivations on an algebra associated to your space, e.g. an algebra of germs of functions, if your goal is to, well, differentiate functions.

There are maps between the various tangent spaces, and under certain assumptions the maps are isomorphisms.

Andrew, do you know about the metric tangential calculus of Elisabeth Burroni and Jacques Penon?

The central definition — and it’s amazing if this is new — is as follows. Let $W$ and $X$ be metric spaces, and let $f, g: W \to X$ be two maps. By “map” I just mean map of sets for the moment. Let $w \in W$. Then $f$ and $g$ are tangent at $w$ if $f(w) = g(w)$ and

$$\frac{d(f(u), g(u))}{d(u, w)} \to 0 \quad as u \to w \quad (u \neq w).$$

This is an equivalence relation on maps $W \to X$. You could define the “$(W, w)$-tangent bundle” of $X$ to be the set of all equivalence classes of maps $W \to X$. (I guess we want to restrict to just nice enough maps: continuous or Lipschitz or something. Also, maybe we don’t want actual maps $W \to X$, but germs at $w$ of maps $W \to X$.)

If you’re also given a point $x \in X$, you could define the “$(W, w)$-tangent space” of $X$ at $x$ to be the set of equivalence classes $[f]$ such that $f(w) = x$. I haven’t read Burroni and Penon’s paper, but from the abstract it looks as if they make some such definitions (using “locally lipschitzian” maps).

Particularly interesting cases would be where $W = \mathbb{R}$ (and $w = 0$, say), or where $W = [0, \infty)$ and $w = 0$.

Hi,
I usually read you guys but this is my first post here.
In the past, I’ve came a cross infinite dimensional bundles in which it would be nice to do some Cartan calculus: given a connection, define the curvature, Lie derivative, contraction, etc… and obtain the well known identities (Cartan’s formula, Bianchi, etc…). Can you do this with any notion of tangent space on your infinite dimensional manifold?