### Tangency

#### Posted by Urs Schreiber

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This is a **guest post** by ** Andrew Stacey** (NTNU, currently on sabbatical at Oxford .

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## Quiz Time

Here’s a question for you all. What is the tangent space at $0$ of the unit interval, $T_{0} [0,1]$? To make it easier, I’ll make it multiple choice:

$\{ 0\}$,

$[0,\infty )$,

$\mathbb{R}$,

None of the above.

What about $\mathbb{R} \vee \mathbb{R}$, the space formed by gluing two copies of $\mathbb{R}$ together at their origins (you can think of this as the union of the $x$ and $y$ axes if it helps). Here are your options for this space.

$\{ 0\}$,

$\mathbb{R} \vee \mathbb{R}$,

$\mathbb{R} ^{2}$,

None of the above.

There’s no particular “right answer” to these (though your answers to the two questions should match up). I can justify all three of the concrete answers. I shan’t, yet, because I want to know what others think and why without tainting the survey.

## Background

Those who know me will know that I like loop spaces. I’m pretty happy to meet them in any guise, but if I had to express a preference then it would be as a differential topologist (me, that is, not the loop spaces). This means that I’m considering them as some sort of infinite dimensional manifold.

It’s not a long, nor a difficult, path (ha ha) from loop spaces to more general smooth spaces. Spaces that are almost, but not completely, unlike manifolds. We’ve had many discussions here about what a generalised smooth space should look like. Right now, I don’t want you to think too deeply about that. I just want you to be aware of the fact that there are smooth spaces beyond manifolds. They don’t have charts, but they have a strong family resemblance to manifolds so a lot of intuition and ideas can be extended from manifolds to these more general spaces.

This is what I’m trying to do with tangent spaces: extend them from manifolds to generalised smooth spaces.

## Tangent Spaces

The problem is that there is not a unique definition of “tangent space” in differential topology. There are several equivalent definitions, but they do not remain equivalent when one generalises them. That’s okay because actually I’m not after a *unique* definition. I’m after a characterisation. Thus the question I really want to ask is the following:

Suppose I gave you two smooth spaces, $X$ and $Y$, and told you that $Y$ was a tangent space for $X$ (I’d probably better give you the projection $\pi \colon Y \to X$ as well). What would you expect that to tell you about $Y$?

Note that I’m using “tangent space” here to mean all the pointwise tangent spaces put together into a new smooth space. I can’t say *tangent bundle* because they may not form a bundle. Note also that, following from what I said about the different definitions, I’m using the indefinite article: *a* tangent space.

## Conclusion

My desired conclusion from this is to be able to give a characterisation of a *tangent structure* on a category of generalised smooth spaces. It will consist of an endofunctor, and one or two natural transformations, where the functor assigns to a smooth space a tangent space. But before I can characterise such functors, I need to know what characterises a tangent space. Hence the question.

The longer term goal is that I want to use tangent spaces as a tool to study smooth spaces. A finite dimensional manifold is actually modelled on its pointwise tangent spaces and this turns out to be a very important property in studying mapping spaces. More general smooth spaces will not have as close a relationship, but nonetheless there might still be enough of a relationship to be able to exploit it.

Posted at October 9, 2012 5:59 PM UTC
## Re: Tangency

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