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Stream: theory: category theory

Topic: Tangent space on a hom-object

Bruno Gavranović (Jan 29 2023 at 20:06):

If I've got the base space $\mathbb{R}$, we usually say its tangent space is $\mathbb{R}$ itself, for every point in the base. For an arbitrary manifold $M$, the tangent space $T_mM$ might be different for every point.

If my base space is a function space (for instance $[\mathbb{R}^2, \mathbb{R}]$) are there any theories telling me what's the tangent space of a point in it?

Fabrizio Genovese (Jan 29 2023 at 20:08):

Well it depends on what differential structure you put on it, no?

Bruno Gavranović (Jan 29 2023 at 20:12):

Isn't there a canonical one? At least for the base space $\mathbb{R}$ there seems to be one.

Fabrizio Genovese (Jan 29 2023 at 20:13):

I don't remember a whole lot about diffgeo, but:

• Start with a topological space X
• Put an atlas on it: For each open U, provide a local omeomorphism X -> R^n
• Require that 'chaging charts on the intersections' is a differentiable function.

Now you have a diffeentiable manifold, and you can define the derivative on a point x. If I remember correctly the way to do so was to first consider functions R -> X and defining the derivative on x by composing each of these functions with the relevant chart and then doing the derivative. Having done this you can take equivalence classes of these things and use them to build your tangent space

Fabrizio Genovese (Jan 29 2023 at 20:14):

This is as handwavy as it can get. If you have, say, 6 hours, I suggest you this: https://www.youtube.com/watch?v=uGEV0Wk0eIk&list=PLPH7f_7ZlzxTi6kS4vCmv4ZKm9u8g5yic&index=6

Fabrizio Genovese (Jan 29 2023 at 20:15):

These are among the best lectures I've ever watched that talk about this stuff. In any case, as soon as you can spell out what differentiable structure you want on your manifold, then you $\text{should}^{\dagger}$ be able to build the tangent space at a point.

$\dagger$ Modulo the fact that the calculations may as well be absolutely unfeasible

Bruno Gavranović (Jan 29 2023 at 20:16):

Uh, I might wait before spending 6 hours on this :big_smile:
I'm really expecting a definition theorem that tells me that in a given category $T([X, Y]) \coloneqq \text{something}$.

Fabrizio Genovese (Jan 29 2023 at 20:17):

BTW this is a very much classical treatment of diffgeo. If you want something more categorically oriented then best to look elsewhere!

Bruno Gavranović (Jan 29 2023 at 20:17):

Fabrizio Genovese said:

In any case, as soon as you can spell out what differentiable structure you want on your manifold, then you $\text{should}^{\dagger}$ be able to build the tangent space at a point.

$\dagger$ Modulo the fact that the calculations may as well be absolutely unfeasible

Yeah, I'm really not sure what I want to get. In automatic differentiation a big problem seems to be differentiating through function spaces, and I can't understand whether it's impossible or not.

Fabrizio Genovese (Jan 29 2023 at 20:18):

May it be because spelling out concretely what your differential structure is ends up being very difficult?

Bruno Gavranović (Jan 29 2023 at 20:22):

I believe so. Usual AD paper often mix in various operational aspects as well, and it gets hard to see the essence of the idea. I also maybe just haven't found the right paper. I was hoping there is some nice theory developed.

James Deikun (Jan 29 2023 at 20:43):

The calculus of variations might be relevant here.

Emilio Minichiello (Jan 29 2023 at 23:36):

This paper might be helpful https://arxiv.org/pdf/1411.5425.pdf

JS PL (he/him) (Jan 30 2023 at 00:38):

Bruno Gavranovic said:

Uh, I might wait before spending 6 hours on this :big_smile:
I'm really expecting a definition theorem that tells me that in a given category $T([X, Y]) \coloneqq \text{something}$.

So in tangent categories (in the Cockett Cruttwell sense/differential category version), Jonathan Gallagher studied this in his PhD Thesis https://prism.ucalgary.ca/handle/1880/108793 maybe you will find an answer in there (Chapter 5)

John Baez (Jan 30 2023 at 06:24):

Bruno Gavranovic said:

If I've got the base space $\mathbb{R}$, we usually say its tangent space is $\mathbb{R}$ itself, for every point in the base. For an arbitrary manifold $M$, the tangent space $T_mM$ might be different for every point.

If my base space is a function space (for instance $[\mathbb{R}^2, \mathbb{R}]$) are there any theories telling me what's the tangent space of a point in it?

If your base space is a vector space $V$, each tangent space $T_x V$ comes with a god-given isomorphism to $V$. So we usually sloppily say $T_x V$ "is" $V$.

John Baez (Jan 30 2023 at 06:24):

This is what you're doing here:

If I've got the base space $\mathbb{R}$, we usually say its tangent space is $\mathbb{R}$ itself, for every point in the base.

John Baez (Jan 30 2023 at 06:27):

and since $[\mathbb{R}^2, \mathbb{R}]$ is a vector space you can do it there too, modulo issues related to the fact that you haven't said what you mean by $[\mathbb{R}^2, \mathbb{R}]$ since I don't know what kinds of maps you're allowing, so maybe it's an infinite-dimensional vector space, and you haven't said what theory of tangent spaces of infinite-dimensional manifolds you want to use.

Matteo Capucci (he/him) (Jan 30 2023 at 11:35):

Also relevant [[diffeological spaces]], [[smooth spaces]] (EDIT: there's also an nLab page [[manifold structure of mapping spaces]] with more traditional constructions, you get a [[Frechet manifold]] in this way)

Bruno Gavranović (Jan 30 2023 at 11:40):

John Baez said:

If your base space is a vector space $V$, each tangent space $T_x V$ comes with a god-given isomorphism to $V$. So we usually sloppily say $T_x V$ "is" $V$.

I knew I was going to get corrected for this (:

Bruno Gavranović (Jan 30 2023 at 11:52):

John Baez said:

and since $[\mathbb{R}^2, \mathbb{R}]$ is a vector space you can do it there too, modulo issues related to the fact that you haven't said what you mean by $[\mathbb{R}^2, \mathbb{R}]$ since I don't know what kinds of maps you're allowing, so maybe it's an infinite-dimensional vector space, and you haven't said what theory of tangent spaces of infinite-dimensional manifolds you want to use.

I have a feeling this might not end up doing what I want it to. I just used $\mathbb{R}$ as an example, and am really interested in $T([X, Y])$ for arbitrary $X$ and $Y$.

But working with your suggestion, if I have a higher-order function $f : [\mathbb{R}^2, \mathbb{R}] \to \mathbb{R} \coloneqq g \mapsto g(2,3)$, then this means I can compute its reverse derivative as a map $f^\sharp : (\mathbb{R}^2 \to \mathbb{R}) \to (\mathbb{R} \multimap [\mathbb{R}^2 \to \mathbb{R}])$ which takes the initial function $g$, and produces a linear function which takes a change in the output $dy : \mathbb{R}$ and produces a change in the input, which ought to be of type $[\mathbb{R}^2, \mathbb{R}]$.

But I'm not sure what linear function this is.

Compare this to the usual case, of, say, $f : \mathbb{R^2} \to \mathbb{R} \coloneqq (x, y) \mapsto x^2y$ where the resulting reverse derivative is a function $f^\sharp : \mathbb{R^2} \to (\mathbb{R} \multimap \mathbb{R}^2) \coloneqq (x, y) \mapsto (dz \mapsto (2xydz, x^2dz)$

Fernando Yamauti (Jan 30 2023 at 20:39):

@Bruno Gavranovic In what setting are you working? In the synthetic setting, the tangent space functor is corepresentable since there exists an infinitesimal affine line. Depending on your setting, you might also try using the evaluation map and mimick some sort of transgression.

Bruno Gavranović (Jan 31 2023 at 21:29):

I'm not sure what setting I'm working in. Could you suggest me some? I'm really trying to wrap my mind around what the simple example ought to be, and then figure out what that tells me CT-wise

Bruno Gavranović (Jan 31 2023 at 21:31):

I've boiled it down to the essential question of how $T([X, Y])$ is defined. I'm open to seeing it in many different settings, but I'm not sure if a simple answer of the form $T([X, Y]) \coloneqq X \times List Z$ was provided (obviously, I just came up with this one)

Fernando Yamauti (Feb 08 2023 at 00:54):

Bruno Gavranovic said:

I'm not sure what setting I'm working in. Could you suggest me some? I'm really trying to wrap my mind around what the simple example ought to be, and then figure out what that tells me CT-wise

Sorry for the late reply. I was thinking about a smooth topos. In any case, if one is willing to talk about tangent spaces of "curved" stuff, then some notion of infinitesimality will likely be required (for instance, differential cohesion).