I started making some notes on my personal web:

If anything becomes nLab-worthy, I’ll move it over, but it is still pretty rough.

Since diffeological spaces are defined as maps from open subsets of $\mathbb{R}^n$, it is pretty easy to go down the list of differential geometric concepts and define them on smooth spaces.

My approach is to

- Define the space of chains on a smooth space by pushing forward chains on the domains via plots
- Define the space of cochains on a smooth space as the dual space of chains
- Define differential forms on smooth space to be those cochains that pull back to forms on the domains via plots
- Define boundary and coboundary on smooth spaces
- Prove Stokes theorem for smooth spaces

Armed with this, I hope to start looking at relationships between chains/cochains on smooth spaces $M$,$N$ and chains/cochains on the mapping space $[M,N]$, since $[M,N]$ is also a smooth space.

If I get anywhere with this (which would be a miracle!) I hope to see how it relates to transgression.

PS: I added some references to generalized smooth space and transgression.

]]>Some questions:

- If $M$, $N$ are Frölicher spaces, is the mapping space $[M,N]$ a Frölicher space? (I hope so!)
- Are smooth manifolds with boundaries Frölicher spaces?
- Are piecewise-smooth manifolds with boundary Frölicher spaces?
- In general, is there a way to define integration on any Frölicher space?
- Is it always possible to define spaces of chains and cochains on Frölicher spaces?
- Is there a relation between the space of chains $C(M)$, $C(N)$, and $C([M,N])$?
- Is there a relation between the space of cochains $\Omega(M)$, $\Omega(N)$, and $\Omega([M,N])$?
- Is there a such thing as a directed Frölicher space?

Thinking about that last question reminded me of an explanation Andrew gave me a long time ago about recovering the tangent space from the collection of curves. That explanation allayed some of my fears and helped me embrace the concept of Frölicher space. Now, for a “directed Frölicher space”, you wouldn’t have a full tangent space. It would be like a “tangent space without negatives”. The direction would splits the tangent space into two half spaces, i.e. a forward tangent space and a backward tangent space, so all curves in a directed Frölicher space passing through a point $x$ would have to be all in the forward direction. I know about “rings without negatives”, i.e. rigs, but how about vector spaces without negatives? In a vector space without negatives, you can only multiply vectors by a scalar $s\ge 0$ and you can only add such vectors together, i.e. no subtractions.

It is too bad “vector” doesn’t have an “n” in it, so I thought we might call them “vectors without cancellation” or “vetors” :)

Although my last comment was presented in a whacky way, my question is serious. I’d be curious to see a definition of directed Frölicher space and its tangent space would probably be a “taget vetor space”, i.e. a tangent vector space without negatives :)

]]>Right!

]]>I didn’t link directly to manifold because I wanted to link to a page that was a little more traditional in style. I like Todd’s stuff on manifold but for my purposes, I needed the “standard treatment”. Not linking directly is sort-of a note to myself to sort that lot out a little.

But in that case, please be sure to eventually add a section along the lines that you imagine to manifold. I think we said before, elsewhere that

we do want entries to reflect different approaches and different point of views, if reasonably desireable. Add another section with another point of view, and be it a bit repetitive. If it gets too much for a single entry, we should create one top-level entry that explains that there are different ways of looking at it and then branches off via links to the corresponding subentries. For instance the material that we already have could become an entry “manifolds in terms of pseudogroups”.

Maybe I should also clarify concerning my creation of the stub smooth manifold: I think we also said elsewherre that for long entries it serves to split off seperate entries on subtopics which they do cover. The reader who may know what a topological manifold is and just wonders what exactly a smooth manifold might be (are there such readers?? but anyway) needs just the information on what a smooth transition function is, and will appreciate it to see this quickly defined in a dedicated entry. The reader who arrives at “smooth manifold” and discovers that even the basic concept of manifold is not clear to him can follow the link to the more in-depth article.

In other words: a bit of redundancy shouldn’t hurt and in fact be desireable.

Right?

]]>The idea section often gets bloated and actively detracts from the article. I'm packing up to go home, but I'll find some examples tomorrow. The absolute worst is when you say something in the formal definition like "use the construction from the idea part", which is completely useless.

]]>Whew! Nice lot of comments!

Yes, I know it should be more structured; but as I hope is clear, it’s not finished, and it is actual research-in-progress!

I didn’t link directly to manifold because I wanted to link to a page that was a little more traditional in style. I like Todd’s stuff on manifold but for my purposes, I needed the “standard treatment”. Not linking directly is sort-of a note to myself to sort that lot out a little.

Whilst my son was at his swimming lesson, I figured out the correct compactness condition and I want to call it **curvaceously compact**. “Functionally compact” is actually already defined on Froelicher space; at the time, I couldn’t come up with what I thought the corresponding curvaceous version was but now I know. The condition is:

Every sequence $(x_n)$ in $X$ has a subsequence $(x_{n_k})$ for which there is a smooth curve $c \colon \mathbb{R} \to X$ and a sequence $(t_k) \subseteq \mathbb{R}$ with $(t_k) \to 0$ and $c(t_k) = x_{n_k}$.

If you replace “smooth curve” by “continuous function $\{0\} \cup \{\frac1n\} \to X$” then this is the definition of “sequential compactness” in a topological space.

(I’ll add that when I next get a chance, and go through expanding some of the stubs. Hey, working on the nLab is fun! Why didn’t anyone tell me it was as more fun to do maths on the nLab than hacking all the programs needed to make it work?)

]]>Yes, I know. But now we have a page smooth manifold that links to manifold.

]]>Urs, there’s also a page manifold which includes the notion of smooth manifold and much more (essentially all notions of manifold which are described by pseudogroups: topological, PL, foliated, and many more).

]]>Yes, very good. I think this is a good practice. We should start adding “Summary” or “Abstract” sections to all longer entries.

]]>Since the unbounded operator page is long and will grow longer, I added a summary paragraph, its terse - but its mainly a showcase. Is it what you have in mind?

]]>Andrew,

I tried to make “Functional compactness” a formal definition. Please check if I got it right.

]]>Isn’t the “Idea” paragraph supposed to be a summary?

Most of the idea paragraphs that we have don’t summarize the main points but motivate the main definition.

Maybe we should have another default section “Summary” or “Abstract”.

Idea

Abstract

Definition

Properties

Examples

References

Isn't the "Idea" paragraph supposed to be a summary?

]]>Oh, I see, it is any fixed map?

]]>Andrew, one question:

what is “$g$” in the first displayed math in the section Functional compactness?

]]>figure out some other way to grab Urs’ attention

I am increasingly finding that there are not enough hours in a day. We need summaries. It should be possible to read an entry by “zooming into it” not just by reading it linearly. Ideally.

]]>to get rid of some of the unsaturated links on Andrew’s page, I created stubs for

and made Chen space redirect to diffeological space.

]]>What a relieve to see that you both have a sense of humor that I can understand.

Andrew, you'll need to figure out some other way to grab Urs' attention, because meanwhile there are guys like me who add tics, tocs and tracs while looking for a distraction during a coffee break. For example, I made a stub about convenient vector spaces.

]]>apart from that: thanks for the page, i had already looked at it earlier today or so. very useful stuff.

but why the title? shouldn’t it be “mappng spaces that are manifolds”?

and let’s make sure to cross-link this prominently, so that people can find it, even if they don’t know that they are looking for it

]]>Also, haven’t put a table of contents at manifolds of mapping spaces since I’ve learnt that the best way to get Urs to read something is to not put a toc in.

oh no, that’s the wrong take-home message!!

i think that if you want your stuff to be read, then structure it nicely and provide a toc. all too often do i see you guys write lengthy entries with lengthy introductions and no tocs. make yourself clear what the user experience is for someone arriving at that page: on the screen there is only a big chunk of blah-blah and no indication what gems are to follow as one scrolls down. life is short. the typical user reaction is to close that page. on the other hand, if there is a well-structured toc, then the user can see at one glance what the page has to offer for him. or her

]]>Split off the mapping spaces stuff from local addition into manifolds of mapping spaces. Still plenty to do and things to check (particularly on the linear stuff, and particularly figuring out what “compact” means). Haven’t actually deleted the relevant bit from local addition yet. Also, haven’t put a table of contents at manifolds of mapping spaces since I’ve learnt that the best way to get Urs to read something is to *not* put a toc in.