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Started thinking about smooth paths.
(Incidentally, David, do you want query boxes added to your web? And would you like to change the CSS for off-web links from those boxes to some nice colour?)
CSS changed. Feel free to ask if you want the colours tweaked.
The discussion rumbles on ... (in case anyone else is interested in following along)
Sketched out the definition of the manifold structure on the space of 2-tracks in a manifold.
(Stuck it in the page which smooth paths do I use (davidroberts) as I wasn't sure if this was what David wants at the manifold of 2-arrows (davidroberts) so played safe.)
Question for David:
Do the fibre categories of 2-tracks fit together into one big smooth manifold (over M), or are they separate?
So when building the space of 2-tracks in a manifold, we have the three layers:
2-tracks -> paths -> end-points
When building a neighbourhood of a 2-track, we want to allow the paths to vary a little. Do we also allow the end-points to vary?
Clearly, if we can do it with end-points varying then we can do it with them fixed. I think that it is possible to do it with them varying, but requires a little more work (I think it needs the fact that the loop fibration, is a locally trivial fibre bundle, not just a fibration) and the construction will be just a little more opaque.
Which do you want?
(aarg - still can't get LaTeX to work here - bear with me)
Do the fibre categories of 2-tracks fit together into one big smooth manifold (over M), or are they separate?
They most definitely fit together into one manifold, else there would not be the map associating to a path the identity 2-track
When building a neighbourhood of a 2-track, we want to allow the paths to vary a little. Do we also allow the end-points to vary?
We do want the endpoints to vary - if are points in , and are charts (in ) around them, paths between them and , charts around them (in ) and a 2-track from to , then a chart around should look like . (edit: It just occurred to me that this so-called chart could look crazy - not like our model TVS at all. Hmmm Is this where you need to use the local triviality of the loop fibration?)
Also, re your previous comment all the stuff on my web can go on one page, and be formatted/distributed later into a more user-friendly version
The trick with LateX here is
always include in double dollar signs (so a total of 4! :-)
no line breaks inside the dollar sings
Used to be difficult for me, as by default I usually put a line break after and before a double dollar sign.
You just have to remember that the double dollar signs are really single dollar signs; there is no support for displayed equations as such. You just have to double them for the sake of the Forum's parser for some reason.
(I see that David has figured it out between my comment and Urs's above.)
@David: Great! That's certainly the more interesting case, and it will use the locally trivial structure in a non-trivial way (though I'm not completely sure of the details). Imagine moving the end-point of (a path that's the end-point of ) a track a little bit: you want the whole track to move with you.
@Everyone else: yes, I suppose you're right. You should complain to the idiot who designed the LaTeX capability on this forum and demand your money back. Slightly more seriously, tips like those should go on the FAQ (linked from the menu on the bar at the top of these pages), if they aren't already there.
Finally (gosh, over a month later!) written up the basics of the chart maps for 2-tracks with allowing endpoints to vary.
Annoyingly, the SVG-editor wasn't playing nicely with me so there's an unfinished diagram. I'll clean it up when I can get to a better browser.
This is, I gather from your comments in other places, not the same as the compact-open topology.
Let me just comment on that quickly to clear it up. There’s a difference between paths on $(0,1)$ and paths on $[0,1]$. With paths on $(0,1)$, if one wants a manifold structure then one has to consider a slightly bizarre topology on $P M$. The easiest way to show how bizarre it is is to point out that it has uncountably many connected components: $f$ and $g$ are in the same component if $f = g$ outside some compact set. However, the conclusion of that is not that we use $P M$ with a different topology, but that we say “Oh well, it’s not a manifold, it’s an X space.” (Frölicher, diffeological, what-have-you). With $[0,1]$, though, there’s no difficulty because $[0,1]$ is compact and it is a manifold and it has the right topology. The problem is merely that “free path space” is ambiguous and can refer to both open or closed paths depending on context.
I’ve also split off smooth fundamental bigroupoid (davidroberts) from the page which smooth paths do I use (davidroberts) to make it easier to focus on.
re #15 ok - we won’t look at open paths :)
And thanks for splitting the pages up.
As I threatened to do elsewhere, I’ve extended the construction of the smooth fundamental bigroupoid to a suitable pair $(N, \partial N)$. Although I write $\partial N$, I don’t assume that Frolicher spaces have natural boundaries, this is a statement about pairs. I still have more checking to do on the details, and I guess I should expand on the transition maps. The conditions that the pair $(N, \partial N)$ have to satisfy are:
I guess that 3 and 4 could be simplified slightly. If 3 is weakened to the existence of $U$ and a smooth map $U \to \partial N$ which is homotopic (rel $\partial N$) to the identity on $U$, then we can use the bump function from 4 to extend this to all of $N$.
One advantage for me in this generalised approach is that it helps me separate out all the uses of the interval! When considering maps from $I^3$ then it’s hard to keep track of which $I$ is which.
The page in question is smooth+fundamental+bigroupoid (davidroberts)
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