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I’ve started porting my notes “differential topology of loop spaces” over to the nlab, starting at differential topology of mapping spaces. As part of the transfer, I intend to map out the theory for general mapping spaces, not just loop spaces (that’ll give me a bit more motivation to do the transfer since just cut-and-paste is boring!). I’ve just copied over the contents and the introduction so far and haven’t edited them as yet. In particular, although I’ve wikilinked all the original section names, these will get changed as they currently focus on loop spaces.
The introduction to the original document ended as follows (not copied over to the new version):
This document began life as notes from talks given at NTNU and at Sheffield so I would like to thank the topologists at those institutions, and in particular Nils Baas, for letting me talk about my favourite mathematical subject. I would also like to thank Ralph Cohen and the “loop group” at Stanford.
This is by no means a ﬁnished document, as an example it is somewhat sparse on references. Any comments, suggestions, and constructive criticism will be welcomed.
The second paragraph is sort-of stating the obvious as it holds to some extent for any nLab page! And I would love to be able to add some more names to the list in the first paragraph. Again, I hope it goes without saying but I’ll say it anyway: although I anticipate being the main contributor to these pages, it is not my project! I would love it if people read it, add comments, add other stuff, write (constructive) graffiti, link it to other stuff.
Thanks, Andrew. it is awesome that you are carrying out this project on the nLab! Just the kind of thing one would hope for.
I went through your entry differential topology of mapping spaces and added plenty of hyperlinks. Hopefully some of the unsaturated one inspire you to create at least stubs for the corresponding entries.
Andrew, one minor comment: I suggest you rename your entry using capital letters. We have the convention that genuine keywords appear lower case in the title, while page titles that are really titles of some text are capitalized. For instance Structured Spaces versus structured (infinity,1)-topos.
I personally in nlab follow this convention for titles just in the first word. So if I created it it would be Structured spaces. Still for less famous books and notes it may be even better to have more descriptive page names like
conference: algebraic geometry of cubics
seminar: algebraic geometry of cubics
lecture notes: algebraic geometry for nlabers
Of course Higher Topos Theory does not need pagename like
though it is in principle a better convention with some redirects for famous titles.
Created stub for bornological topological vector space and redirected everything that refers to bornological from linear mapping spaces over there.
Excellent! Thanks.
I thought that on topological vector spaces the list of types of TVS should be short and sweet, maybe with a slogan for each type but without any more detail; the detail can go on the relevant page as can the relationships between the various types.
Speaking of topological vector space: what’s TVR? It appears at least twice on that page.
Ah, that’ll be me being confused between English and Norwegian. “Space” translates to “rom” in Norwegian.
Err, more likely, a typo by whoever wrote that bit. I’ve certainly never heard of a TVR and the page makes sense with TVR replaced by TVS.
Not Norwegian “rom” , but German: “topologischer Vektorraum – TVR”. It appears with Tim’s revision 9.
Yes, sorry :-) Will try to correct it whenever I see it…
Andrew wrote:
…the list of types of TVS should be short and sweet, maybe with a slogan for each type but without any more detail…
That is precisely what I had in mind, when learning about TVS I was overwhelmed (and still am) by all the different types and what they are useful for…
I had/have exactly the same feeling! That’s what led me to ask this question. Now that I’m getting back into adding mathematical content to the nLab, I hope to import Greg’s answer in some fashion.
Wow. As a start I uploaded Greg’s chart to TVS.
I’ve converted it to SVG. Well, actually I converted it to dot (the language for graphviz) which I then exported to SVG. I’ve left the original there as well just so someone else can take a quick look and confirm that the two diagrams are the same.
Advantages of SVG: We can add to it later (I’ve uploaded the source file), we can style it so it looks better, and we can add hyperlinks to the relevant pages (I’ve added “basic” hyperlinks: everything links to “X topological vector space”, clearly not all of these are the best choice but I figured enough would be that it would be simple to change the others.).
Query: everything there is a LCTVS so should this diagram go at LCTVS instead? Or should we add TVS at the top!
Completed the linear section of linear mapping spaces.
I’ve converted it to SVG.
Cool, I did not know that that could be done so easily. Could we have a HowTo-entry that explains what software to use etc.?
should this diagram go at LCTVS instead?
I’d look at the root entry TVS for such a chart, but that’s just me. Counterquestion: What non-lc spaces could be added to the chart?
I’ve created lctvs dot source (probably not the best page name) to hold the source of that diagram (so that we can track changes, the system doesn’t keep old versions of uploaded files) and added basic instructions on how to generate it.
I’m fine with keeping it at TVS rather than LCTVS but we should at least add “locally convex” at the appropriate place! I guess we should go through Schaefer’s book and look at which concepts assume convexity and which he defines before that.
As to your final question, it could be read as being about specific examples in which case I would answer “none” as this is a chart of properties. If, as I guess is more probable, you mean types of non-lc spaces, then there are some variants on convexity, if I remember aright, which the $L^p$-spaces satisfy for $p \lt 1$.
Actually, I have Schaefer’s book to hand so let me stop being lazy and have a look!
Chapter I is the pre-lc section and in it he discusses:
(plus a load about basic categorical constructions)
I seem to remember that Jarchow has a bit more on non-lc spaces, but my library doesn’t have that.
I’ve created lctvs dot source (probably not the best page name) …
And I see you added links to all the boxes of the graph, again: cool!
If, as I guess is more probable, you mean types of non-lc spaces…
Yes, that’s what I meant, and now you mention it, it occurs to me that $L^p ([0, 1])$ spaces are F-spaces for $0 \lt p \lt 1$, i.e. they admit complete translation-invariant metrics with respect to which the vector space operations are continuous, and they are not locally convex.
Therefore some of the boxes on the chart contain spaces that are not locally convex, and e.g. the arrow from metrizable to bornological needs the lc assumption…
That way madness lies …
More seriously, can you think of a way to include information of the form “X + Y => Z”? One way might be to have mini-nodes for “X + Y”. So there’d be an empty circle linked to X and to Y which represented “X + Y” (not sure whether the arrows should go from X and Y to “X + Y” or the other way; logic suggests the other way as “X + Y” implies both X and Y, but somehow it feels as though it would be clearer the other way around) and then an arrow from the empty circle to Z.
The important thing is to keep the diagram clear!
Just thinking aloud … if we had “locally convex” as a property, then technically it would appear at the bottom of the chart, since any property that requires it in the definition would imply it. But I feel that there’s a difference between “implies by theorem” and “implies by definition”! I don’t know whether or not it’s possible to make this distinction without cluttering up the graph, though.
Those links in the table are pretty awsome.
Maybe eventually one could make them look like links, so that people can notice that there are links to be found. Maybe just by underlining them? Maybe by giving them color code as for the links in the text?
There is a nice monograph about F-spaces that concentrates on the non-lc aspects, its mentioned by wikipedia,too: Kalton, N.J.; Peck, N.T.; Roberts, James W.: “An F-space sampler.” Never felt the need to look into it, but there are connections with algebraic analysis, because some sort of hyperfunctions live in Hardy spaces that are not-lc.
…can you think of a way to include information of the form…
The graphical language that I use from time to time is UML , and what comes to my mind here is a entity relationship where the relationship has attributes.
“Implies by definition” and “implies by theorem” could be distinguished by the kind of arrow (like dashed and not dashed).
You could do a pair of pants-type arrow from the two (or more!) types of vector space to the one which it implies (or the other way around). Think: string diagrams!
By the way, I think you all are doing great work here on adding entries about topology, TVSs and functional analysis. In fact, there are so many entries on these topics now, that it is hard to see what material there is and how to navigate it. Personally, I would think it is time to create a floating table of contents that gives an overview of what the nLab has to say about the subject, and add it to the side of the entries in question. I think this will greatly improve the visibility of the material to the outside, and also greatly help to get an overview of what we have and what we have not yet.
…it is time to creating a floating table of contents…
Sorry, what is that?
…it is time to create a floating table of contents…
Sorry, what is that?
One of these columns with keywords on the right of an entry, such as at cohomology.
I started one for topology, so that you can see how it works:
the page for the table of contents itself is topology - contents.
To make another page include this on the right, insert at the beginning of that page the lines
<div class="rightHandSide toc">
[[!include topology - contents]]
</div>
See how I did it at topological space.
You can see how it works from that. By some straightforward copy-and-pasting you can create such tocs for “functional analysis” for instance, or any other topic cluster.
i think it is useful to have these floating tocs in order to bind together collections of entries on a given topic. It is unlikely that every nLab entry that one needs shows up in a Google search. One also wants to have an overview of which entries there are on a given topic.
I agree that a floating table of contents for functional analysis would be good, but I’d also like to point out that this diagram that we are talking about is an attempt to achieve a similar end. The drawback of the diagram is that it’s a bit unwieldy to be included on all the pages.
Yes, that diagram is very good as a toc. It’s actually amazing. The floating toc could point in its very first line to a page just containing that diagram.
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