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pasted Todd's n-forum comment on triangulations into triangulation
(thanks to the awseome now functionality that Andrew has added, I could just literally get the source code and paste it in. Only added some hyperlinks.)
Here is the link to Todd’s comment.
Oh no! What is wrong with my comment above? Is there a problem linking to other n-Forum comments now? (Edit: Got it. Thanks Andrew. “Source” is a lifesaver :))
By the way, could we do something similar for n-Forum comments as nLab links, e.g. [ [nforum:#7866] ], or something?
The problem is with the ampersand: it should be &
.
Hmm, perhaps I should add a step to the validator that automatically converts entities in URLs. It's going to be a pain having to keep remembering to do that by hand.
Ah, a little experimenting reveals that the problem is because you put the HTML code in by hand. If you'd used the right Markdown syntax for links then it would have been okay.
So here is a link to the comment, or http://www.math.ntnu.no/~stacey/Vanilla/nForum/comments.php?DiscussionID=1042&Focus=7819#Comment_7819 if you want the whole address visible.
As mentioned at my last comment in the Surface Diagram thread, I have some questions about triangulations. The standard definition is that a triangulation of a space $Y$ is a simplicial complex $X$ together with a homeomorphism $h: R(X) \to Y$, where $R$ here denotes geometric realization of simplicial complexes. On the other hand, I was using a similar notion but with simplicial sets in place of simplicial complexes. One question I had is: does it make any difference? That is: is the class of spaces which admits a triangulation in the sense of simplicial complexes the same as for the other sense using simplicial sets? Is this standard?
The other question I had has to do with the actual notion of realization of simplicial complexes. Over at that article, the description was given in terms of a metric space topology. Other articles I have seen seem to imply that the realization $R(X)$ works out to be the directed colimit of $R(X')$ where $X'$ ranges over finite simplicial subcomplexes. I am worried that the metric topologies and the filtered colimit topologies might not work out to be the same in general, if $X$ is some uncountably large simplicial complex.
@Tod I seem to remember that Curtis discussed the geometric realisation and the subdivision constructions (which go from simplicial sets to simplicial complexes) towards the end of his 1972 survey article. I have not got my copy to hand (I saw it about two months ago, probably halfway down a metre high pile in our dining room at home in North Wales. I am sitting in a small appartment in Lyon. I am not volunteering to go an look for the copy today!) But I think you will find the construction there. (It was in the Advances very early on.)
There is discussion in Spanier about two topologies for the realisation. The question is not phrased in categorical terms,but I think it gets near to providing an answer to your second question.
There is some interesting stuff on triangulations that has appeared recently. http://www.emis.de/journals/ZPOMI/v267/p046.ps.gz
I mention this because it is nice stuff, not that I think it is relevant to your entry. :-)
I pasted Todd’s discussion of the cosimplicial cubical set into triangulation, with a brief comment.
Grrr… I just went through a massive edit here, but internal errors occurred while processing. I have the version saved on my hard drive, though.
Thanks, Tim. I’ll try to get hold of the Curtis article (but if anyone knows a ’yes’ or ’no’ answer to my first question, are the classes of spaces the same, then I’d appreciate hearing it – my own guess is that under the coherent-topology realization of simplicial complexes, the class of spaces is indeed the same).
Spanier indicates that the metric space topology is in general weaker (fewer open sets) than the coherent topology, which makes sense to me. He also seems to favor the coherent topology, which also makes sense to me. So unless there are objections, I think a slight adjustment should be made at simplicial complex.
Dammit… I keep trying to submit my edit, and I keep getting “an application error occurred while processing your request”. Any idea what the problem may be?
You mean on the nLab? That sounds like you may have an unintentional filtered word, e.g “cialis” appears in the word “specialist”.
PS: If you send me the edit, I can play around and see if I can figure out what’s going on. Actually you can now paste it here :)
Thanks for the offer, Eric. Mike tracked down the problem (an unmatched dollar sign, or rather a double dollar bracketed with a single one). But I’ve done that sort of thing before, and it’s the first time I’ve gotten such an error message. Anyway, many thanks to Mike – I thought the problem was in some sort of communication with my PC.
That page is coming out as blank at the moment. It is not rendering at all, yet the source is there.
Sorry, fixed now.
Thanks.
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