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• CommentRowNumber1.
• CommentAuthorTobyBartels
• CommentTimeDec 11th 2009

I started classical mathematics to link to from internal logic.

• CommentRowNumber2.
• CommentAuthorHarry Gindi
• CommentTimeDec 11th 2009
I know we're not on wikipedia, but NPOV violation! NPOV violation!
• CommentRowNumber3.
• CommentAuthorzskoda
• CommentTimeDec 11th 2009

Toby, Who uses term "classical mathematics" in any consistent way ? For every person this is something else.

Harry, why do you use strange unknown abbreviations ? It is confusing. It is like writing an empty space or "#\$%&/U()I sign.

• CommentRowNumber4.
• CommentAuthorHarry Gindi
• CommentTimeDec 11th 2009
NPOV=Neutral point of view. I was merely remarking that the page in question is very "charged". I guess that's the word I'll use.
• CommentRowNumber5.
• CommentAuthorTodd_Trimble
• CommentTimeDec 11th 2009

Zoran, I believe the abbreviation is well-known to frequent users of the English Wikipedia, but perhaps not well-known to others.

There is something charged about the article, but I find it hard to disagree with anything Toby actually wrote -- it's all true one way or another. Maybe "ignoring the lessons of higher category theory" could be perceived as a slap on the hands, or at least admonition to mathematicians at large that they should be paying attention to these lessons. It's opinionated, but defensibly so.

While I don't mind the presence of this article, others (looking at you, Zoran! (-: ) should feel free to add their own meanings. Toby did say that the phrase could mean any of the following, and didn't seem to be pretending he was saying anything exhaustive.

• CommentRowNumber6.
• CommentAuthorTobyBartels
• CommentTimeDec 11th 2009

I hereby repeat Todd's last sentence. And if anybody wants to rephrase things so that it's clearer that Todd's last sentence is correct, please do so.

As for neutrality of point of view, the nLab has a POV, so … (^_^).

• CommentRowNumber7.
• CommentAuthorUrs
• CommentTimeDec 11th 2009

Yes, we are not interested in being neutral. We are interested in being right. :-)

• CommentRowNumber8.
• CommentAuthorHarry Gindi
• CommentTimeDec 11th 2009
Yeah, I wasn't disagreeing, merely stating the facts. =)
• CommentRowNumber9.
• CommentAuthorzskoda
• CommentTimeDec 11th 2009

I like Urs's statement.

• CommentRowNumber10.
• CommentAuthorAndrew Stacey
• CommentTimeDec 11th 2009

Just to be facetious, the nLab of course enforces the nPOV.

(That's the n-Point of View)

• CommentRowNumber11.
• CommentAuthorMike Shulman
• CommentTimeDec 12th 2009

I'm in favor of being right too, but I'm also in favor of not unnecessarily offending people who don't (yet) share our point of view. The word "lessons" in particular is a bit too opinionated for my comfort, since it implies that people who don't adhere to non-evil etc. are "failing to learn something they should." We may believe that to be true, but it takes some people a while to realize it, and (especially in the U.S.) category theory is still recovering from decades of ignominy due (partly, I am told) to an initially overzealous attitude towards changing the world. So I tried to soften the phrase a bit.

PS. I think there needs to be a page called nPOV now.

• CommentRowNumber12.
• CommentAuthorHarry Gindi
• CommentTimeDec 12th 2009
Yeah, I mean, that's kinda what I was getting at. If we want higher categories to be taught in elementary school, we shouldn't go around talking trash (even if we're justified in doing so).
• CommentRowNumber13.
• CommentAuthorUrs
• CommentTimeDec 12th 2009
• (edited Dec 12th 2009)
This comment is invalid XHTML+MathML+SVG; displaying source. <div> <blockquote> PS. I think there needs to be a page called nPOV now. </blockquote> <p>I gave it a try: <a href="http://ncatlab.org/nlab/show/nPOV">nPOV</a>.</p> <p>This is just a suggestion, using what came to mind on a Saturday afternoon. Feel free to edit to your heart's content.</p> </div>
• CommentRowNumber14.
• CommentAuthorHarry Gindi
• CommentTimeDec 12th 2009
I don't know how wise it is to take a confrontational stand like that, but that having been said, I think it's pretty damn funny.
• CommentRowNumber15.
• CommentAuthorTobyBartels
• CommentTimeDec 12th 2009

I wrote a comment, rather than an edit, for now.

• CommentRowNumber16.
• CommentAuthorUrs
• CommentTimeDec 13th 2009

I agree with your and Mike's comment. Would be great if you could work it into the text.

• CommentRowNumber17.
• CommentAuthorMike Shulman
• CommentTimeDec 14th 2009

I made an attempt at nPOV.

• CommentRowNumber18.
• CommentAuthorzskoda
• CommentTimeDec 14th 2009

I think that we also like to have multiple points of view, if I look back. When somebody disagrees guys, specially various elves try to find a way to accomodate that viewpoint as well (as long as it stands the elementary persuasion by counterreasoning).

• CommentRowNumber19.
• CommentAuthorHarry Gindi
• CommentTimeDec 14th 2009
• (edited Dec 14th 2009)
Some of us even hold multiple points of view and keep everything in line by means of cognitive dissonance!
• CommentRowNumber20.
• CommentAuthorzskoda
• CommentTimeDec 14th 2009

Sounds like Schoenberg's sonatas.

• CommentRowNumber21.
• CommentAuthorUrs
• CommentTimeDec 14th 2009
This comment is invalid XHTML+MathML+SVG; displaying source. <div> <blockquote> I made an attempt at nPOV. </blockquote> <p>I like that.</p> <p>Eventually the list of exaples deserves to be expanded. I imagine there is potential among the regular contributors to add to my attempt of indicating the power of category theory in geometry an example of its power in logic and foundations.</p> </div>
• CommentRowNumber22.
• CommentAuthorHarry Gindi
• CommentTimeDec 14th 2009
The chain rule and formation of tangent bundles is an example of functoriality and also a grothendieck fibration.
• CommentRowNumber23.
• CommentAuthorTobyBartels
• CommentTimeDec 14th 2009

Then put that in at nPOV!

• CommentRowNumber24.
• CommentAuthorzskoda
• CommentTimeDec 15th 2009
• (edited Dec 15th 2009)

I think that MacLane wrote a whole paper concentrating on category theory in differential geometry about almost half a century ago. Things like tangent bundles are among main examples. Or was it someone else ?

But in differential geometry things concentrate on functoriality and naturality of constructions. This is easy category theory. With groupoids, algebraoids and homotopy ideas, the things become much more sophisticated.

• CommentRowNumber25.
• CommentAuthorUrs
• CommentTimeDec 15th 2009
This comment is invalid XHTML+MathML+SVG; displaying source. <div> <blockquote> This is easy category theory. </blockquote> <p>Right, I think there is a distinction to be made between</p> <ul> <li><p>on the one hand just identifying a category, a functor, a natural transformation</p></li> <li><p>on the other applying category theory to that, usually by invoking universal constructions,</p></li> </ul> </div>
• CommentRowNumber26.
• CommentAuthorHarry Gindi
• CommentTimeDec 15th 2009
• (edited Dec 15th 2009)
I don't think Diff is even closed under pullbacks, is it?
• CommentRowNumber27.
• CommentAuthorUrs
• CommentTimeDec 15th 2009

Right, only under pullback along transversal maps.

This observation is usually the starting point for realizing that differential geometry is in need of a bit of category theory in the form of higher geometry.

In all notions of generalized smooth spaces all pullbacks do exist,.But they may still not be the "right" pullbacks. For instance cohomology of pullback objects may not have the expected properties. This is solved by passing to smooth derived stacks, such as derived smooth manifolds.

(I'll copy this remark to the entry nPOV now...)