The n-Category Café

Thanks John! Fascinating as always:

To do this, we take two sets of figures, say $X=G/H$ and $Y=G/K$ and find all the invariant relations between them: that is, subsets of $X\times Y$ preserved by all the symmetries. I’ll say more about this next time: we can use something called “double cosets”.

These are exactly the things which crop up when you consider morphisms between 2-representations of $G$. Can’t wait for next time.

Thanks go to Robin Houston and Lukas-Fabian Moser for creating PDF and DjVu versions of the English translation of Klein’s Erlangen program! These make it much easier to read.

Also thanks go to James Dolan for catching some (but undoubtedly far from all) errors in the first version of week249.

Namely: the morally correct group of symmetries in projective geometry is not $\mathrm{PSL}$ but $\mathrm{PGL}$.

And: it’s not true in Klein geometry that each subgroup $H$ of our symmetry group $G$ is ‘a figure’. I think it’s better to say that each set $X$ on which $G$ acts transitively is ‘a type of figure’. Each figure $x\in X$ has a stabilizer $H$, but often different figures will have the same stabilizer. Indeed, the stabilizer of $gx\in X$ is $gH{g}^{-1}$, and this can be the same as $H$ even when $gx$ is different from $x$. The point is that the normalizer of $H$ can be bigger than $H$!

Note also that different sets on which $G$ acts transitively can be isomorphic while still not equal — so we have a vital concept of “isomorphic” types of figures. In particular, the types of figures corresponding to $X=G/H$ and $Y=G/K$ are isomorphic if the subgroups $H$ and $K$ are conjugate in $G$. This issue was lurking around the edges near the end of week249.

I am not contributing much at the moment, but at least here I can help with spotting a typo in the amazingly typo-free TWFs:

somewhere towards the last third, in the discussion of transitive actions it currently says:

“So, $G$ acts transitively on $X$ on precisely when $X//G$ is equivalent to a group!”

One superfluous “on”.

Thanks for catching that typo, Urs.

And guess what? Nitin C. Rughoonauth has taken the marvelous step of putting Klein’s Erlangen program paper into LaTeX! You can see a PDF file of it here. If anyone catches typos — I see one already — please email me or notify me here, and I’ll fix them.

(I don’t have the LaTeX source code yet, but I’ll get it and make it available here. Eventually I’ll try to make sure this paper gets onto the arXiv.)

A very elementary introduction to this topic may be found at:

Is Beauty Truth and Truth Beauty?
How Keats’s famous line applies to math and science
By Martin Gardner
Book Review
Scientific American
April 2007

Image: ANDY ROUSE Corbis
SYMMETRY was once a synonym for beauty–recall William Blake’s “Tyger! Tyger! burning bright/In the forests of the night,/ What immortal hand or eye/Could frame thy fearful symmetry?”

WHY BEAUTY IS TRUTH: A HISTORY OF SYMMETRY
by Ian Stewart
Basic Books, 2007

[whoops, form input won’t accept the long URL, but it can be easily found at sciam.com]

Of course, both Ian Stewart and Martin Gardner have been leading popularizers of mathematics for a very long time.