n-categories: logic and geometry
The 2006 Reith Lectures are being aired at the moment. This year they're on the importance of music, and are written and delivered by the musician Daniel Barenboim. I've heard only the first so far, in which there's a wonderful passage where he explains how he came to understand in terms of a badly timed piece of music why the Oslo peace process would never work. The current domination of the visual over the aural, and the anaestheticisation of the later through exposure to muzak are other themes he touches on, and are to be developed in later lectures. In sum, we are letting down our children by failing to help them to understand and make serious music, and so learn to integrate discipline and passion.
This seems reminiscent of the teachings of Plato and Aristotle, who had plenty to say about the place of music in education. Later in a person's education they must meet mathematics, something Plato in particular stresses is vital for the formation of political leaders. I hope that a mathematician will be chosen as a future Reith lecturer to talk to us about the transformational role of mathematics. Perhaps John Baez would be a good choice. Recently he has been delivering some Lectures on n-categories and cohomology in Chicago. These notes include an appendix by Michael Shulman which moves the program on a little in an interesting direction. It's fascinating to see how logic and topology are so thoroughly interwoven in the n-categories setting. For something else on this see the end of Categorification as a Heuristic Device, where a link to modal logic is touched upon. (I know there's a mistake in that equation explaining permutations as sets of cycles.)
Here's a brief exchange I had with Baez in recent days:
Hi,
Here's a thought: gaining new insights by categorifying the very simplest entities seems a good way to bring in the punters. You've treated various kinds of number, natural, rational, etc. [E.g., From finite sets to Feynman diagrams] Now, if you ask anyone what else they first learned at school, they'll say 2-dimensional Euclidean geometry (if they're old enough to have been taught properly). Where, they may ask, are categorified lines and circles? If all interesting equations are lies, what of Pythagoras' theorem? Or, when we say the intersection of any pair of altitudes of a triangle is the *same* point as that of any other, is there room for weakening?
I suppose you might give two responses:
1) Euclidean geometry although it came first is actually a complicated affair. First you need to categorify a stripped down 'geometry' such as differential topology.
2) OK. 2d-Euclidean space is a homogeneous space, the points corresponding to cosets of the quotient of the Lie group of Euclidean transformations by the stabilizer of a point. All we need is a Lie 2-group version.
Either way what prevents a Erlangen program for 2-groups?
Best, David
Hi -
> Where, they may ask, are categorified lines and
> circles?
Interesting idea; one could take it in various directions, I suppose.
> 1) Euclidean geometry although it came first is actually a
> complicated affair. First you need to categorify a stripped
> down 'geometry' such as differential topology.
Mainly you need to see where the categories are, so you can see if there are interesting n-categories lurking beneath them.
> 2) OK. 2d-Euclidean space is a homogeneous space, the
> points corresponding to cosets of the quotient of the Lie
> group of Euclidean transformations by the stabilizer of a
> point. All we need is a Lie 2-group version.
> Either way what prevents a Erlangen program for 2-groups?
Nothing! Especially since the Erlangen program is just the flip side of Galois theory:
http://math.ucr.edu/home/baez/namboodiri/nam1.pdf
(see especially the slide about the icosahedron), and Galois theory has already been n-categorified to powerful and still growing effect.
But you're right - nobody seems to have thought hard about Klein geometry with Lie 2-groups (or higher) replacing Lie groups. Somehow people have skipped straight to categorifying principal bundles, even though principal bundles are a stripped-down way of thinking about Cartan geometries, which generalize Klein geometries! Sometimes ontogeny fails to recapitulate phylogeny. So, maybe the "punters" should be handed a nice specific Lie 2-group, some 2-spaces on which it acts, and be asked to study the "incidence relations" between these figures. Incidence geometry could be given a whole new lease on life!
Best, jb
In a pleasanter world I'd be funded to think longer about such things. For those with the leisure time, you can read about incidence geometry at TWF 178, which treats incidence relations in projective geometry in terms of the Dynkin diagrams An. Perhaps we should start with projective rather than Euclidean geometry. Is there an obvious candiate for a Lie 2-group one step up from SL(n, C)? What then is projective 2-geometry? What are Dynkin 2-diagrams?
Or doing things axiomatically, perhaps we can categorify the axioms of projective geometry, such as those for the projective plane, taken from week 145:
A) Given two distinct points, there exists a unique line that both points lie on.
B) Given two distinct lines, there exists a unique point that lies on both lines.
C) There exist four points, no three of which lie on the same line.
D) There exist four lines, no three of which have the same point lying on them.
This seems reminiscent of the teachings of Plato and Aristotle, who had plenty to say about the place of music in education. Later in a person's education they must meet mathematics, something Plato in particular stresses is vital for the formation of political leaders. I hope that a mathematician will be chosen as a future Reith lecturer to talk to us about the transformational role of mathematics. Perhaps John Baez would be a good choice. Recently he has been delivering some Lectures on n-categories and cohomology in Chicago. These notes include an appendix by Michael Shulman which moves the program on a little in an interesting direction. It's fascinating to see how logic and topology are so thoroughly interwoven in the n-categories setting. For something else on this see the end of Categorification as a Heuristic Device, where a link to modal logic is touched upon. (I know there's a mistake in that equation explaining permutations as sets of cycles.)
Here's a brief exchange I had with Baez in recent days:
Hi,
Here's a thought: gaining new insights by categorifying the very simplest entities seems a good way to bring in the punters. You've treated various kinds of number, natural, rational, etc. [E.g., From finite sets to Feynman diagrams] Now, if you ask anyone what else they first learned at school, they'll say 2-dimensional Euclidean geometry (if they're old enough to have been taught properly). Where, they may ask, are categorified lines and circles? If all interesting equations are lies, what of Pythagoras' theorem? Or, when we say the intersection of any pair of altitudes of a triangle is the *same* point as that of any other, is there room for weakening?
I suppose you might give two responses:
1) Euclidean geometry although it came first is actually a complicated affair. First you need to categorify a stripped down 'geometry' such as differential topology.
2) OK. 2d-Euclidean space is a homogeneous space, the points corresponding to cosets of the quotient of the Lie group of Euclidean transformations by the stabilizer of a point. All we need is a Lie 2-group version.
Either way what prevents a Erlangen program for 2-groups?
Best, David
Hi -
> Where, they may ask, are categorified lines and
> circles?
Interesting idea; one could take it in various directions, I suppose.
> 1) Euclidean geometry although it came first is actually a
> complicated affair. First you need to categorify a stripped
> down 'geometry' such as differential topology.
Mainly you need to see where the categories are, so you can see if there are interesting n-categories lurking beneath them.
> 2) OK. 2d-Euclidean space is a homogeneous space, the
> points corresponding to cosets of the quotient of the Lie
> group of Euclidean transformations by the stabilizer of a
> point. All we need is a Lie 2-group version.
> Either way what prevents a Erlangen program for 2-groups?
Nothing! Especially since the Erlangen program is just the flip side of Galois theory:
http://math.ucr.edu/home/baez/namboodiri/nam1.pdf
(see especially the slide about the icosahedron), and Galois theory has already been n-categorified to powerful and still growing effect.
But you're right - nobody seems to have thought hard about Klein geometry with Lie 2-groups (or higher) replacing Lie groups. Somehow people have skipped straight to categorifying principal bundles, even though principal bundles are a stripped-down way of thinking about Cartan geometries, which generalize Klein geometries! Sometimes ontogeny fails to recapitulate phylogeny. So, maybe the "punters" should be handed a nice specific Lie 2-group, some 2-spaces on which it acts, and be asked to study the "incidence relations" between these figures. Incidence geometry could be given a whole new lease on life!
Best, jb
In a pleasanter world I'd be funded to think longer about such things. For those with the leisure time, you can read about incidence geometry at TWF 178, which treats incidence relations in projective geometry in terms of the Dynkin diagrams An. Perhaps we should start with projective rather than Euclidean geometry. Is there an obvious candiate for a Lie 2-group one step up from SL(n, C)? What then is projective 2-geometry? What are Dynkin 2-diagrams?
Or doing things axiomatically, perhaps we can categorify the axioms of projective geometry, such as those for the projective plane, taken from week 145:
A) Given two distinct points, there exists a unique line that both points lie on.
B) Given two distinct lines, there exists a unique point that lies on both lines.
C) There exist four points, no three of which lie on the same line.
D) There exist four lines, no three of which have the same point lying on them.
2 Comments:
David asks (or rather, has a hypothetical character ask) what categorified lines and circles are. John points out that this could be taken in various directions. Here's a possible beginning of an answer.
We have to decide which aspects of lines and circles we're interested in. Let's treat them as metric spaces. Then the "categorified circle" should be a categorified metric space. OK, so what's a categorified metric space?
Here we can follow Lawvere, who's done a lot to develop the thesis that everything is a category. (I'm exaggerating, but see the first page proper of his metric spaces paper.) If a thing can be regarded some kind of category, that increases our chances of being able to perform some useful sort of categorification. This is ironic, as Lawvere seems to disapprove of categorification...
Anyway, Lawvere interprets metric spaces as being categories enriched in R, the poset of non-negative reals ordered by >=, made monoidal by its additive structure. So it looks as if our task is to categorify R - for then a categorified metric space could be defined as a category (weakly) enriched in the categorified R.
As far as I know, there's no really compelling answer yet to "what is the categorification of the reals?"
Could you also recapitulate Descartes' coordinate-based approach to geometry? With categorified reals you could carve out subcategories of R^2 in terms of isomorphisms such as X^2 + Y^2 is isomorphic to 1. Seems like you wouldn't be too far from Joyal's species.
Doing things the Kleinian way, my conversation with Baez proceeded:
> Is there an obvious candidate
>for a nice specific Lie 2-group?
I list a bunch of nice 2-groups in HDA5, and if I were going to categorify Klein geometry I would I would look at a bunch in parallel. The fun of course is seeing the effect and significance of the 2-morphisms (sorry, now I'm thinking of a 2-group as a 1-object 2-groupoid). We've got the ordinary groups, with no 2-morphisms to speak of; then we've got the 2-groups with only one morphism and an abelian group of 2-morphisms.In between these extremes, and more interesting, are groups built from a group G acting on an abelian group A; a great example is the "Euclidean 2-group" where G = SO(n), A = R^n. Or, it might be nice to let G be the whole Euclidean group and A some abelian group on which it acts: this would decategorify to the Euclidean group.
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