# March 28, 2007 {#week248}

This week I'll continue the Tale of Groupoidification. But first: some
new views of the Sun!

Here's a cool movie of the Moon passing in front of the Sun, as viewed
from the "STEREO B" spacecraft. Click on it:

2) Astronomy Picture of the Day, March 3 2007, "Lunar transit from STEREO", `http://antwrp.gsfc.nasa.gov/apod/ap070303.html`

As the name hints, there's a pair of STEREO satellites in orbit around
the Sun. One is leading the Earth a little, the other lagging behind a
bit, to provide a stereoscopic view of coronal mass ejections.

What's a "coronal mass ejection"? It's an event where the Sun shoots
off a blob of ionized gas --- billions of tons of it --- at speeds around
1000 kilometers per second!

That sounds cataclysmic... but it happens between once a day and 5-6
times a day, depending on where we are in the 11-year solar cycle, also
known as the "sunspot cycle". Right now we're near the minimum of
this cycle. Near the maximum, coronal mass ejections can really screw up
communication systems here on Earth. For example, in 1998 a big one
seems to have knocked out a communication satellite called Galaxy 4,
causing 45 million people in the US to lose their telephone pager
service:

3) Gordon Holman and Sarah Benedict, "Solar Flare Theory: Coronal mass ejections, solar flares, and the Earth-Sun connection", `http://www.agu.org/sci_soc/articles/eisbaker.html`

So, it's not only fun but also practical to understand coronal mass
ejections. Here's a movie of one taken by the Solar and Heliospheric
observatory (SOHO):
$$\includegraphics[max width=0.65\linewidth]{../images/coronal_mass_ejection.png}$$

4) NASA, "Cannibal coronal mass ejections", `http://science.nasa.gov/headlines/y2001/ast27mar_1.htm`

As I mentioned in ["Week 150"](#week150), SOHO is a satellite
orbiting the Sun right in front of the Earth, at an unstable equilibrium
--- a "Lagrange point" --- called L1. SOHO is bristling with detectors and
telescopes of all sorts, and this movie was taken by a coronagraph,
which is a telescope specially designed to block out the Sun's disk and
see the fainter corona.

If a coronal mass ejection hits the Earth, it does something like this:

5) NASA, "What is a CME?", `http://www.nasa.gov/mpg/111836main_what_is_a_cme_NASA%20WebV_1.mpg`

In this artist's depiction you can see the plasma shoot off from the
Sun, hit the Earth's magnetic field --- this actually takes one to five
days --- and squash it, pushing field lines around to the back side of the
Earth. When the magnetic field lines
[reconnect](http://en.wikipedia.org/wiki/Magnetic_reconnection) in back,
trillions of watts of power come cascading down through the upper
atmosphere, producing auroras. Here's a nice movie of what *those* can
look like:

6) YouTube, "Aurora (Northern Lights)", `http://www.youtube.com/watch?v=qIXs6Sh0DKs`

I wish I understood this magnetic field line trickery better!
Magnetohydrodynamics --- the interactions between electromagnetic fields
and plasma --- is a branch of physics that always gave me the shivers. The
Navier-Stokes equations describing fluid flow are bad enough --- if you
can prove they have solutions, you'll win \$1,000,000 from the Clay
Mathematics Institute. Throw in Maxwell's equations and you get a real
witches' brew of strange phenomena.

In fact, this subject is puzzling even to experts. For example, why is
the Sun's upper atmosphere --- the corona --- so hot? Here's a picture of
the Sun in X-rays taken by another satellite:
$$\href{http://trace.lmsal.com/POD/TRACEpodarchive26.html}{\includegraphics[max width=0.65\linewidth]{../images/solar_corona_TRACE.jpg}}$$

7) Transition Region and Coronal Explorer (TRACE), "Images of the sun", `http://trace.lmsal.com/POD/TRACEpodarchive26.html`

This lets you see plasma in the corona with temperatures between 1
million kelvin (shown as blue) and 2 million kelvin (red). By
comparison, the visible surface of the Sun is a mere 5800 kelvin!

Where does the energy come from to heat the corona? There are lots of
competing theories. It could even be due to "magnetic field
reconnection", the same topological phenomenon that triggers auroras
when coronal mass ejections smash into the Earth's magnetic field, as
in that movie above. For more, try this:

8) Andrew L. Haynes, Clare E. Parnell, Klaus Galsgaard and Eric R. Priest, "Magnetohydrodynamic evolution of magnetic skeletons", _Proc. Roy. Soc. Lond. A_ **463** (2007) 1097--1115. Also available as [`astro-ph/0702604`](http://arxiv.org/abs/astro-ph/0702604).

A new satellite called Hinode is getting a good look at what's going
on, and it seems the magnetic field on the Sun's surface is much more
dynamic than before thought:
$$\href{http://www.nasa.gov/mission_pages/solar-b/}{\includegraphics[max width=0.65\linewidth]{../images/solar_flare_hinode.jpg}}$$

9) NASA, "Hinode: investigating the Sun's magnetic field", `http://www.nasa.gov/mission_pages/solar-b/`

In fact, weather on the Sun may be more complex than on the Earth.
There's "rain" when plasma from the corona cools and falls back down
to the Sun's surface... and sometimes there are even tornados! You
think tornados on Earth are scary? Check out this movie made during an
8-hour period in August 2000, near the height of the solar cycle:

10) TRACE, "Tornados and fountains in a filament on 2 Aug. 2000", movie 13, `http://trace.lmsal.com/POD/`

Besides the tornados, near the end you can see glowing filaments of
plasma following magnetic field lines!

Now for something simpler: the Tale of Groupoidification.

I don't want this to be accessible only to experts, since a bunch of it
is so wonderfully elementary. So, I'm going to proceed rather slowly.
This may make the experts impatient, so near the end I'll zip ahead and
sketch out a bit of the big picture.

Last time I introduced spans of sets. A span of sets is just a set $S$
equipped with functions to $X$ and $Y$:
$$
  \begin{tikzcd}
    &S\drar["F"]\dlar[swap,"G"]&
  \\X&&Y
  \end{tikzcd}
$$
Simple! But the important thing is to understand this thing as a
"witnessed relation".

Have you heard how computer scientists use the term "witness"? They
say the number 17 is a "witness" to the fact that the number 221
isn't prime, since 17 evenly divides 221.

That's the idea here. Given a span $S$ as above, we can say an element $x$
of $X$ and an element $y$ of $Y$ are "related" if there's an element $s$ of $S$
with
$$F(s) = x \quad\mbox{and}\quad G(s) = y$$
The element $s$ is a "witness" to the relation.

Last week, I gave an example where a Frenchman $x$ and an Englishwoman $y$
were related if they were both the favorites of some Russian $s$.

Note: there's more information in the span than the relation it
determines. The relation either holds or fails to hold. The span does
more: it provides a set of "witnesses". The relation holds if this set
of witnesses is nonempty, and fails to hold if it's empty.

At least, that's how mathematicians think. When I got married last
month, I discovered the state of California demands *two* witnesses
attend the ceremony and sign the application for a marriage license.
Here the relation is "being married", and the witnesses attest to that
relation --- but for the state, one witness is not enough to prove that
the relation holds! They're using a more cautious form of logic.

To get the really interesting math to show up, we need to look at other
examples of "witnessed relations" --- not involving Russians or
marriages, but geometry and symmetry.

For example, suppose we're doing $3$-dimensional geometry. There's a
relation "the point $x$ and the line $y$ lie on a plane", but it's pretty
dull, since it's always true. More interesting is the witnessed
relation "the point $x$ and the line $y$ lie on the plane $z$". The reason
is that sometimes there will be just *one* plane containing a point and
a line, but when the point lies on the line, there will be *lots*.

To think of this "witnessed relation" as a span
$$
  \begin{tikzcd}
    &S\drar["F"]\dlar[swap,"G"]&
  \\X&&Y
  \end{tikzcd}
$$
we can take $X$ to be the set of points and $Y$ to be the set of lines.

Can we take $S$ to be the set of planes? No! Then there would be no way to
define the functions $F$ and $G$, because the same plane contains lots of
different points and lines. So, we should take $S$ to be the set of
triples $(x,y,z)$ where $x$ is a point, $y$ is a line, and $z$ is a plane
containing $x$ and $y$. Then we can take
$$F(x,y,z) = x$$
and
$$G(x,y,z) = y$$
A "witness" to the fact that $x$ and $y$ lie on a plane is not just a
plane containing them, but the entire triple.

(If you're really paying attention, you'll have noticed that we need
to play the same trick in the example of witnesses to a marriage.)

Spans like this play a big role in "incidence geometry". There are
lots of flavors of incidence geometry, with "projective geometry"
being the most famous. But, a common feature is that we always have
various kinds of "figures" --- like points, lines, planes, and so on.
And, we have various kinds of "incidence relations" involving these
figures. But to really understand incidence geometry, we need to go
beyond relations and use spans of sets.

Actually, we need to go beyond spans of sets and use spans of groupoids!
The reason is that incidence geometries usually have interesting
symmetries, and a groupoid is like a "set with symmetries". For
example, consider lines in $3$-dimensional space. These form a set, but
there are also symmetries of $3$-dimensional space mapping one line to
another. To take these into account we need a richer structure: a
groupoid!

Here's the formal definition: a groupoid consists of a set of
"objects", and for any objects $x$ and $y$, a set of "morphisms"
$$f\colon x \to y$$
which we think of as symmetries taking $x$ to $y$. We can compose a morphism
$f\colon x \to y$ and a morphism $g\colon y \to z$ to get a morphism $fg\colon x \to z$. We think
of $fg$ as the result of doing first $f$ and then $g$. So, we demand the
associative law
$$(fg)h = f(gh)$$
whenever either side is well-defined. We also demand that every object $x$
has an identity morphism
$$1_x\colon x \to x$$
We think of this as the symmetry that doesn't do anything to $x$. So,
given any morphism $f\colon x \to y$, we demand that
$$f 1_y = f = 1_x f$$
So far this is the definition of a "category". What makes it a
"groupoid" is that every morphism $f\colon x \to y$ has an "inverse"
$$f^{-1}\colon y \to x$$
with the property that
$$f f^{-1} = 1_x$$
and
$$f^{-1} f = 1_y$$
In other words, we can "undo" any symmetry.

So, in our spans from incidence geometry:
$$
  \begin{tikzcd}
    &S\drar["F"]\dlar[swap,"G"]&
  \\X&&Y
  \end{tikzcd}
$$
$X$, $Y$ and $S$ will be groupoids, while $F$ and $G$ will be maps between
groupoids: that is, "functors"!

What's a functor? Given groupoids $A$ and $B$, clearly a functor
$$F\colon A \to B$$
should send any object $x$ in $A$ to an object $F(x)$ in $B$. But also, it
should send any morphism in $A$:
$$f\colon x \to y$$
to a morphism in $B$:
$$F(f)\colon F(x) \to F(y)$$
And, it should preserve all the structure that a groupoid has, namely
composition:
$$F(fg) = F(f) F(g)$$
and identities:
$$F(1_x) = 1_{F(x)}$$
It then automatically preserves inverses too:
$$F(f^{-1}) = F(f)^{-1}$$
Given this, what's the meaning of a span of groupoids? You could say
it's a "invariant" witnessed relation --- that is, a relation with
witnesses that's *preserved* by the symmetries at hand. These are the
very essence of incidence geometry. For example, if we have a point and
a line lying on a plane, we can rotate the whole picture and get a new
point and a new line lying on a new plane. Indeed, a "symmetry" in
incidence geometry is precisely something that preserves all such
"incidence relations".

For those of you not comfy with groupoids, let's see how this actually
works. Suppose we have a span of groupoids:
$$
  \begin{tikzcd}
    &S\drar["F"]\dlar[swap,"G"]&
  \\X&&Y
  \end{tikzcd}
$$
and the object $s$ is a witness to the fact that $x$ and $y$ are related:
$$F(s) = x \quad\mbox{and}\quad G(s) = y$$
Also suppose we have a symmetry sending $s$ to some other object of $S$:
$$f\colon s \to s'$$
This gives morphisms
$$F(f)\colon F(s) \to F(s')$$
in $X$ and
$$G(f)\colon G(s) \to G(s')$$
in $Y$. And if we define
$$F(s') = x' \quad\mbox{and}\quad G(s') = y'$$
we see that $s'$ is a witness to the fact that $x'$ and $y'$ are related.

Let me summarize the Tale so far:

- Spans of groupoids describe "invariant witnessed relations".
- Invariant witnessed relations are the essence of incidence geometry.
- There's a way to turn spans of groupoids into matrices of numbers,
    so that multiplying matrices corresponds to some nice way of
    "composing" spans of groupoids (which I haven't really explained
    yet).

From all this, you should begin to vaguely see that starting from any
sort of incidence geometry, we should be able to get a bunch of
matrices. Facts about incidence geometry will give facts about linear
algebra!

"Groupoidification" is an attempt to reverse-engineer this process. We
will discover that lots of famous facts about linear algebra are
secretly facts about incidence geometry!

To prepare for what's to come, the maniacally diligent reader might
like to review ["Week 178"](#week178), ["Week 180"](#week180),
["Week 181"](#week181), ["Week 186"](#week186) and
["Week 187"](#week187), where I explained how any Dynkin diagram
gives rise to a flavor of incidence geometry. For example, the
simplest-looking Dynkin diagrams, the $\mathrm{A}_n$ series, like this for $n = 3$:
$$
  \begin{tikzpicture}
    \node[label=below:{points}] at (0,0) {$\bullet$};
    \node[label=below:{lines}] at (2,0) {$\bullet$};
    \node[label=below:{planes}] at (4,0) {$\bullet$};
    \draw[thick] (0,0) to (4,0);
  \end{tikzpicture}
$$
give rise to $n$-dimensional projective geometry. I may have to review
this stuff, but first I'll probably say a bit about the theory of group
representations and Hecke algebras.

(There will also be other ways to get spans of groupoids, that don't
quite fit into what's customarily called "incidence geometry", but
still fit very nicely into our Tale. For example, Dynkin diagrams become
"quivers" when we give each edge a direction, and the "groupoid of
representations of a quiver" gives rise to linear-algebraic structures
related to a quantum group. In fact, I already mentioned this in item E
of ["Week 230"](#week230). Eventually this will let us groupoidify
the whole theory of quantum groups! But, I don't want to rush into
that, since it makes more sense when put in the right context.)

By the way, some of you have already pointed out how unfortunate it is
that *last* Week was devoted to $\mathrm{E}_8$, instead of *this* one. Sorry.

------------------------------------------------------------------------

**Addendum:** I thank logopetria for catching typos. For more
discussion, go to the
[$n$-Category Café](http://golem.ph.utexas.edu/category/2007/03/this_weeks_finds_in_mathematic_9.html).

------------------------------------------------------------------------

> *Science is the only news. When you scan through a newspaper or
magazine, all the human interest stuff is the same old he-said-she-said,
the politics and economics the same sorry cyclic dramas, the fashions a
pathetic illusion of newness, and even the technology is predictable if
you know the science.*
> 
> --- Stewart Brand