# May 27, 2007 {#week252}

Today I want to tell you about the electromagnetic snake at the center
of our Galaxy, and continue the Tale of Groupoidification.

But first, the long-range weather forecast. There's a 40% chance of
rain on Neptune in 8 billion years! More precisely, that's the chance
these authors give for the formation of an ocean on Neptune when its
interior cools down enough:

1) Sloane J. Wiktorowica and Andrew P. Ingersoll, "Liquid water oceans in ice giants", available as [`astro-ph/0609723`](https://arxiv.org/abs/astro-ph/0609723).

Right now, even though Neptune is named after the Roman god of seas and
has a nice blue appearance:
$$\href{http://en.wikipedia.org/wiki/Neptune}{\includegraphics[max width=0.65\linewidth]{../images/neptune.jpg}}$$
it's mighty dry --- at least on top. Spectroscopy detects no water at all
in its upper atmosphere! But that's consistent with the presence of
water down below, since water vapor is a lot heavier than hydrogen and
helium, which make up most of Neptune's upper atmosphere, and the pull
of gravity there is mighty fierce.

In fact, scientists believe Neptune has a core of molten metal and rock
surrounded by more rock, methane ice, ammonia ice, and water ice --- all
solid due to high pressures. The overall density of Neptune makes sense
if there's a lot of water down there. But, Wiktorowica and Ingersoll
argue that the planet can't have an ocean of liquid water.

Surprisingly, this is because Neptune is too *hot* --- even though its
upper atmosphere is a chilly 50 kelvin! I think their argument goes
roughly like this, though I don't understand it well. If you fell down
through the atmosphere of Neptune, you'd find that it gets hotter and
hotter as you go down, and moister and moister too --- but always too hot,
given the amount of moisture, for liquid water to be more stable than
water vapor.

However, they also indulge in some predictions about the far future!

First let's set the stage:

- In 1.1 billion years the Sun will become 10% brighter than now, and
    the Earth's atmosphere will dry out.
- In 3 billion years the Andromeda Galaxy will collide with our
    galaxy. Many solar systems will be destroyed.
- In 3.5 billion years the Sun will become 40% brighter than today. If
    the Earth is still orbiting the sun, its oceans will evaporate.
- In 5.4 billion years from now the Sun's core will run out of
    hydrogen. It will enter its first red giant phase, becoming 1.6
    times bigger and 2.2 times brighter than today.
- In 6.5 billion years from now the Sun will become a full-fledged red
    giant, 170 times bigger and 2400 times brighter than today. The
    Republican Party will finally admit the existence of global warming,
    but point out that it's not human-caused.
- In 6.7 billion years from now the Sun will start fusing helium and
    shrink back down to 10 times bigger and 40 times brighter than
    today.
- In 6.8 billion years from now the Sun will run out of helium. Being
    too small to start fusing carbon and oxygen, it'll enter a second
    red giant phase, growing 180 times bigger and 3000 times brighter
    than today.

But then, about 6.9 billion years from now, the Sun will start
pulsating, ejecting half of its mass in the form of solar wind! It'll
become what they call a "planetary nebula". Eventually only its inner
core will be left. In ["Week 223"](#week223) I quoted Bruce
Balick's eloquent description:

> The remnant Sun will rise as a dot of intense light, no larger than
> Venus, more brilliant than 100 present Suns, and an intensely hot
> blue-white color hotter than any welder's torch. Light from the
> fiendish blue "pinprick" will braise the Earth and tear apart its
> surface molecules and atoms. A new but very thin "atmosphere" of
> free electrons will form as the Earth's surface turns to dust.

This is where Wiktorovika and Ingersoll *begin* their story. So far,
Neptune will have warmed up a lot --- assuming for the sake of argument
that it wasn't thrown out of the Solar System when the Milky Way hit
Andromeda. But when the Sun loses mass, Neptune will either collide with
Uranus, be ejected from the Solar System, or assume a stable orbit about
twice its current size.

In the latter two scenarios, Neptune will chill out, at least when the
remnant Sun cools down and becomes a white dwarf. When the surface
temperature of Neptune reaches 30 kelvin, they estimate it has a 41.5%
plus or minus 4.2% chance of forming oceans.

So, they say: "Billions of years from now, after the Sun has gone,
Neptune may therefore become the only object in the Solar system with
liquid water oceans".

This sounds nice --- but don't buy your beachfront property there yet.

First, they don't study how the atmospheric composition of Neptune may
change when the Sun gets 3000 times brighter than now! Maybe this will
*help* Neptune form oceans. After all, light gases like molecular
hydrogen and helium, which dominate Neptune's upper atmosphere now, are
more likely than water vapor to be driven off into outer space when it
gets hot. But, they don't even check to see if Neptune will have *any*
atmosphere left after this era.

Second, the estimate of a "41.5% plus or minus 4.2% chance" seems
strangely precise, given the uncertainties involved. The error bars
should probably have big error bars themselves!

Third, it's worth admitting that the atmosphere of Neptune is a bit
mysterious. For example, nobody knows why it's bright blue. It's
probably because of methane --- but Uranus also has methane in its
atmosphere, and it's not as blue:
$$\href{http://en.wikipedia.org/wiki/Uranus}{\includegraphics[max width=0.65\linewidth]{../images/uranus.jpg}}$$
Also, despite being the coldest planet in our Solar System, Neptune has
the fastest winds: up to 2100 kilometers per hour, almost supersonic!
Nobody knows what powers these winds.

When the Voyager 2 spacecraft flew by Neptune in 1992, it saw a storm
system the size of Eurasia, which was dubbed "The Great Dark Spot":
$$\href{http://en.wikipedia.org/wiki/Great_Dark_Spot}{\includegraphics[max width=0.65\linewidth]{../images/neptune_great_dark_spot.jpg}}$$
It seemed to resemble the Great Red Spot on Jupiter, which has been
around for at least 177 years. But when the Hubble Space Telescope took
another good look at Neptune in 1994, the Great Dark Spot was gone!
Meanwhile, other storms had formed.

So, the weather on Neptune is dynamic and poorly understood. Doing
forecasts for the next 8 billion years seems pretty risky... though
fun.

Of course, liquid water oceans are nice if you're looking for life. And
while there probably isn't life on Neptune, there could be life on
similar planets in other solar systems. So far people have found 233 of
these "exoplanets", most of them heavier than Jupiter but close to
their suns --- because such planets are the easiest to detect by how they
pull on their sun.

Here's a chart showing the masses and orbital radii of known exoplanets
as of 2004:
$$\href{http://en.wikipedia.org/wiki/Extrasolar_planet}{\includegraphics[max width=0.65\linewidth]{../images/exoplanets.jpg}}$$

2) P. R. Lawson, S. C. Unwin, and C. A. Beichman, "Precursor Science for the Terrestrial Planet Finder", _JPL Pub_ **04-014**, Oct. 2004, page 21, fig. 5. Chart at `http://en.wikipedia.org/wiki/Image:Extrasolar_Planets_2004-08-31.png`. Report at `http://planetquest.jpl.nasa.gov/documents/RdMp272.pdf`

For comparison, the letters V, E, M, J, S, U, N, stand for planets in
our Solar System, not counting Mercury or the subsequently dethroned
Pluto.

As you'll see, there are lots of "hot Jupiters" --- planets as big as
Jupiter, or even up to 200 times heavier, but closer to their sun than
we are to ours. Recently a postdoc at UCLA found evidence for water in
the atmosphere of one of these planets:

3) Travis Barman, "Identification of absorption features in an extrasolar planet atmosphere", available as [`arxiv:0704.1114`](http://www.arxiv.org/abs/0704.1114).

This planet is only 7 million kilometers away from its yellow-white Sun,
much closer than Mercury is to ours. Its year is only 3.5 of our days!
It's bigger in size than Jupiter, but only 0.7 times as heavy. Its
surface temperature must be about 1000 kelvin. That's one really hot
Jupiter.

People have also been finding "hot Neptunes". In fact, I read about a
nice one in the newspaper while writing this! It's called Gliese 436 b.
It's the size of Neptune and it's orbiting the red dwarf star Gliese
436, which is 33 light-years from Earth. This star is only 1% as bright
as our Sun --- but the planet is so close that its year lasts less than
three of our days! So, its surface temperature is high: higher than the
melting point of lead.

However, because this planet is so big, the pressure down below can
still make water into a solid. In fact, its density suggests that it's
mainly made of ice!

I think this is the paper that triggered the newspaper reports:

4) M. Gillon et al, "Detection of transits of the nearby hot Neptune GJ 436 b", available as [`arxiv:0705.2219`](http://www.arxiv.org/abs/0705.2219).

It seems hot Neptunes like this could have started as hot Jupiters and
then lost a lot of their atmosphere:

5) I. Baraffe, G. Chabrier, T. S. Barman, F. Selsis, F. Allard, and P. H. Hauschildt, "Hot-Jupiters and hot-Neptunes, a common origin?", available at [`astro-ph/0505054`](http://www.arxiv.org/abs/astro-ph/0505054).

There could also be cold Neptunes, perhaps with liquid water oceans. We
haven't seen these yet, but they'd be hard to see. So, while
Wiktorowica and Ingersoll's paper doesn't convince me about the future
of *our* Neptune, it raises some interesting possibilities.

Next: the snake at the center of our galaxy!

Gregory Benford is best known for his science fiction, which spans the
galaxy, but he's also an astrophysicist at U. C. Irvine. Recently he
spent a week at my school, U. C. Riverside, which has one of the
world's best SF libraries: the [Eaton Collection](http://eaton-collection.ucr.edu/). Since my wife is involved
in the SF program at the comparative literature department here, he had
dinner at our place one night. The conversation drifted all over the
place, with a heavy focus on fruit flies that have been bred to live
twice as long as usual. But when I asked him about his research, he said
he'd written some papers about enormous glowing filaments near the
center of the Milky Way.

I hadn't known about these! He said the biggest one could be a million
years old, perhaps formed by some energetic event, maybe a star falling
into the central black hole. Here's an expository paper he wrote about
it:

6) Gregory Benford, "The electromagnetic snake at the galactic center", `http://www.ps.uci.edu/physics/news3/benford.html`

I'll just quote a little:

> Five years ago radio astronomy revealed the oddest and longest
> filament yet discovered at our galactic center: a uniquely kinked
> structure about 150 light years long and two to three light years wide
> --- the Snake. Its large kinks are its brightest parts. There is
> energetic activity at one end and a supernova bubble at the other,
> which the Snake appears to penetrate unharmed.
>
> How does nature form stable, long-lived magnetic structures which
> display considerable polarization (about 60% at 10.55 GHz in the
> Snake)? In 1988 I had modeled others of the dozens of filaments seen
> uniquely at the galactic center in terms of an electrodynamic view, in
> which currents set up coherent magnetic pinches. Such self-organizing
> filaments can exist in laboratory plasmas for long times; the galactic
> ones could be at least a million years old, as estimated by the time
> that shear forces would disrupt them.
>
> The electrodynamic view uses pinch forces of currents to form
> filaments, driven by the $E = v \times B$ of conducting molecular clouds
> moving across a strong milliGauss ambient, ordered field. A return
> current must then flow at larger radii, making a closed loop which has
> a springy flexibility, able to withstand the turbulent velocity fields
> known near the galactic center. The picture then anticipates that
> aberrant molecular clouds, moving contrary to the general galactic
> rotation, should accompany each filament. This prediction has held up
> as more filaments were found.

I'd like to learn more about these!

Back in ["Week 248"](#week248), I mentioned some of the complex
things that electromagnetic fields and plasma do in the Sun. The center
of the Galaxy is another place where electromagnetohydrodynamics runs
rampant. There's a black hole there, of course, but also these
filaments, and a fairly strong magnetic field that contains about
$4\times10^{47}$ joules of energy within about 300 light years of the galactic
center. By comparison, a supernova emits a mere $10^{44}$ joules. So,
there's a lot of energy around....

The big picture here, created by Farhad Yusef-Zadeh and collaborators,
shows the galactic center quite nicely, as viewed in radio frequencies:
$$\href{http://www.nrao.edu/pr/2004/filaments/}{\includegraphics[max width=0.65\linewidth]{../images/galactic_center.jpg}}$$

7) National Radio Astronomy Observatory, "Origin of enigmatic galactic-center filaments revealed", `http://www.nrao.edu/pr/2004/filaments/`

You can see some supernova remnants (SNRs), the region Sgr A which
contains the black hole at the galactic center, various nonthermal radio
filaments (NRFs), and the Snake. Some of these filaments come from
regions where stars are forming... that could be important.

But, this article discusses another piece of the puzzle --- the possible
role of turbulence in winding up the galactic magnetic field:

8) Stanislav Boldyrev and Farhad Yusef-Zadeh, "Turbulent origin of the galactic-center magnetic field: nonthermal radio filaments", available as [`astro-ph/0512373`](https://arxiv.org/abs/astro-ph/0512373).

It's a complicated stew. I don't hope to understand it, just admire
it.

And finally: the Tale of Groupoidification! In
["Week 250"](#week250) we reached the point of seeing how spans of
groupoids over a fixed group $G$ subsume the theory of $G$-sets and
invariant relations between these --- which are traditionally studied
using "double cosets".

There is a lot more we could say about this. But, our most urgent goal
is to see how spans of groupoids act like twice categorified matrices ---
matrices whose entries are not just *numbers*, and not just *sets*, but
*groupoids*! This will expose the secret combinatorial underpinnings of
a lot of fancy linear algebra. Once we've categorified linear algebra
this way, we'll be in a great position to tackle fashionable topics
like categorified quantum groups, invariants of higher-dimensional
knots, and the like.

But, we should restrain ourselves from charging ahead too fast!
Everything will hang together better if we lay the groundwork properly.
For this, it pays to re-examine the history of mathematics a bit. If
we're trying to understand linear algebra using groupoids, it pays to
ask: how did people connect linear algebra and group theory in the first
place?

This book is very helpful:

9) Charles W. Curtis, _Pioneers of Representation Theory: Frobenius, Burnside, Schur and Brauer_, History of Mathematics vol. **15**, AMS, Providence, Rhode Island, 1999.

Back in 1897, a mathematician named William Burnside wrote the first
book in English on finite groups. It was called Theory of Groups of
Finite Order.
$$\href{http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Burnside.html}{\includegraphics[max width=0.65\linewidth]{../images/burnside.jpg}}$$
In the preface, Burnside explained why he studied finite groups by
letting them act as permutations of sets, but not as linear
transformations of vector spaces:

> Cayley's dictum that "a group is defined by means of the laws of
> combination of its symbols" would imply that, in dealing with the
> theory of groups, no more concrete mode of representation should be
> used than is absolutely necessary. It may then be asked why, in a book
> that professes to leave all applications to one side, a considerable
> space is devoted to substitution groups \[permutation groups\], but
> other particular modes of representation, such as groups of linear
> transformations, are not even referred to. My answer to this question
> is that while, in the present state of our knowledge, many results in
> the pure theory are arrived at most readily by dealing with properties
> of substitution groups, it would be difficult to find a result that
> could most directly be obtained by the consideration of groups of
> linear transformations.

In short, he didn't see the point of representing groups on vector
spaces --- at least as a tool in the "pure" theory of finite groups, as
opposed to their applications.

However, within months after this book was published, he discovered the
work of Georg Frobenius, who used linear algebra very effectively to
study groups!
$$\href{http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Frobenius.html}{\includegraphics[max width=0.65\linewidth]{../images/frobenius.jpg}}$$
So, Burnside started using linear algebra in his own work on finite
groups, and by the time he wrote the second edition of his book in 1911,
he'd changed his tune completely:

> Very considerable advances in the theory of groups of finite order
> have been made since the appearance of the first edition of this book.
> In particular the theory of groups of linear substitutions has been
> the subject of numerous and important investigations by several
> writers; and the reason given in the original preface for omitting any
> account of it no longer holds good. In fact it is now more true to say
> that for further advances in the abstract theory one must look largely
> to the representation of a group as a group of linear transformations.

It's interesting to see exactly how representing finite groups on
vector spaces lets us understand them better. By now almost everyone
agrees this is true. But how much of the detailed machinery of linear
algebra is really needed? How much we could do purely combinatorially,
using just spans of groupoids?

I don't really know the full answer to this question. But, it quickly
leads us into the fascinating theory of "Hecke operators", which will
play a big role in the Tale of Groupoidification. So, let's pursue it a
bit.

Suppose we have two guys, William and Georg, who are studying a finite
group $G$.

William says, "I'm studying how $G$ acts on sets."

Georg replies, "I'm studying how it acts on complex vector spaces, as
linear transformations. Mere sets are feeble entities. I can do anything
you can do --- but I have the tools of linear algebra to help me!"

William says, "But, you're paying a moral price. You're getting the
complex numbers --- a complicated infinite thing --- involved in what should
be a completely finitary and combinatorial subject: the study of a
finite group. Is this really necessary?"

Georg replies, "I don't know, but it's sure nice. For example,
suppose I have $G$ acting on vector space $V$. Then I can always break down
$V$ into a direct sum of subspaces preserved by $G$, which can't themselves
be broken down any further. In technical terms: every representation of
$G$ is a direct sum of irreducible representations. And, this
decomposition is unique! It's very nice: it's a lot like how every
natural integer has a unique prime factorization."

William says, "Yes, it's always fun when we can break things down into
'atoms' that can't be further decomposed. It's very satisfying to
our reductionist instincts. But, I can do even better than you!"

Georg raises an eyebrow. "Oh?"

"Yeah," William says. "Suppose I have our group $G$ acting on a set $S$.
Then I can always break down $S$ into a disjoint union of subsets
preserved by $G$, which can't themselves be broken down any further. In
technical terms: every action of $G$ is a disjoint union of transitive
actions. And, this decomposition is unique!"

Embarrassed, Georg replies, "Oh, right --- we heard that back in
["Week 249"](#week249). I wasn't paying enough attention. But how
is what you're doing better than what I'm doing? It sounds just the
same."

William hesitates. "Well, first of all, a rather minor point, which I
can't resist mentioning ... when you said your decomposition of
representations into irreducibles was unique, you were exaggerating a
little. It's just unique up to isomorphism, and not a canonical
isomorphism either.

For example, if you have an irreducible representation of $G$ on $V$, there
are lots of *different ways* to slice the direct sum $V\oplus V$ into two
copies of the representation V. It's a sort of floppy business. On the
other hand, when I have a transitive action of $G$ on $S$, there's exactly
*one way* to chop the disjoint union $S\sqcup S$ into two copies of the $G$-set
$S$. I just look at the two orbits."

Georg says, "Hmm. This is actually a rather delicate point. There's
not really a *canonical* isomorphism in your case either, since $S$ may be
isomorphic to itself in lots of ways, even as a $G$-set. There's
something in what you say, but it's a bit hard to make precise, and
it's certainly not enough to worry me."

"Okay, but here's my more serious point. Given that I can break apart
any set on which $G$ acts into 'atoms', my job is to classify those
atoms: the transitive $G$-sets. And there's a very nice classification!
Any subgroup $H$ of $G$ gives a transitive $G$-set, namely $G/H$, and all
transitive $G$-sets look like this. More precisely: isomorphism classes of
transitive $G$-sets correspond to conjugacy classes of subgroups of $G$.

Even better, this has a nice meaning in terms of Klein geometry. Any
type of figure in Klein geometry gives a transitive $G$-set, with H being
the stabilizer of a chosen figure of that type.

You, on the other hand, need to classify irreducible representations of
$G$. And this is not so conceptual. What do these irreducible
representations *mean* in terms of the group $G$?"

Georg replies, "Well, there's one irreducible representation for each
conjugacy class in $G$..."

At this, William pounces. "You mean the *number* of isomorphism classes
of irreducible representations of $G$ equals the *number* of conjugacy
classes in $G$! But as you know full well, there's no god-given
correspondence. You can't just take a conjugacy class in $G$ and cook up
an irreducible representation, or irrep. So, you've just made my point.
You've shown how mysterious these irreps actually are!"

Georg replies, "Actually in some cases there *is* a nice way to read
off irreps from conjugacy classes. For example, you can do it for the
symmetric groups $S_n$. But, I admit you can't in general... or at
least, I don't know how."

William laughs, "So, I win!"

"Not at all!" replies Georg. "First, there are lots of theorems about
groups that I can prove using representations, which you can't prove
using actions on sets. For example, nobody knows how to prove that
[every group with an odd number of elements is solvable](http://en.wikipedia.org/wiki/Odd_order_theorem) without using
the tools of linear algebra."

William nods. "I admit that linear algebra is very practical. But just
give us time! I proved back in 1904 that every group of size $p^aq^b$ is
solvable if $p$ and $q$ are prime. To do it, I broke down and used linear
algebra. But then, in 1972, Helmut Bender found a proof that avoids
linear algebra."

Georg said, "Okay, struggle on then. So far, without using linear
algebra, nobody can even prove my famous theorem on '[Frobenius groups](http://en.wikipedia.org/wiki/Frobenius_group)'. The details
don't matter here: the point is, this is a result on group actions,
which seems to need linear algebra for its proof.

But if practicality won't sway you, maybe this conceptual argument
will. My atoms are more fine-grained than yours!"

"What do you mean?" asks William.

"You can decompose any action of $G$ into 'atoms', namely transitive
$G$-sets. Similarly, I can decompose any representation of $G$ into one of
my 'atoms', namely irreps. But, there's an obvious way to turn $G$-sets
into representations of $G$, and if we do this to one of your atoms, we
don't get one of my atoms! We can usually break it down further! So, my
atoms are smaller than yours."

"How does this work, exactly?"

"It's easy," says Georg, getting a bit cocky. "Say you have a group
$G$ acting on a set $S$. Then I can form the vector space $\mathbb{C}[S]$ whose
elements are formal linear combinations of elements of $S$. In other
words, it's the vector space having $S$ as basis. If we're feeling
sloppy we can think of guys in $\mathbb{C}[S]$ as functions on $S$ that vanish
except at finitely many points. It's better to think of them as
measures on $S$. But anyway: since $G$ acts on $S$, it acts linearly on
$\mathbb{C}[S]$!

So, any $G$-set gives a representation of $G$. But, even when $G$ acts
transitively on $S$, its representation on $\mathbb{C}[S]$ is hardly ever
irreducible."

William glowers. "Oh yeah?"

"Yeah. Suppose for example that $S$ is finite. Then the constant
functions on $S$ form a $1$-dimensional subspace of $\mathbb{C}[S]$ that's invariant
under the action of $G$. So, at the very least, we can break $\mathbb{C}[S]$ into
two pieces."

"Well," replies William defensively, "That's pretty obvious. But
it's also not such a big deal. So you can break up any my transitive
$G$-sets into two of your irreps, one being the 'trivial' irrep. So
what???"

"It wouldn't be a big deal if that's *all* that ever happened," says
Georg. "In fact, we can break $\mathbb{C}[S]$ into precisely two irreps whenever
the action of $G$ on $S$ is 'doubly transitive' --- meaning we can send any
*pair* of distinct elements of $S$ to any other using some element of $G$.
But, there lots of transitive actions aren't doubly transitive! And
usually, one your atoms breaks down into a *bunch* of my atoms. In fact
I'd like to show you how this works, in detail, for the symmetric
groups."

"Maybe next week," says William. "But, I see your point. Your atoms
are more atomic than my atoms."

Georg seems to have won the argument. But, William wouldn't have
conceded the point quite so fast, if he'd thought about invariant
relations!

The point is this. Suppose we have two $G$-sets, say $X$ and $Y$. Any $G$-set
map from $X$ to $Y$ gives an intertwining operator from $\mathbb{C}[X]$ to $\mathbb{C}[Y]$.
But, even after taking linear combinations, there are typically plenty
of intertwining operators that don't arise this way. It's these extra
intertwining operators that let us chop representations into smaller
atoms.

But where do these extra intertwining operators come from? They come
from *invariant relations* between $X$ and $Y$!

And, what are these extra intertwining operators called? In some famous
special cases, like in study of modular forms, they're called "[Hecke operators](http://en.wikipedia.org/wiki/Hecke_operator)". In some other
famous special cases, like in the study of symmetric groups, they form
algebras called "[Hecke algebras](http://en.wikipedia.org/wiki/Hecke_algebra)".

A lot of people don't even know that Hecke operators and Hecke algebras
are two faces of the same idea: getting intertwining operators from
invariant relations. But, we'll see this is true, once we look at some
examples.

I think I'll save those for future episodes. But if you've followed
the Tale so far, you can probably stand a few extra hints of where
we're going. Recall from ["Week 250"](#week250) that invariant
relations between $G$-sets are just spans of groupoids equipped with some
extra stuff. So, invariant relations between $G$-sets are just a warmup
for the more general, and simpler, theory of spans of groupoids. I said
back in ["Week 248"](#week248) that spans of groupoids give linear
operators. What I'm trying to say now is that these linear operators
are a massive generalization --- but also a simplification --- of what
people call "Hecke operators".

Finally, for students trying to learn a little basic category theory,
I'd like to cast the argument between William and Georg into the
language of categories, just to help you practice your vocabulary.

A $G$-set is the same as a functor
$$A\colon G \to \mathsf{Set}$$
where we think of $G$ as a $1$-object category. There's a category of
$G$-sets, namely
$$\operatorname{Hom}(G,\mathsf{Set})$$
This has functors $A\colon G \to \mathsf{Set}$ as objects, and natural transformations
between these as morphisms. Usually the objects are called "$G$-sets",
and the morphisms are called "maps of $G$-sets".

We can also play this whole game with the category of vector spaces
replacing the category of sets. A representation of $G$ is the same as a
functor
$$A\colon G \to \mathsf{Vect}$$
As before, there's a category of such things, namely
$$\operatorname{Hom}(G,\mathsf{Vect})$$
This has functors $A\colon G \to \mathsf{Vect}$ as objects, and natural transformations
between these as morphisms. Now the objects are called "representations
of $G$" and the morphisms are called "intertwining operators".

We could let any groupoid take the place of the group $G$. We could also
let any other category take the place of $\mathsf{Set}$ or $\mathsf{Vect}$.

Part of what William and Georg were debating was: how similar are
$\operatorname{Hom}(G,\mathsf{Set})$ and $\operatorname{Hom}(G,\mathsf{Vect})$? How are they related?

First of all, there's a functor
$$F\colon \mathsf{Set} \to \mathsf{Vect}$$
sending each set $S$ to the vector space $\mathbb{C}[S]$ with that set as basis.
So, given an action of $G$ on a set:
$$A\colon G \to \mathsf{Set}$$
we can compose it with $F$ and get a representation of $G$:
$$FA\colon G \to \mathsf{Vect}$$
This kind of representation is called a "permutation representation".
And, this trick gives a functor from $G$-sets to representations of $G$:
$$
  \begin{aligned}
    \operatorname{Hom}(G,\mathsf{Set}) &\to \operatorname{Hom}(G,\mathsf{Vect})
  \\A &\mapsto FA
  \end{aligned}
$$
If this functor were an equivalence of categories, it would have to be
essentially surjective, full and faithful. But, not every representation
of $G$ is isomorphic to a permutation representation! In other words, the
functor
$$\operatorname{Hom}(G,\mathsf{Set}) \to \operatorname{Hom}(G,\mathsf{Vect})$$
is not "essentially surjective".

Moreover, not every intertwining operator between permutation
representations comes from a map between their underlying $G$-sets! In
other words, the functor
$$\operatorname{Hom}(G,\mathsf{Set}) \to \operatorname{Hom}(G,\mathsf{Vect})$$
is not "full".

But, given two different maps from one $G$-set to another, they give
different intertwining operators. So, at least our functor is
"faithful".

Maps of $G$-sets are a special case of invariant relations. So, to get a
category that more closely resembles $\operatorname{Hom}(G,\mathsf{Vect})$, while remaining purely
combinatorial, we can replace $\operatorname{Hom}(G,\mathsf{Set})$ by the category with $G$-sets as
objects and invariant binary relations as morphisms. This is the basic
idea of "Hecke operators".

Or, even better, we can try a weak $2$-category, with

- groupoids over $G$ as objects
- spans of groupoids over $G$ as morphisms
- maps between spans of groupoids over $G$ as $2$-morphisms

This is where groupoidification comes into its own.

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**Addendum:** For more discussion, go to the [$n$-Category Café](http://golem.ph.utexas.edu/category/2007/05/this_weeks_finds_in_mathematic_13.html).

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> *The present treatise is intended to introduce to the reader the main
outlines of the theory of groups of finite order apart from any
applications. The subject is one which has hitherto attracted but little
attention in this country; it will afford me much satisfaction if, by
means of this book, I shall arouse interest among English mathematicians
in a branch of pure mathematics which becomes the more fascinating the
more it is studied*
> 
> --- William Burnside