This Week's Finds in Mathematical Physics (Week 287)
John Baez
Also available at http://math.ucr.edu/home/baez/week287.html
December 19, 2009
This Week's Finds in Mathematical Physics (Week 287)
John Baez
This week: a fascinating history of categorical logic, and more about
rational homotopy theory. But first, guess what this is a picture of:
http://math.ucr.edu/home/baez/014379_0925.jpg
If you give up, go to the bottom of this article.
Next, here's an incredibly readable introduction to the revolution
that happened in logic starting in the 1960s:
1) Jean-Pierre Marquis and Gonzalo Reyes, The history of categorical
logic, 1963-1977. Available at
https://www.webdepot.umontreal.ca/Usagers/marquisj/MonDepotPublic/HistofCatLog.pdf
It's a meaty but still bite-sized 116 pages. It starts with the
definitions of categories, functors, and adjoint functors. But it
really takes off in 1963 with Bill Lawvere's thesis, which
revolutionized universal algebra using category theory. It then moves
on through Lawvere and Tierney's introduction of the modern concept of
topos, and it ends in 1977, when Makkai and Reyes published their book
on categorical logic, and Johnstone published his book on topos theory.
The world has never been the same since!
One great thing about this paper is that it discusses the history in a
blow-by-blow way, including conferences and unpublished but
influential writings. It also gives a great summary of the key ideas
in Lawvere's thesis. I'll quote it, since everyone should know or at
least *have seen* these ideas:
1. To use the category of categories as a framework for mathematics,
i.e. the category of categories should be the foundations of
mathematics;
2. Every aspect of mathematics should be representable in one way or
another in that framework; in other words, categories constitute the
background to mathematical thinking in the sense that, in this
framework, essential features of that thinking are revealed;
3. Mathematical objects and mathematical constructions should be
thought of as functors in that framework;
4. In particular, sets always appear in a category, there are no
such thing as sets by themselves, in fact there is no such thing as
a mathematical concept by itself;
5. But sets form categories and the latter categories play a key role
in the category of categories, i.e. in mathematics;
6. Adjoint functors occupy a key position in mathematics and in the
development of mathematics; one of the guiding principles of the
development of mathematics should be "look for adjoints to given
functors"; in that way foundational studies are directly linked to
mathematical practice and the distinction between foundational
studies and mathematical studies is a matter of degree and
direction, it is not a qualitative distinction;
7. As the foregoing quote clearly indicates, Lawvere is going back
to the claim made by Eilenberg and Mac Lane that the "invariant
character of a mathematical discipline can be formulated in these
terms" [i.e. in terms of functoriality]. But now, in order to
reveal this invariant character, extensive use of adjoint functors
is made.
8. The invariant content of a mathematical theory is the
"objective" content of that theory; this is expressed at various
moments throughout his publications. To wit:
As posets often need to be deepened to categories to accurately
reflect the content of thought, so should inverses, in the sense
of group theory, often be replaced by adjoints. Adjoints retain
the virtue of being uniquely determined reversal attempts, and
very often exist when inverses do not.
9. Not only sets should be treated in a categorical framework, but
also logical aspects of the foundations of mathematics should be
treated categorically, in as much as they have an objective
content. In particular, the logical and the foundational are
directly revealed by adjoint functors.
If this sounds mysterious, well, read the paper!
Now I want to dig a little deeper into rational homotopy theory,
and start explaining this chart:
RATIONAL SPACES
/ \
/ \
/ \
/ \
/ \
DIFFERENTIAL GRADED ------- DIFFERENTIAL GRADED
COMMUTATIVE ALGEBRAS LIE ALGEBRAS
Last time I described rational spaces and the category they form: the
"rational homotopy category". I actually described this category in
three ways. But there are two other ways to think about this category
that are much cooler!
That's what the other corners of this triangle are. The lower left
corner involves a drastic generalization of differential forms on
smooth manifolds. The lower right corner involves a drastic
generalization of Lie algebras coming from Lie groups.
Today I'll explain the road to the lower left corner. Very roughly,
it goes like this. If you give me a space, I can replace it with a
space made of simplices that has the same homotopy type, and then take
the differential forms on this replacement. Voila! A differential
graded commutative algebra!
But I want to avoid talking to just the experts. Among other things,
I want to use rational homotopy theory as an excuse to explain lots of
good basic math. So, first I'll remind you about differential forms
on a manifold, and why they're a "differential graded commutative
algebra", or "DGCA" for short. Then I'll show you how to define
something like differential forms starting from any topological space!
Again, they're a DGCA. And it turns out that for rational spaces,
this DGCA knows everything there is to know about our space - at least
as far as homotopy theory is concerned.
So: what are differential forms? Differential forms are a basic tool
for doing calculus on manifolds. We use them throughout physics:
they're the grownup version of the "gradient", "divergence" and "curl"
that we learn about as kids. There are lots of ways to define them,
but the most rapid is this.
The smooth real-valued functions
f: X -> R
on a manifold X form an algebra over the real numbers. In other
words: you can add and multiply them, and multiply them by real
numbers, and a bunch of familiar identities hold, which are the axioms
for an algebra. Moreover, this algebra is commutative:
fg = gf
Starting from this commutative algebra, or any commutative algebra A,
we can define differential forms as follows. For each algebra element
f make up a symbol df. Then, freely let yourself add and multiply
these new symbols along with your algebra elements. Then impose the
axioms for an algebra, together with two rules that make "d" act like
a derivative:
d(f + g) = df + dg
d(fg) = (df) g + f dg
Finally, impose these rules:
f dg = (dg) f
(df) (dg) = - (dg) (df)
We now have the algebra of differential forms for A, which I'll call
Omega(A).
If you first learned about differential forms some other way, this
should be compatible with that... but this way is probably quicker.
Thanks to the funny rule
(df) (dg) = - (dg) (df)
the algebra Omega(A) is not commutative. However, it's "graded
commutative", meaning roughly that it's commutative except for some
systematically chosen minus signs.
A bit more precisely: every differential form can be written as a
linear combination of guys like this:
v = f dg_1 dg_2 ... dg_p
where p ranges over all natural numbers. Linear combinations of guys
of this sort for a particular fixed p are called "p-forms". We also
say they're "of degree p". And the algebra of differential forms obeys
v w = (-1)^{pq} w v
whenever v is of degree p and w is of degree q. This is what we
mean by saying Omega(A) is "graded commutative".
But the algebra of differential forms is better than a mere graded
commutative algebra! We've already introduced df when f is an element
of our original algebra. But we can define "d" for *all* differential
forms simply by saying that d is linear and saying that d of
f dg_1 dg_2 ... dg_p
is
df dg_1 dg_2 ... dg_p
This definition implies three facts. First, it implies that d of a
p-form is a (p+1)-form. That's pretty obvious. Second, it implies
that
d(dw) = 0
for any differential form w. Why? Well, I'll let you check it, but
I'll give you a hint: the key step is to show that d1 = 0. And third,
it implies this version of the product rule:
d(v w) = (dv) w + (-1)^p v dw
for any p-form v and q-form w. Again the proof is a little calculation.
We can summarize these three facts, together with the linearity of d, by
saying that differential forms are a "differential graded commutative
algebra", or "DGCA".
You can do lots of wonderful stuff with differential forms. After you
learn a bunch of this stuff, it becomes obvious that you should
generalize them to apply to spaces of many kinds.
It's easy to generalize them from manifolds to spaces X where you
have a reasonable idea of when a real-valued function
f: X -> R
counts as "smooth". Just take the commutative algebra A of smooth
real-valued functions on X and construct Omega(A) following my
instructions!
There are many examples of such spaces, including manifolds with
boundary, manifolds with corners, and infinite-dimensional manifolds.
In fact, there are general theories of "smooth spaces" that systematically
handle lots of these examples:
2) Andrew Stacey, Comparative smootheology, available as
arXiv:0802.2225.
3) Patrick Iglesias-Zemmour, Diffeology. Available at
http://math.huji.ac.il/~piz/Site/The%20Book/The%20Book.html
4) John Baez and Alexander Hoffnung, Convenient categories of smooth
spaces, to appear in Trans. Amer. Math. Soc.. Also available as
arXiv:0807.1704.
But here's a tougher question: how can we generalize differential
forms to an arbitrary *topological* space X?
You could take A to be the algebra of *continuous* functions on X and
form Omega(A). There's no law against it... go ahead... but I bet no
good will come of it. (What goes wrong?)
But there's a better approach, invented by Dennis Sullivan in this
famous paper:
5) Dennis Sullivan, Infinitesimal computations in topology,
Publications Mathimatiques de l'IHES 47 (1977), 269-331. Available
at http://www.numdam.org/item?id=PMIHES_1977__47__269_0
We start by turning our topological space into a simplicial set.
Remember, a simplicial set is a bunch of
0-simplices (vertices)
1-simplices (edges)
2-simplices (triangles)
3-simplices (tetrahedra)
and so on, all stuck together. Given a topological space X, we can
form an enormous simplicial set whose n-simplices are all possible
maps
f: Delta^n -> X
where Delta^n is the standard n-simplex, that is, the intersection
of the hyperplane
x_0 + x_1 + ... + x_n = 0
with the set where all the coordinates x_i are nonnegative.
This enormous simplicial set is called the "singular nerve" of X,
Sing(X). Like any simplicial set, we can think of Sing(X) as a purely
combinatorial gadget, but we can also "geometrically realize" it and
think of it as a topological space in its own right. The resulting
space is called |Sing(X)|.
(For more details on the singular nerve and its geometric realization,
see items E and F of "week116".)
This space |Sing(X)| is made of a bunch of simplices stuck together
along their faces. So, we can say a real-valued function on |Sing(X)|
is "simplex-wise smooth" if it's continuous and smooth on each
simplex. And this is enough to set up a theory of differential forms!
We just take the algebra A of simplex-wise smooth functions on
|Sing(X)|, and use this to build our algebra of differential forms
Omega(A) as I've described!
But Sullivan noted that we can go even further. Thanks to how we've
defined the standard n-simplex, it makes sense to talk about
polynomial functions on this simplex. We can even sidestep the need
for real numbers, by looking at polynomial functions with *rational*
coefficients. And that's just right for rational homotopy theory.
So, let's focus our attention on functions on |Sing(X)| that when
restricted to any simplex give polynomials with rational coefficients.
This is a commutative algebra over the *rational* numbers. Call it A.
We can copy the construction of Omega(A) working with rational
numbers. Let's call guys in here "rational differential forms".
Now, you may complain that we're not really studying differential
forms on X: we're studying them on this other space |Sing(X)|. At one
point in my life this really annoyed me. It seemed like a cheat. But
for the purposes of homotopy theory it's perfectly fine, since
|Sing(X)| has the same homotopy type as X.
And even better, when X is a rational space, the rational differential
forms on |Sing(X)| will know *everything* about the homotopy type of
X. This is amazing! It means that for rational spaces, we can reduce
homotopy theory to a souped-up version of the theory of differential
forms!
In particular, Sullivan was able to use this trick to compute
the *homotopy groups* of a rational space X, starting from the
rational differential forms on |Sing(X)|.
Since X and |Sing(X)| have the same homotopy type, they have the same
homotopy groups, and cohomology groups, and so on. And it's not
surprising that we can read off the *cohomology groups* of |Sing(X)|
starting from the rational differential forms on this space - this is
just a slight twist on the usual idea of deRham cohomology. But it's
surprising that we can compute the *homotopy groups*, which are
usually a lot harder. This is the magic of rational homotopy theory.
I won't explain how to compute this magic, at least not today. For
that read Sullivan's paper, or this paper I recommended last time:
6) Kathryn Hess, Rational homotopy theory: a brief introduction, in
Interactions Between Homotopy Theory and Algebra, ed. Luchezar
L. Avramov, Contemp. Math 436, AMS, Providence, Rhode Island, 2007.
Also available as arXiv:math.AT/0604626.
Instead, I want to go over what I just said in a slightly more formal
way. This will give me an excuse to introduce a bunch of beautiful
concepts that everyone should know... and maybe demonstrate a tiny bit
of what Lawvere was talking about: the power of categories.
First of all, what's a DGCA, really? It's a commutative monoid in the
symmetric monoidal category of cochain complexes!
Let me explain.
A "cochain complex", for us, will be a list of vector spaces and
linear maps
d d d
C_0 ---> C_1 ---> C_2 --->
with d^2 = 0. We can use vector spaces over any field we like; let's
use the rational numbers to be specific.
Just as you can tensor vector spaces, you can tensor cochain complexes.
The tensor product of cochain complexes C and C' will have
(C tensor C')_n = sum_{p+q = n} C_p tensor C'_q
and we define
d(c tensor c') = dc tensor c' + (-1)^p c tensor dc'
when c is in C_p and c' is in C'_q.
(You've seen a similar "product rule" earlier in this article. There's
a general principle at work here. Physicists know that whenever you
exchange two fermions, their phase gets multiplied by -1. In math, we
should stick in a minus sign whenever we switch two "odd" things. The
map d counts as odd since it sends guys in our cochain complex to guys
whose degree is 1 more, and the number 1 is odd. The element c in C_p
counts as "odd" whenever p is odd. In the equation above, we're switching
d past c and getting a minus sign whenever c is odd.)
Just as you can define a commutative algebra to be a vector space
V with a product
V tensor V -> V
that's associative and commutative, you can define a "differential
graded commutative algebra", or DGCA, to be a cochain complex C with a
product
C tensor C -> C
that's associative and graded-commutative. By "graded-commutative",
I mean you need to remember to put in a sign (-1)^{pq} whenever you
switch a guy in C_p and a guy in C_q.
We can systematize all this by checking that, just like the category
of vector spaces with its usual tensor product, the category of
cochain complexes with its tensor product is a "symmetric monoidal
category":
7) nLab, Symmetric monoidal category,
http://ncatlab.org/nlab/show/symmetric+monoidal+category
So is the category of sets with its cartesian product. We can define
a "commutative monoid" in any symmetric monoidal category. In the
category of sets, this is just a commutative monoid in the traditional
sense. In the category of vector spaces, it's a commutative algebra.
And in the category of cochain complexes, it's a DGCA!
Notice: a DGCA where only C_0 is nonzero is just a plain old
commutative algebra. So, DGCAs really are a generalization of
commutative algebras. So whenever anyone tells you something about
DGCAs, you should check to see what it says about commutative
algebras. And whenever anyone tells you something about commutative
algebras, you should try to generalize it to DGCAs!
This should keep you pretty busy, since commutative algebras are the
playground of the simplest kind of algebraic geometry: the kind where
you look at solutions of polynomial equations in a bunch of variables,
like this:
x^2 + y^2 + z^2 = 0
xyz - 1 = 0
If you take your polynomials and count them as zero when they satisfy
your equations, you get a commutative algebra. Even better, you can
get any sufficiently small commutative algebra this way - the
technical term is "finitely presented". And if you allow infinitely
many variables and infinitely many equations, you can drop that
technical fine print.
So, the study of commutative algebras is really just the study of
polynomial equations. And if we think about their solutions as
forming curves or surfaces or the like, we're doing algebraic geometry
- so-called "affine" algebraic geometry.
This means that we can - and in fact should! - generalize all of
affine algebraic geometry from commutative algebras to DGCAs. I'd
like to say more about this someday... but not today. This is just
a digression. I got distracted from my real goal.
Before I got distracted, I was telling you how commutative algebras
are the same as DGCA's with only C_0 being nonzero. And here's why I
mentioned this. We can take *any* DGCA and violently kill C_p for all
p > 0, leaving a commutative algebra C_0. We can think of this as a
forgetful functor
[DGCAs] -> [commutative algebras]
And this functor has a left adjoint, which freely generates a DGCA
starting from a commutative algebra:
[commutative algebras] -> [DGCAs]
And this left adjoint is something I already told you about! I called
it "Omega": it's the process that takes a commutative algebra and
creates the DGCA of differential forms for this algebra!
This may be a good place to stop reading, if you don't already love
category theory up to and including "weighted colimits". But I can't
resist saying a bit more. And if you've never understood weighted
colimits, maybe this will make you want to.
I already told you how we turn any topological space X into a
simplicial set Sing(X) and then back into a bigger topological space
|Sing(X)| and then into a DGCA.
But if you know homotopy theory well, you know this subject regards
topological spaces and simplicial sets as two different views of "the
same thing". So turning a topological space into a simplicial set is
no big deal. So in fact, the the core of the above construction is
the process that takes a simplicial set and turns it into a DGCA. And
I'd like to explain this process a bit more efficiently now.
Here's the point: this process is a lot like "geometric realization".
In geometric realization we start with a simplicial set S, which is
really a functor
S: Delta^{op} -> Set
where Delta is the category of simplices. And we know how to turn
any simplex into a topological space, so we also have a functor
F: Delta -> Top
We can then take the "weighted colimit" of F with S as our "weight".
This creates a topological space |S|, the "geometric realization" of
S.
The idea is that we took each simplex in our simplicial set, turned
it into a space, and then glued all these spaces together. For this
trick to work, all we needed is that the category Top has colimits.
Similarly, we know how to turn any simplex into a DGCA, namely the
rational differential forms on that simplex! So we also have a
functor
F': Delta -> [DGCAs]^{op}
There's an "op" here because of the usual contravariant relation
between algebra and geometry. But never mind: what matters is that
DGCA^{op} has colimits. So we can copy what we did before, and take
the weighted colimit of F' with S as our weight. And now this creates
a DGCA: the "rational differential forms" on our simplicial set S.
The idea is that we took each simplex in our simplicial set, turned
it into a DGCA, and then glued all these DGCAs together. But perhaps
I should say "coglued", because of that "op".
While we're playing these games, I should point out a simpler version.
We also have a functor
F": Delta -> [commutative algebras]^{op}
and we can pull the same stunt to turn our simplicial set into a
commutative algebra, which is the algebra of functions that restrict
to polynomials with rational coefficients on each simplex!
But in this case, there's a super-famous name for the category
[commutative algebras]^{op}
It's called the category of "affine schemes". And so we can think
of this stunt more geometrically as the process of taking an affine
scheme for each simplex and gluing them together to get an affine
scheme for our simplicial set S! So we're doing a kind of "geometric
realization" with affine schemes replacing topological spacs.
This leads up to a question for the experts. Is there a famous name
for the category
[DGCAs]^{op} ?
It's *related* to the category of "simplicial affine schemes", no?
But it's not quite the same. Can we think of this category as
consisting of simplicial affine schemes with an extra property? You
see, this bears heavily on the idea that rational homotopy theory is a
generalization of algebraic geometry, with DGCAs replacing commutative
algebras.
Finally: the picture at the start of This Week's Finds shows dry ice -
frozen carbon dioxide - on the south pole of Mars:
8) HiRISE (High Resolution Imaging Science Experiments),
South polar residual cap monitoring: rare stratigraphic contacts,
http://hirise.lpl.arizona.edu/ESP_014379_0925
This dry ice forms quite a variety of baroque patterns. I don't
know how it happens! Here are couple more good pictures:
9) HiRISE (High Resolution Imaging Science Experiments),
South polar residual cap monitoring: rare stratigraphic contacts,
http://hirise.lpl.arizona.edu/PSP_004687_0930
10) HiRISE (High Resolution Imaging Science Experiments),
South polar residual cap monitoring: rare stratigraphic contacts,
http://hirise.lpl.arizona.edu/ESP_014261_0930
Patrick Russell wrote a description of the last one:
This HiRISE image is of a portion of Mars' south polar residual ice
cap. Like Earth, Mars has concentrations of water ice at both poles.
Because Mars is so much colder, however, the seasonal ice that gets
deposited at high latitudes in the winter and is removed in the
spring (generally analogous to winter-time snow on Earth) is
actually carbon dioxide ice. Around the south pole there are areas
of this carbon dioxide ice that do not disappear every spring, but
rather survive winter after winter. This persistent carbon dioxide
ice is called the south polar residual cap, and is what we are
looking at in this HiRISE image.
Relatively high-standing smooth material is broken up by
semi-circular depressions and linear, branching troughs that make a
pattern resembling those of your fingerprints. The high-standing
areas are thicknesses of several meters of carbon dioxide ice. The
depressions and troughs are thought to be caused by the removal of
carbon dioxide ice by sublimation (the change of a material from
solid directly to gas). HiRISE is observing this carbon dioxide
terrain to try to determine how these patterns develop and how fast
the depressions and troughs grow.
While the south polar residual cap as a whole is present every
year, there are certainly changes taking place within it. With the
high resolution of HiRISE, we intend to measure the amount of
expansion of the depressions over multiple Mars years. Knowing the
amount of carbon dioxide removed can give us an idea of the
atmospheric, weather, and climate conditions over the course of a
year.
In addition, looking for where carbon dioxide ice might be being
deposited on top of this terrain may help us understand if there is
any net loss or accumulation of the carbon-dioxide ice over time,
which would be a good indicator of whether Mars' climate is in the
process of changing or not.
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Quote of the Week:
"We live on an island surrounded by a sea of ignorance. As our island
of knowledge grows, so does the shore of our ignorance." - John Wheeler
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