Also available at http://math.ucr.edu/home/baez/week202.html
February 21, 2004
This Week's Finds in Mathematical Physics - Week 202
John Baez
This week I'll deviate from my plan of discussing number theory, and
instead say a bit about something else that's been on my mind lately:
structure types. But, you'll see my fascination with Galois theory
lurking beneath the surface.
Andre Joyal invented structure types in 1981 - he called them "especes
de structure", and lots of people call them "species". Basically, a
structure type is just any sort of structure we can put on finite sets:
an ordering, a coloring, a partition, or whatever. In combinatorics
we count such structures using "generating functions". A generating
function is a power series where the coefficient of x^n keeps track of
how many structures of the given kind we can put on an n-element set.
By playing around with these functions, we can often figure out the
coefficients and get explicit formulas - or at least asymptotic formulas -
that count the structures in question.
The reason this works is that operations on generating functions come
from operations on structure types. For example, in "week190", I
described how addition, multiplication and composition of generating
functions correspond to different ways to get new structure types from
old.
Joyal's great contribution was to give structure types a rigorous
definition, and use this to show that many calculations involving
generating functions can be done directly with structure types. It
turns out that just as generating functions form a *set* equipped with
various operations, structure types form a *category* with a bunch of
completely analogous operations. This means that instead of merely
proving *equations* between generating functions, we can construct
*isomorphisms* between their underlying structure types - which imply
such equations, but are worth much more. It's like the difference
between knowing two things are equal and knowing a specific reason WHY
they're equal!
Of course, this business of replacing equations by isomorphisms is called
"categorification". In this lingo, structure types are categorified power
series, just as finite sets are categorified natural numbers.
A while back, James Dolan and I noticed that since you can use power
series to describe states of the quantum harmonic oscillator, you can
think of structure types as states of a categorified version of this
physical system! This gives new insights into the combinatorial
underpinnings of quantum physics.
For example, the discrete spectrum of the harmonic oscillator
Hamiltonian can be traced back to the discreteness of finite sets!
The commutation relations between annihilation and creation operators
boil down to a very simple fact: there's one more way to put a ball in
a box and then take one out, than to take one out and then put one in.
Even better, the whole theory of Feynman diagrams gets a simple
combinatorial interpretation. But for this, one really needs to go
beyond structure types and work with a generalization called "stuff
types".
I've been thinking about this business for a while now, so last fall
I decided to start giving a year-long course on categorification and
quantization. The idea is to explain bunches of quantum theory,
quantum field theory and combinatorics all from this new point of view.
It's fun! Derek Wise has been scanning in his notes, and a bunch of
people have been putting their homework online. So, you can follow
along if you want:
1) John Baez and Derek Wise, Categorification and Quantization.
Fall 2003 notes: http://math.ucr.edu/home/baez/qg-fall2003/
Winter 2004 notes: http://math.ucr.edu/home/baez/qg-winter2004/
Spring 2004 notes: http://math.ucr.edu/home/baez/qg-spring2004/
I'd like to give you a little taste of this subject now. But, instead
of explaining it in detail, I'll just give some examples of how structure
types yield some far-out generalizations of the concept of "cardinality".
This stuff is a continuation of some themes developed in "week144",
"week185", "week190", so I'll start with a review.
Suppose F is a structure type. Let F_n be the *set* of ways we can put
this structure on a n-element set, and let |F_n| be the *number* of ways
to do it. In combinatorics, people take all these numbers |F_n| and pack
them into a single power series. It's called the generating function of
F, and it's defined like this:
|F_n|
|F|(x) = sum ----- x^n
n!
It may not converge, so in general it's just a "formal" power series -
but for interesting structure types it often converges to an interesting
function.
What's good about generating functions is that simple operations on them
correspond to simple operations on structure types. We can use this to count
structures on finite sets. Let me remind you how it works for binary trees!
There's a structure type T where a T-structure on a set is a way of
making it into the leaves of a binary tree drawn in the plane. For example,
here's one T-structure on the set {a,b,c,d}:
b d a c
\ \ / /
\ \/ /
\ / /
\/ /
\ /
\/
Thanks to the choice of different orderings, the number of T-structures
on an n-element set is n! times the number of binary trees with n
leaves. Annoyingly, the latter number is traditionally called the (n-1)st
Catalan number, C_{n-1}. So, we have:
|T|(x) = sum C_{n-1} x^n
where the sum starts at n = 1.
There's a nice recursive definition of T:
"To put a T-structure on a set, either note that it has one element,
in which case there's just one T-structure on it, or chop it into
two subsets and put a T-structure on each one."
In other words, any binary tree is either a degenerate tree with just one
leaf:
X
or a pair of binary trees stuck together at the root:
----- -----
| | | |
| T | | T |
| | | |
----- -----
\ /
\ /
\ /
\/
We can write this symbolically as
T = X + T^2
Here's why: X is a structure type called "being the one-element set",
+ means "exclusive or", and squaring a structure type means you chop your set
in two parts and put that structure on each part. (I explained these rules
more carefully in "week190".)
I should emphasize that the equals sign here is really an *isomorphism*
between structure types - I'm only using "equals" because the isomorphism
key on my keyboard is stuck. But if we take the generating function of
both sides we get an actual equation, and the notation is set up to make
this really easy:
|T| = x + |T|^2
In "week144" I showed how you can solve this using the quadratic equation:
|T| = (1 - sqrt(1 - 4x))/2.
and then do a Taylor expansion to get
|T| = x + x^2 + 2x^3 + 5x^4 + 14x^5 + 42x^6 + ...
Lo and behold! The coefficient of x^n is the number of binary trees
with n leaves!
There's also another approach where we work directly with the
structure types themselves, instead of taking generating functions.
This is harder because we can't subtract structure types, or divide
them by 2, or take square roots of them - at least, not without
stretching the rules of this game. All we can do is use the
isomorphism
T = X + T^2
and the basic rules of category theory. It's not as efficient, but it's
illuminating. It's also incredibly simple: we just keep sticking in
"X + T^2" wherever we see "T" on the right-hand side, over and over again.
Like this:
T = X + T^2
T = X + (X + T^2)^2
T = X + (X + (X + T^2)^2)^2
and so on. You might not think we're getting anywhere, but if
you stop at the nth stage and expand out what we've got, you'll
get the first n terms of the Taylor expansion we had before!
At least, you will if you count "stages" and "terms" correctly.
I won't actually do this, because it's better if you do it yourself.
When you do, you'll see it captures the recursive process of building
a binary tree from lots of smaller binary trees. Each time you see a "T"
and replace it with an "X + T^2", you're really taking a little binary
tree:
-----
| |
| T |
| |
-----
and replacing it with either a degenerate tree with just a single leaf:
X
or a pair of binary trees:
----- -----
| | | |
| T | | T |
| | | |
----- -----
\ /
\ /
\ /
\/
So, each term in the final result actually corresponds to a specific
tree! This is a good example of categorification: when we calculate the
coefficient of x^n this way, we're not just getting the *number* of binary
planar trees with n leaves - we're getting an actual explicit description
of the *set* of such trees.
Now, what happens if we take the generating function |T|(x) and
evaluate it at x = 1? On the one hand, we get a divergent series:
|T|(1) = 1 + 1 + 2 + 5 + 14 + 42 + ...
This is the sum of all Catalan numbers - or in other words, the number
of binary planar trees. On the other hand, we can use the formula
|T| = (1 - sqrt(1 - 4x))/2
to get
|T|(1) = (1 - sqrt(-3))/2
It may seem insane to conclude
1 + 1 + 2 + 5 + 14 + 42 + ... = (1 - sqrt(-3))/2
but Lawvere noticed that there's a kind of strange sense to it.
The trick is to work not with generating function |T| but with the structure
type T itself. Since |T|(1) is equal to the *number* of planar binary trees,
T(1) should be naturally isomorphic to the *set* of planar binary trees.
And it is - it's obvious, once you think about what it really means.
The number of binary planar trees is not very interesting, but the set of
them is. In particular, if we take the isomorphism
T = X + T^2
and set X = 1, we get an isomorphism
T(1) = 1 + T(1)^2
which says
"a planar binary tree is either the tree with one leaf or
a pair of planar binary trees."
Starting from this, we can derive lots of other isomorphisms involving
the set T(1), which turn out to be categorified versions of equations
satisfied by the number
|T|(1) = (1 - sqrt(-3))/2
For example, this number is a sixth root of unity. While there's no
one-to-one correspondence between 6-tuples of trees and the 1 element
set, which would categorify the formula
|T|(1)^6 = 1
there *is* a very nice one-to-correspondence between 7-tuples of
trees and trees, which categorifies the formula
|T|(1)^7 = |T|(1)
Of course the set of binary trees is countably infinite, and so is the
set of 7-tuples of binary trees, so they can be placed in one-to-one
correspondence - but that's boring. When I say "very nice", I mean
something more interesting: starting with the isomorphism
T = x + T^2
we get a one-to-one correspondence
T(1) = 1 + T(1)^2
which says that any binary planar tree is either degenerate or a
pair of binary planar trees... and using this we can *construct*
a one-to-one correspondence
T(1)^7 = T(1)
The construction is remarkably complicated. Even if you do it
as efficiently as possible, I think it takes 18 steps, like this:
T(1)^7 = T(1)^6 + T(1)^8
= T(1)^5 + T(1)^7 + T(1)^8
.
.
.
= 1 + T(1) + T(1)^2 + T(1)^4
= 1 + T(1) + T(1)^3
= 1 + T(1)^2
= T(1)
I'll let you fill in the missing steps - it's actually quite fun if you
like puzzles.
If you get stuck, you can find the answer online in a couple of places:
2) Andreas Blass, Seven trees in one, Jour. Pure Appl. Alg. 103 (1995),
1-21. Also available at http://www.math.lsa.umich.edu/~ablass/cat.html
3) Marcelo Fiore, Isomorphisms of generic recursive polynomial
types, to appear in 31st Symposium on Principles of Programming
Languages (POPL04). Also available at
http://www.cl.cam.ac.uk/~mpf23/papers/Types/recisos.ps.gz
Or, take a peek at the "Addenda" down below.
Robbie Gates, Marcelo Fiore and Tom Leinster have also proved some very
general theorems about this sort of thing. Gates focused on "distributive
categories" (categories with products and coproducts, the former distributing
over the latter), while the work of Fiore and Leinster applies to more
general "rig categories":
4) Robbie Gates, On the generic solution to P(X) = X in distributive
categories, Jour. Pure Appl. Alg. 125 (1998), 191-212.
5) Marcelo Fiore and Tom Leinster, Objects of categories as complex numbers,
available as arXiv:math.CT/0212377.
A rig category is basically the most general sort of category in which
we can "add" and "multiply" as we do in a ring - but without negatives,
hence the missing letter "n". It turns out that whenever we have an object
Z in a rig category and it's equipped with an isomorphism
Z = P(Z)
where P is a polynomial with natural number coefficients, we can associate
to it a "cardinality" |Z|, namely any complex solution of the equation
|Z| = P(|Z|)
Which solution should we use? Well, for simplicity let's consider the
case where P has degree at least 2 and the relevant Galois group acts
transitively on the solutions of this equation, so "all roots are created
equal". Then we can pick *any* solution as the cardinality |Z|. Any
polynomial equation with natural number coefficients satisfied by one
solution will be satisfied by *all* solutions, so it doesn't matter
which one we choose.
Now suppose the cardinality |Z| satisfies such an equation:
Q(|Z|) = R(|Z|)
where neither Q nor R is constant. Then the results of Fiore and
Leinster say we can construct an isomorphism
Q(Z) = R(Z)
in our distributive category! In other words, a bunch of equations satisfied
by the object's cardinality automatically come from isomorphisms involving the
object itself.
This explains why the set T(1) of binary trees acts like it has cardinality
|T|(1) = (1 - sqrt(-3))/2
or equally well,
|T|(1) = (1 + sqrt(-3))/2
(Since the relevant Galois group interchanges these two numbers, we can
use either one.) More generally, the set T(n) consisting of binary trees
with n-colored leaves acts a lot like the number |T|(n).
This has gotten me interested in trying to find a nice example of a
"Golden Object": an object G in some distributive category that's
equipped with an isomorphism
G^2 = G + 1
The Golden Object doesn't fit into Fiore and Leinster's formalism,
since this isomorphism is not of the form G = P(G) where P has natural
number coefficients. But, it still seems that such an object deserves
to have a "cardinality" equal to the golden number:
|G| = (1 + sqrt(5))/2 = 1.618033988749894848204586834365...
James Propp came up with an interesting idea related to the Golden Object:
consider what happens when we evaluate the generating function for binary
trees at -1. On the one hand we get an alternating sum of Catalan numbers:
|T|(-1) = -1 + 1 - 2 + 5 - 14 + 42 + ...
On the other hand, we can use the formula
|T| = (1 - sqrt(1 - 4x))/2
to get
|T|(-1) = (1 - sqrt(5))/2
which is -1 divided by the golden number. Of course, it's possible we
should use the other sign of the square root, and get
|T|(-1) = (1 + sqrt(5))/2
which is just the golden number! Galois theory says these two roots are
created equal. Either way, we get a bizarre and fascinating formula:
- 1 + 1 - 2 + 5 - 14 + 42 + ... = (1 +- sqrt(5))/2
Can we fit this into some clear and rigorous framework, or is it just nuts?
We'd like some generalization of cardinality for which "the set of binary
trees with -1-colored leaves" has cardinality equal to the golden number.
James Propp suggested one avenue. A while back, Steve Schanuel made an
incredibly provocative observation: if we treat "Euler measure" as a
generalization of cardinality, it makes sense to treat the real line
as a "space of cardinality -1":
6) Stephen H. Schanuel, What is the length of a potato?: an introduction
to geometric measure theory, in Categories in Continuum Physics, Spring
Lecture Notes in Mathematics 1174, Springer, Berlin, 1986, pp. 118-126.
7) Stephen H. Schanuel, Negative sets have Euler characteristic and
dimension, Lecture Notes in Mathematics 1488, Springer Verlag, Berlin,
1991, pp. 379-385.
James Propp has developed this idea in a couple of fascinating papers:
8) James Propp, Euler measure as generalized cardinality, available as
arXiv:math/0203289.
9) James Propp, Exponentiation and Euler measure, available as
arXiv:math/0204009.
Using this idea, it seems reasonable to consider the space of binary
trees with leaves labelled by real numbers as a rigorous version of
"the set of binary trees with -1-colored leaves". So, we just need
to figure out what generalization of Euler characteristic gives this
space an Euler characteristic equal to the golden number. It would
be great if we could make this space into a Golden Object in some
rig category, but that may be asking too much.
Whew! There's obviously a lot of work left to be done here. Here's
something easier: a riddle. What's this sequence?
un, dos, tres, quatre, cinc, sis, set, vuit, nou, deu,...
The answer is at the end of this article.
Now I'd like to mention some important papers on n-categories. You
may think I'd lost interest in this topic, because I've been talking
about other things. But it's not true!
Most importantly, Tom Leinster has come out with a big book on n-categories
and operads:
5) Tom Leinster, Higher Operads, Higher Categories, Cambridge U. Press,
Cambridge, 2003. Also available as arXiv:math.CT/0305049.
As you'll note, he managed to talk the press into letting him keep his book
freely available online! We should all do this. Nobody will ever make much
cash writing esoteric scientific tomes - it takes so long, you could earn
more per hour digging ditches. The only *financial* benefit of writing such
a book is that people will read it, think you're smart, and want to hire you,
promote you, or invite you to give talks in cool places. So, maximize your
chances of having people read your books by keeping them free online! People
will still buy the paper version if it's any good....
And indeed, Leinster's book has many virtues besides being free. He
gracefully leads the reader from the very basics of category theory
straight to the current battle front of weak n-categories, emphasizing
throughout how operads automatically take care of the otherwise mind-numbing
thicket of "coherence laws" that inevitably infest the subject. He doesn't
take well-established notions like "monoidal category" and "bicategory"
for granted - instead, he dives in, takes their definitions apart, and
compares alternatives to see what makes these concepts tick. It's this sort
of careful thinking that we desperately need if we're ever going to reach the
dream of a clear and powerful theory of higher-dimensional algebra. He
does a similar careful analysis of "operads" and "multicategories" before
presenting a generalized theory of operads that's powerful enough to support
various different approaches to weak n-categories. And then he describes
and compares some of these different approaches!
In short: if you want to learn more about operads and n-categories, this is
*the* book to read.
Leinster doesn't say too much about what n-categories are good for, except for
a nice clear introduction entitled "Motivation for Topologists", where he
sketches their relevance to homology theory, homotopy theory, and cobordism
theory. But this is understandable, since a thorough treatment of their
applications would vastly expand an already hefty 380-page book, and diffuse
its focus. It would also steal sales from *my* forthcoming book on higher-
dimensional algebra - which would be really bad, since I plan to retire on
the fortune I'll make from this.
Secondly, Michael Batanin has worked out a beautiful extension of his ideas
on n-categories which sheds new light on their applications to homotopy
theory:
6) Michael A. Batanin, The Eckmann-Hilton argument, higher operads and
E_n spaces, available as math.CT/0207281.
Michael A. Batanin, the combinatorics of iterated loop spaces, available as
arXiv:math.CT/0301221.
Getting a manageable combinatorial understanding of the space of loops
in the spaces of loops in the space of loops... in some space has
always been part of the dream of higher-dimensional algebra. These
"k-fold loop spaces" or have been important in homotopy theory since
the 1970s - see the end of "week199" for a little bit about them.
People know that k-fold loop spaces have k different products that
commute up to homotopy in a certain way that can be summarized by
saying they are algebras of the E_k operad, also called the "little
k-cubes operad". However, their wealth of structure is still a bit
mind-boggling. James Dolan and I made some conjectures about their
relation to k-tuply monoidal categories in our paper "Categorification"
(see "week121"), and now Batanin is making this more precise using
his approach to n-categories - which is one of the ones described in
Leinster's book.
There's also been a lot of work applying higher-dimensional algebra to
topological quantum field theory - that's what got me interested in
n-categories in the first place, but a lot has happened since then.
For a highly readable introduction to the subject, with tons of great
pictures, try:
7) Joachim Kock, Frobenius Algebras and 2D Topological Quantum Field
Theories, Cambridge U. Press, Cambridge, 2003.
This is mainly about 2d TQFTs, where the concept of "Frobenius algebra"
reigns supreme, and everything is very easy to visualize.
When we go up to 3-dimensional spacetime life gets harder, but also more
interesting. This book isn't so easy, but it's packed with beautiful math
and wonderfully drawn pictures:
8) Thomas Kerler and Volodymyr L. Lyubashenko, Non-Semisimple Topological
Quantum Field Theories for 3-Manifolds with Corners, Lecture Notes in
Mathematics 1765, Springer, Berlin, 2001.
The idea is that if we can extend the definition of a quantum field
theory to spacetimes that have not just boundaries but *corners*, we
can try to build up the theory for arbitrary spacetimes from its
behavior on simple building blocks - since it's easier to chop
manifolds up into a few basic shapes if we let those shapes have
corners. However, it takes higher-dimensional algebra to describe all
the ways we can stick together manifolds with corners! Here Kerler
and Lyubashenko make 3-dimensional manifolds going between 2-manifolds
with boundary into a "double category"... and make a bunch of famous
3d TQFTs into "double functors".
Closely related is this paper by Kerler:
9) Thomas Kerler, Towards an algebraic characterization of 3-dimensional
cobordisms, Contemp. Math. 318 (2003) 141-173. Also available as
arXiv:math/0106253.
It relates the category whose objects are 2-manifolds with a circle as
boundary, and whose morphisms are 3-manifolds with corners going
between these, to a braided monoidal category "freely generated by a
Hopf algebra object". (I'm leaving out some fine print here, but
probably putting in more than most people want!) It comes close to
showing these categories are the same, but suggests that they're not
quite - so the perfect connection between topology and higher categories
remains elusive in this important example.
Answer to the riddle: these are the Catalan numbers - i.e., the natural
numbers as written in Catalan. This riddle was taken from the second
volume of Stanley's book on enumerative combinatorics (see "week144").
-----------------------------------------------------------------------
Addenda: Long after this issue was written, we had a discussion on the
n-Category Cafe about the "seven trees in one" problem. Let B be the
set of binary planar trees - the set I was calling T(1) above.
Starting from the isomorphism
B = B^2 + 1
we want to construct an isomorphism
B = B^7
Here is a picture of the proof in Marcelo Fiore's paper:
http://math.ucr.edu/home/baez/seven_trees_in_one_fiore.jpg
At each step he either replaces B^n by B^{n-1} + B^{n+1}, The
underlined portion shows where this will be done. Over at the n-Cafe,
Stuart Presnell made a beautiful picture of this proof:
http://math.ucr.edu/home/baez/seven_trees_in_one_presnell_fiore.gif
He also made a picture of another proof, which is on page
29 of Pierre Ageron's book Logiques, Ensembles, Catégories:
Le Point de Vue Constructif:
http://math.ucr.edu/home/baez/seven_trees_in_one_presnell_bell.gif
You can watch a *movie* of a proof here:
15) Dan Piponi, Arboreal isomorphisms from nuclear pennies, A
Neighborhood of Infinity, September 30, 2007. Available at
http://blog.sigfpe.com/2007/09/arboreal-isomorphisms-from-nuclear.html
It was in the ensuing discussion on this blog that George Bell came
up with his more efficient proof. For a bit more discussion, see:
16) John Baez, Searching for a video proof of "seven trees in one",
n-Category Cafe, July 16, 2009. Available at
http://golem.ph.utexas.edu/category/2009/07/searching_for_a_video_proof_of.html
Now, on to some older addenda!
My pal Squark pointed out that if we try to compute the
generating function for binary trees by making an initial guess for
|T|(x), say t, and repeatedly improving this guess via
t -> x + t^2
the guess will converge to the right answer if t is small - but the
process will fail miserably, with t approaching infinity, if and only
if the complex number x lies outside the Mandelbrot set!
After an earlier version of this Week appeared on the category theory
mailing list, Steve Schanuel posted some corrections. I've tried to
correct the text above as much as possible without making it too
technical - for example, by citing the important work of Robbie Gates,
and distinguishing more clearly between his work on distributive
categories and the paper by Fiore and Leinster, which applies to rig
categories. I tend to talk about 3 different sorts of ring-like
categories in This Week's Finds:
* Rig categories. A "rig category" is one equipped with a symmetric
monoidal structure called + and a monoidal structure called x, with
all the usual rig axioms holding up to natural isomorphism, and these
isomorphisms satisfying a set of coherence laws worked out by Laplaza
and Kelly:
17) M. Laplaza, Coherence for distributivity, Lecture Notes in
Mathematics 281, Springer Verlag, Berlin, 1972, pp. 29-72.
18) G. Kelly, Coherence theorems for lax algebras and distributive
laws, Lecture Notes in Mathematics 420, Springer Verlag, Berlin,
1974, pp. 281-375.
(These authors spoke of "ring categories", but the term "rig category"
is more appropriate since, as in a rig, there need be no additive inverses.)
* 2-Rigs. A "2-rig" is a symmetric monoidal cocomplete category where
the monoidal structure, which we call x, distributes over the colimits,
which we think of as a generalized form of addition. For more on rigs,
2-rigs and structure types see "week191". In a 2-rig, distributivity
is just a property of the monoidal structure, rather than a structure,
as it is in a rig category. However, by choosing a particular coproduct
for each pair of objects, and a particular initial object, we can promote
any 2-rig to a rig category. To get an example of a rig category that's
not a 2-rig, just take any rig and think of it as a discrete category
(a category with only identity morphisms). Another example would be
the category of finite-dimensional vector spaces, since this only has
finite colimits. (Of course, we could make up some sort of "finitary
2-rig" that only had finite colimits, but the profusion of terminology
is already annoying.)
* Distributive categories. A "distributive category" is a category
with finite products and coproducts, the products distributing over
the coproducts. Here again, distributivity is just a property. But,
by choosing specified products and coproducts for every pair of objects,
and choosing terminal and initial objects, we can promote any distributive
category into a rig category. A good example of a 2-rig that is not a
distributive category is the category Vect, with direct sum and tensor
product as + and x. Another example is the discrete category on a rig.
By not distinguishing these, the original version of "week202" made it
sound as if Fiore and Leinster had simply redone Gates' work on distributive
categories. I hope this is a bit clearer now. Schanuel's remarks are
still worth reading for their description of what Gates actually did:
Dear colleagues,
For those who read the most recent long discursion of John Baez, a few
of the errors in the section on distributive categories merit correction:
(1) J. B. suggests that Blass published what Lawvere had already worked
out. In fact, Lawvere (partly to counteract some incorrect uses of
infinite series in analyses of 'data types' in computer science) had
worked out the algebra of the rig presented by one generator X and one
relation X=1+X^2, roughly by the method in (3) below, and conjectured that
this rig could be realized as the isomorphism classes in a distributive
(even extensive) category, which conjecture Blass then proved (and a bit
more) in "Seven Trees...".
(2) The generalization of Blass's theorem to one generator ond one
polynomial relation of the 'fixed-point' form X=p(X), where p is a
polynomial with natural number coefficients and nonzero constant term is
not, as J. B. seems to suggest, due to Fiore and Leinster; it was part of
the prize-winning doctoral thesis of Robbie Gates, who (using a calculus
of fractions) described explicitly the free distributive category on one
object X together with an isomorphism from p(X) to X, proving that this
category is extensive and that its rig of isomorphism classes satisfies no
further relations, i.e. is the rig R presented by one generator and the
one relation above.
(3) If p is as in (2) and of degree at least 2, the algebra of the rig R
is made by J. B. to seem mysterious. It is more easily understood in the
way the X=2X+1 case was treated in my "Negative Sets..." paper; just show
that the Euler and dimension homomorphisms, tensoring with Z and with 2
(the rig true/false) respectively, are jointly injective. In this case the
dimension rig has only three elements, which explains why the Euler
characteristic captures almost, but not quite, everything.
Greetings to all,
Steve Schanuel
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Quote of the week:
"A traveller who refuses to pass over a bridge until he personally tests
the soundness of every part of it is not likely to go far; something must
be risked, even in mathematics." - Horace Lamb
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