mm-272

Subject: Re: Poincare Conjecture appears to be
solvedOriginator: tchow@lagrange.mit.edu.mit.edu (Timothy
Chow)Originator: israel@math.ubc.ca (Robert Israel)>According
to a report in today's Boston Globe[...]>it looks like
Perelman has solved the Poincare Conjecture, and>possibly the
stronger Geometrization Conjecture.Specifically, Milnor refers
to a remarkable trio of preprints and tofurther details to be
provided in a fourth preprint. The Los AlamosArXiv currently
has four preprints by Perelman, but the fourth one seemsto be
on a different topic from the first three. Am I right in
assumingthat the fourth preprint that Milnor refers to is not
yet available, andthat therefore Perelman has not yet released
his complete proof to thepublic?-- Tim Chow
tchow-at-alum-dot-mit-dot-eduThe range of our
projectiles---even ... the artillery---however great,
willnever exceed four of those miles of which as many thousand
separate us fromthe center of the earth. ---Galileo, Dialogues
Concerning Two New Sciences===Subject: Re: Poincare Conjecture
appears to be solvedOriginator: tchow@lagrange.mit.edu.mit.edu
(Timothy Chow)Content-Length: 1122Originator:
rusin@vesuvius>The Los Alamos ArXiv currently has four
preprints by Perelman, but>the fourth one seems to be on a
different topic from the first three.>Am I right in assuming
that the fourth preprint that Milnor refers to>is not yet
available, and that therefore Perelman has not yet
released>his complete proof to the public?It has been poin out
to me that the preprint that I said seemsto be on a different
topic is in fact by a different author, whoselast name also is
Perelman. Sorry for the gaffe. Anyway, the questionstill
stands...is it true that the complete proof has not yet been
madeavailable to the public? This would explain why there
hasn't been afinal verdict yet from the experts even though
it's been such a longtime since the result was announced.--
Tim Chow tchow-at-alum-dot-mit-dot-eduThe range of our
projectiles---even ... the artillery---however great,
willnever exceed four of those miles of which as many thousand
separate us fromthe center of the earth. ---Galileo, Dialogues
Concerning Two New Sciences===Subject: Re: Poincare Conjecture
appears to be solvedContent-Length: 1731Originator:
rusin@vesuviusI don't think it is taking any longer than it
did for Wiles's proof of Fermat's Last Theorem. Or for Hales's
proof of the Kepler Conjecture.According to people I've consul,
Perelman's proof pushes the limitsof what is known in geometry
and topology. So it will take some time before people have
worked through the proof carefully enoughto say with
confidence that the proof is correct and complete.The feedback
I get is that there is nothing obviously wrongabout Perelman's
proof and that he has introduced newideas that appear to
overcome much and maybe all of the missingpieces to Hamilton's
program for proving the geometrization conjecture.But that's
still far from enough.Perelman's papers are relatively short,
so it will take some time beforepeople can decide whether they
can fill in all the details.>>The Los Alamos ArXiv currently
has four preprints by Perelman, but>>the fourth one seems to
be on a different topic from the first three.>>Am I right in
assuming that the fourth preprint that Milnor refers to>>is
not yet available, and that therefore Perelman has not yet
released>>his complete proof to the public?> It has been poin
out to me that the preprint that I said seems> to be on a
different topic is in fact by a different author, whose> last
name also is Perelman. Sorry for the gaffe. Anyway, the
question> still stands...is it true that the complete proof
has not yet been made> available to the public? This would
explain why there hasn't been a> final verdict yet from the
experts even though it's been such a long> time since the
result was announced.===Subject: Paper published by Algebraic
and Geometric TopologyOriginator: israel@math.ubc.ca (Robert
Israel)The following paper has been published:Algebraic and
Geometric
TopologyURL:http://www.maths.warwick.ac.uk/agt/AGTVol3/agt-3-
46.abs.htmlTitle:Cell-like resolutions preserving
cohomological dimensionsAuthor(s):Michael LevinAbstract:We
prove that for every compactum X with dim_Z X <= n >= 2 there
is acell-like resolution r: Z --> X from a compactum Z onto X
such thatdim Z <= n and for every integer k and every abelian
group G such thatdim_G X <= k >= 2 we have dim_G Z <=k. The
latter property impliesthat for every simply connec CW-complex
K such that e-dim X <= K wealso have e-dim Z <=
K.Keywords:Cohomological dimension, cell-like
resolutionAuthor(s) address(es):Department of Mathematics Ben
Gurion University of the Negev P.O.B. 653 Be'er Sheva 84105,
ISRAELEmail: mlevine@math.bgu.ac.il===Originator:
israel@math.ubc.ca (Robert Israel)ILLC Scientific
Publications----------------------------This document contains
the titles of the reports that were publishedby the Institute
for Logic, Language and Computation (ILLC) this year.All ILLC
reports are available from the ILLC bureau: ILLC Bureau
University of Amsterdam Plantage Muidergracht 24 NL-1018 TV
Amsterdam The NetherlandsMany reports are also electronically
available, by WWW at http://www.illc.uva.nl/Publications and
or FTP at
ftp://ftp.science.uva.nl/pub/theory/illc/researchreports/The
ILLC bureau may be contac by email, at
illc@science.uva.nlReports are numbered Series-Year-Number,
where `Series' is one of PP = Prepublication Series MoL =
Master of Logic
Thesis--------------------------------------------------------
--------------Title: A Note on Modeling TheoriesAuthor: Johan
van BenthemTitle: Structural Properties of Dynamic
ReasoningAuthor: Johan van BenthemTitle: Is There Still Logic
in Bolzano's Key?Author: Johan van BenthemTitle: The Epistemic
Logic of IF GamesAuthor: Johan van BenthemTitle: What Logic
Games are Trying to Tell UsAuthor: Johan van BenthemTitle:
Rational Dynamics and Epistemic Logic in GamesAuthor: Johan
van BenthemTitle: 'One is a Lonely Number': on the Logic of
CommunicationAuthor: Johan van BenthemTitle: Categorial
Grammar at a Cross-RoadsAuthor: Johan van BenthemTitle:
Conditional Probability and Update LogicAuthor: Johan van
BenthemTitle: Tableaux for Quantified Hybrid LogicAuthor:
Blackburn, Maarten MarxTitle: Silver Measurability and its
Relation to other Regularity PropertiesAuthor: Jorg Brendle,
Lorenz Halbeisen, Benedikt LoweTitle: The Pointwise View of
Determinacy: Arboreal Forcings, Measurability and Weak
MeasurabilityAuthor: Benedikt LoweTitle: Canonical varieties
with no canonical axiomatisationAuthor: Ian Hodkinson, Yde
VenemaTitle: A Hierarchy of norms defined via Blackwell
gamesAuthor: Benedikt LoweTitle: Stone CoalgebrasAuthor:
Clemens Kupke, Alexander Kurz, Yde VenemaTitle: The Pragmatic
Dimension of IndefinitesAuthor: Paul DekkerTitle: Extending
ILM with an operator for $Sigma_1$-nessAuthor: Evan
GorisTitle: The Simulation Technique and its Consequences for
Infinitary Combinatorics under the Axiom of Blackwell
DeterminacyAuthor: Benedikt LoweTitle: Determinacy for
infinite games with more than two players with
preferencesAuthor: Benedikt LoweTitle: The Categorial
Fine-Structure of Natural LanguageAuthor: Johan van
BenthemTitle: Logic and the Dynamics of InformationAuthor:
Johan van BenthemTitle: What One May Come to KnowAuthor: Johan
van BenthemTitle: Optimal Interpolation in ALCAuthor: Stefan
SchlobachTitle: Monotonic Modal LogicsAuthor: Helle Hvid
HansenTitle: All normal extensions of S5-squared are finitely
axiomatizableAuthor: Nick Bezhanishvili, Ian HodkinsonTitle:
Erd.9as graphs resolve Fine's canonicity problemAuthor: R.
Goldblatt, I. Hodkinson, Y. VenemaTitle: Explaining New
Phenomena in Terms of Previous PhenomenaAuthor: Rens BodTitle:
Some Intuitionistic Provability and Preservativity Logics (and
their interrelations)Author: Chunlai ZhouTitle: A Study of
Stemming Effects on Information Retrieval in Bahasa
IndonesiaAuthor: Fadillah TalaTitle: A Combined System for
Update Logic and Belief RevisionAuthor: Guillaume AucherTitle:
Source Code Retrieval using Conceptual GraphsAuthor: Gilad
Mishne===Subject: This Week's Finds in Mathematical Physics
(Week 200)Originator: baez@math-cl-n01.math.ucr.edu (John
Baez)Originator: israel@math.ubc.ca (Robert Israel)Also
available at http://math.ucr.edu/home/baez/week200.htmlThis
Week's Finds in Mathematical Physics - Week 200John Baez Happy
New Year! I'm making some changes in my life. For many years
I've dreamt of writing a book on higher-dimensional algebra
that will explainn-categories and their applications to
homotopy theory, representation theory, quantum physics,
combinatorics, logic - you name it! It's an intimidating goal,
because every time I learn something new about these subjects I
want to put it in this imaginary book, so it keeps getting
longer and longer in my mind! Actually writing it will require
heroic acts of pruning. But, I want to get star. It'll be
freely available online, and it'll show up here as
itmaterializes - but so far I've just got a tentative
outline:1) John Baez, Higher-Dimensional Algebra,
http://math.ucr.edu/home/baez/hda.htmlUnfortunately, I'm very
busy these days. As you get older, duties accumulate like
barnacles on a whale if you're not careful! When I star
writing This Week's Finds a bit more than ten years ago, I was
lonely and bored with plenty of time to spare. My life is very
different now: I've got someone to live with, a house and a
garden that seem to need constant attention, a gaggle of grad
students, and too many invitations to give talks all over the
place.In short, the good news is I'm never bored and there's
always something fun to do. The bad news is there's always TOO
MUCH to do! So, a while ago I decided to shed some duties and
make more time for things I consider really important:
thinking, playing the piano, writing this book... and yes,
writing This Week's Finds. First I quit working for all the
journals I helped edit. Then I star job it's really fun to
quit. But doing so didn't free up nearly enough time. So now
I've also decided to stop moderating the newsgroup This is
painful, because I've learned so much from this newsgroup over
the last 10 yeamet so many interesting people, and had such
fun. I thank everyone on the group. I'll miss you! I'll
probably be backwhenever I get lonely or bored.Ahem. Before I
get weepy and nostalgic, I should talk about some math. This
November in Florence there was a conference in honor of the
40th anniversary of Bill Lawvere's Ph.D. thesis - a famous
thesis calledFunctorial Semantics of Algebraic Theories, which
explored the applications of category theory to algebra, logic
and physics. There are videos of all the talks on the
conference website:2) Ramifications of Category Theory,
http://ramcat.scform.unifi.it/but right now this website seems
to be down.This conference was organized and funded by Michael
Wright, a businessman with a great love of mathematics and
osophy, so it was appropriate that it was held in the old city
of Cosimo de Medici, Renaissance banker and patron of scholars.
And since there were talks both by mathematiciansand osophers -
especially Alberto Peruzzi, a osopher at theUniversity of
Florence who helped run the show - I couldn't help but
remember Cosimo's Platonic Academy, which spearheaded the
rebirth of classical learning in Renaissance Italy. When not
attending talks, I spent a lot of time roaming around twisty
old streets, talking category theory at wonderful restaurants,
reading The Rise and Fall of the House ofMedici, and
desperately trying to soak up the overabundance of
incredibleart and architecture: the Ponte Vecchio, the Piazza
del Duomo, the SantaCroce where everyone from Galileo to Dante
to Machiavelli is buried....Ahem. Math!What was Lawvere's
thesis about? It's never been published, so I've never read it
- though I hear it's going to be. So, my impression of its
contents comes from gossip, rumors and later research that
refers to his work.Lawvere star out as a student of Clifford
Truesdell, working on continuum mechanics, which is the very
practical branch of field theory that deals with fluids,
elastic bodies and the like. In theprocess, Lawvere got very
interes in the foundations of physics, particularly the
notions of continuum and physical theory. Somehow he decided
that only category theory could give him the toolsto really
make progress in understanding these notions. After all,
thiswas the 1960s, and revolution was in the air. So, he
somehow got himself sent to Columbia University to learn
category theory from Sam Eilenberg, In my own education I was
fortunate to have two teachers who used the term foundations
in a common-sense way (rather than in the speculative way of
the Bolzano-Frege-Peano-Russell tradition). This way is
exemplified by their work in Foundations of Algebraic
Topology, published in 1952 by Eilenberg (with Steenrod), and
The Mechanical Foundations of Elasticity and Fluid Mechanics,
published in the same year by Truesdell. The orientation of
these works seemed to be concentrate the essence of practice
and in turn use the result to guide practice. It may seem like
a big jump from the down-to-earth world of continuum mechanics
to category theory, but to Lawvere the connection made perfect
sense - and while I've always found his writings inpenetrable,
after hearing him give four long lectures in Florence I think
it makes sense to me too! Let's see if I can explain it.
Lawvere first observes that in the traditional approach to
physical theories, there are two key players. First, there are
concrete particulars - like specific ways for a violin string
to oscillate, or specific ways for the planets to move around
the sun. Second, there are abstract generals: the physical
laws that govern the motionof the violin string or the
planets. In traditional logic, an abstract general is called a
theory, while a concrete particular is called a model of this
theory. A theory is usually presen by giving some mathematical
language, some rules of deduction, and then some axioms. A
model is typically some sort of map that sends everything in
the theory to something in the world of sets andtruth values,
in such a way that all the axioms get mapped to true. Since
theories involve playing around with symbols according to
fixedrules, the study of theories is often called syntax.
Since themeaning of a theory is revealed when you look at its
models, thestudy of models is called semantics. The details
vary a lot depending on what you want to do, and physicists
rarely bother to formulate theirtheories axiomatically, but
this general setup has been regarded as the ideal of rigor
ever since the work of Bolzano, Frege, Peano and Russell
around the turn of the 20th century.And this is what Lawvere
wan to overthrow! Actually, I'm sort of kidding. He didn't
really want to overthrow this setup: he wan to radically build
on it. First, he wan to free the notion of model from the
chains of set theory. In other words, he wan to consider
models not just in the category of sets, but in
othercategories as well. And to do this, he wan a new way of
describingtheories, which is less tied up in the nitty-gritty
details of syntax.To see what Lawvere did, we need to look at
an example. But thereare so many examples that first I should
give you a vague sense of the*range* of examples. You see, in
logic there are many levels of what you might call strength or
expressive power, ranging from wimpy languages that don't let
you say very much and deduction rules that don't let you prove
very much, to ultra-powerful ones that let you do all sorts of
marvelous things. Near the bottom of this hierarchy there's
the propositional calculus where we only get to say things
like((P implies Q) and (not Q)) implies (not P)Further up
there's the first-order predicate calculus, where we get to
say things likefor all x (for all y ((x = y and P(x)) implies
P(y)))Even further up, there's the second-order predicate
calculus where we get to quantify over predicates and say
things likefor all x (for all y (for all P (P(x) iff P(y))
implies x = y))Etcetera... And, while you might think it's
always best to use the most powerful form of logic you can
afford, this turns out not to be true!One reason is that the
more powerful your logic is, the fewer categoriestheories
expressed in this logic can have models in. This point
maysound esoteric, but the underlying principle should be
familiar. Whichis better: a hand-opera drill, an electric
drill, or a drill press? A drill press is the most powerful.
But I forgot to mention: you're using it to board up broken
windows after a storm. You can't carry adrill press around, so
now the electric drill sounds best. But anotherthing: this is
in rural Ghana! With no electricity, now the hand-opera drill
is your tool of choice.In short, there's a tradeoff between
power and flexibility. Specializedtools can be powerful, but
they only operate in a limi context. These days we're all
painfully aware of this from using computers: fancy software
only works in a fancy environment! Lawvere has even come up
with a general theory of how this tradeoff works in
mathematical logic... he called this the theory of doctrines.
But I'm getting way ahead of myself! He came up with doctrines
in 1969, and I'm still trying to explain his 1963 thesis.Just
like traditional logic, Lawvere's new approach to logic has
been studied at many different levels in the hierarchy of
strength. He began fairly near the bottom, in a realm
traditionally occupied by something called universal algebra,
developed by Garrett Birkhoff in 1935. The idea here was that
a bunch of basic mathematical gadgets can be defined using
very simple axioms that only involve n-ary operations on some
set and equations between different ways of composing these
operations. A theory like this is called an algebraic theory.
The axioms for an algebraic theory aren't even allowed to use
words like and, or, not or implies. Just equations.Okay, now
for an example.A good example is the algebraic theory of
groups. A group is a set equipped with a binary operation
called multiplication, a unary operation called inverse, and a
nullary operation (that is, a constant) called the unit,
satisfying these equational laws: (gh)k = g(hk) ASSOCIATIVITY
1g = g LEFT UNIT LAW g1 = g RIGHT UNIT LAWg^{-1}g = 1 LEFT
INVERSE LAW gg^{-1} = 1 RIGHT INVERSE LAWSuch a primitive
gadget is robust enough to survive in very rugged
environments... it's more like a stone tool than a drill
press!Lawvere noticed that we can talk about models of these
axioms not just in the category of sets, but in any category
with finite products. The point is that to talk about an n-ary
operation, we just need to be able to take the product of an
object G with itself n times and consider a morphismf: G x ...
x G -> G |- n times -|For example, the category of smooth
manifolds has finite products, so we can talk about a group
object in this category, which is just a *Lie group*. The
category of topological spaces has finite products, so we can
talk about a group object in this category too: it's a
*topological group*. And so on. But Lawvere's really big idea
was that there's a certain categorywith finite products whose
only goal in life is to contain a groupobject. To build this
category, first we put in an object G Since our category has
finite products this automatically meansit gets objects 1, G,
G x G, G x G x G, and so on. Next, we put in a binary
operation called multiplication, namely a morphismm: G x G ->
GWe also put in a unary operation called inverse:inv: G ->
Gand a nullary operation called the unit:i: 1 -> GAnd then we
say a bunch of diagrams commute, which express allthe axioms
for a group lis above.Lawvere calls this category the theory
of groups, Th(Grp). The object G is just like a group - but
not any *particular* group, since its operations only satisfy
those equations that hold in *every* group!By calling this
category a theory, Lawvere is suggesting that like a theory of
the traditional sort, it can have models - and indeedit can! A
model of theory of groups in some category X with
finiteproducts is just a product-preserving functorF: Th(Grp)
-> XBy the way things are set up, this gives us an
objectF(G)in C, together with morphismsF(m): F(G) x F(G) ->
F(G)F(inv): F(G) -> F(G)F(i): F(1) -> F(G)that serve as the
multiplication, inverse and identity elementfor F(G)... all
making a bunch of diagrams commute, that expressthe axioms for
a group!So, a model of the theory of groups in X is just a
group object in X.Whew. So far I've just explained the *title*
of Lawvere's PhD thesis: Functorial Semantics of Algebraic
Theories. In Lawvere's approach, an algebraic theory is given
not by writing down a list of axioms, but by specifying a
category C with finite products. And the semantics of such
theories is all about product-preserving functors F: C ->
X.Hence the term functorial semantics.Lawvere did a lot
starting with these ideas. Let me just briefly summarize, and
then move on to his work on topos theory and mathematical
physics. Wise mathematicians are interes not just in models,
but also the homomorphisms between these. So, given an
algebraic theory C,Lawvere defined its category of models in
X, say Mod(C,X), to have product-preserving functors F: C -> X
as objects and natural transformations between these as
morphisms. For example, taking C to be the theory of groups
and X to be the category of sets, we get the usual category of
groups:Mod(Th(Grp),Set) = GrpThat's reassuring, and that's how
it always works. What's less obvious, though, is that one can
always recover C from Mod(C,Set) together with its forgetful
functor to the category of sets. In other words: not only can
we get the models from the theory, but we can also get back
the theory from its category of models!I explained how this
works in week136 so I won't do so again here. This result
actually generalizes an old theorem of Birkhoff on universal
algebra. But fans of the Tannaka-Krein reconstruction theorem
for quantum groups will recognize this duality between
theories and theircategory of models as just another face of
the duality between algebras and their category of
representations - the classic example being the Fourier
transform and inverse Fourier transform! And this gives me an
excuse to explain another bit of Lawvere's jargon: while a
theory is an abstract general, and particular model of itis a
concrete particular, he calls the category of *all* its models
in some category a concrete general. For example, Th(Grp) is an
abstract general, and any particular group is a concrete
particular, but Grp is a concrete general. I mention this
mainly because Lawvere flings around this trio of terms quite
a bit, and some people find them off-putting. There are lots
of reasons to find his work daunting, but this need not be
one.In short, we have this kind of setup: ABSTRACT GENERAL
CONCRETE GENERAL theory models syntax semanticsand a precise
duality between the two columns!I would love to dig deeper in
this direction - I've really justscratched the surface so far,
and I'm afraid the experts will bedisappoin... but I'm even
more afraid that if I went further,the rest of you readers
would drop like flies. So instead, let me say a bit about
Lawvere's work on topos theory and physics. Most practical
physics makes use of logic that's considerably stronger than
that of algebraic theories, but still considerably weaker than
what most of us have been brainwashed into accepting as our
defaultsetting, namely Zermelo-Fraenkel set theory with the
axiom of choice. So if we want, we can do physics in a context
less general than an arbitrary category with finite products,
while still not restricting ourselves to the category of sets.
This is where topoi come in - they're a lot like the category
of sets, but vastly more general. Topos theory was born when
Grothendieck decided to completely rewrite algebraic geometry
as part of a massive plan to prove the Weil conjectures.
Grothendieck was another revolutionary of the early 1960s, and
he arrived at his concept of topos sometime around 1962. In
1963, Lawvere and Myles Tierney took this concept - now called
a Grothendieck topos - and made it both simpler and more
general, arriving at the present definition. Briefly put, a
topos is a category with finite limits, exponentials, and a
subobject classifier. But instead of saying what these words
mean, I'll just say that this lets you do most of what you
normally want to do in mathematics, but without the law of
excluded middle or the axiom of choice. One of the many
reasons this middle ground is so attractive is that it lets
you do calculus with infinitesimals the way physicists enjoy
doingit! Lawvere star doing this in 1967 - he called it
synthetic differential geometry. Basically, he cooked up some
axioms on a toposthat let you do calculus and differential
geometry with infinitesimals. The most famous topos like this
is the topos of schemes - algebraicgeometers use this one a
lot. The usual category of smooth manifolds is not even a
topos, but there are topoi that can serve as a substitute,
which have infinitesimals.I won't list the axioms of synthetic
differential geometry, but themain idea is that our topos needs
to contain an object T called the infinitesimal arrow. This is
a rigorous version of those little arrows physicists like to
draw when talking about vectors: -----> The usual problem with
these little arrows is that they need to bereally tiny, but
still point somewhere. In other words, the headcan't be at a
finite distance from the tail - but they can't be at the same
place, either! This seems like a paradox, but one can neatly
sidestep it by dropping the law of excluded middle - or in
technicaljargon, working with a non-Boolean topos. That sounds
like a drastic solution - a cure worse than the disease,
perhaps! - but it's really not so bad. Indeed, algebraic
geometers are perfectly comfortable with the topos of schemes,
and they don't even raise an eyebrow over the fact that this
topos is non-Boolean - mainly because you're allowed to use
ordinary logic to reason *about*a topos, even if its internal
logic is funny.But enough logic! Let's do some geometry! Let's
say we're in sometopos with an infinitesimal arrow object, T.
I'll call the objectsof this topos smooth spaces and the
morphisms smooth maps. Howdoes geometry work in here?It's very
nice. The first nice thing is that given any smooth space X, a
tangent vector in X is just a smooth map f: T -> Xthat is, a
way of drawing an infinitesimal arrow in X. In general,
themaps from any object A of a topos to any other object B
form an objectcalled B^A - this is part of what we mean when
we say a topos has exponentials. So, the space of all tangent
vectors in X is X^T. And this is what people usually call the
tangent bundle of X! So, the tangent bundle is pathetically
simple in this setup: it's justa space of maps. This means we
can compose a tangent vector f: T -> X with any smooth map g:
X -> Y to get a tangent vector gf: T -> Y. This is what people
usually call pushing forward tangent vectors. This trick gives
a smooth map between tangent bundles, the differential of g,
which it makes sense to callg^T: X^T -> Y^TMoreover, it's
pathetically easy to check the chain rule:(gh)^T = g^T h^TAnd
so far we haven't used *any* axioms about the object T - just
basic stuff about how maps work!We can also define higher
derivatives using T. For second derivativeswe start with T x
T, which looks like an infinitesimal square. Thenwe mod out by
the mapS_{T,T}: T x T -> T x Tthat switches the two factors.
You should visualize this map as reflection across the
diagonal. When we mod out by it, we get a quotient space that
deserves the nameT^2/2!and if we now use some axioms about T,
it turns out that a smooth mapf: T^2/2! -> Xpicks out what's
called a second-order jet in X. This is a conceptfamiliar from
traditional geometry, but not as familiar as it should be.The
information in a second-order jet consists of a point in X,
the first derivative of a curve through X, and also the
*second* derivative of a curve through X. Or in physics lingo:
position, velocity and acceleration! We can go ahead and define
nth-order jets using T^n/n! in a perfectlyanalogous way, and
the visual resemblance to Taylor's theorem is by nomeans an
accident... but let me stick to second derivatives, since
I'mtrying to get to Newton's good old F = ma.Just as the space
of all tangent vectors in X is the tangent bundle X^T, the
space of all 2nd-order jets in X is the 2nd-order jet
bundleX^{T^2/2!}Using some axioms about T, we can show there
is a smooth map T^2/2! -> Twhich throws out the second-order
infinitesimal data and justkeeps the first-order part. This
gives a smooth mapp_X: X^{T^2/2!} -> X^Tfrom the 2nd-order jet
bundle to the tangent bundle. Intuitivelyyou can think of this
as sending any position-velocity-accelerationtriple, say
(q,q',q), to the pair (q,q'). Now for the fun part: Lawvere
defines a dynamical law to be a smooth map going the other
way:s_X: X^T -> X^{T^2/2!}such that s_X followed by p_X is the
identity. In other words, it's a way of mapping any
position-velocity pair (q,q') to a triple (q,q',q). So, it's a
formula for acceleration in terms of position and velocity!
There is a category where an object is a smooth space equipped
with a dynamical law and a morphism is a lawful motion: thatis,
a smooth mapf: X -> Ythat makes the obvious diagram commute:
s_X X^T -------------> X^{T^2/2!} | | | | | | f^T | |
f^{T^2/2!} | | | | | | V s_Y V Y^T -------------> Y^{T^2/2!}In
particular, if we take R to be the real numbers - time - and
equip it with the law saying q = 0 meaning that time ticks at
an unchanging rate, then a lawful motionf: R -> Xis precisely
a trajectory in X that follows the law, meaning that the
acceleration of the trajectory is the desired function of
positionand velocity. This example is a setup for the
classical mechanicsby replacing R by a higher-dimensional
space.I'm sure many of you have the same impression that I had
when seeingthis stuff, namely that it's a bit quixotic for a
high-powered mathematicianto be reformulating the foundations
of classical mechanics here at the turnof the 21st century,
instead of working on something cutting-edge likestring
theory. Even if Lawvere's approach is better, one can't help
but wonder if it gives truly *new* insights, or just a clearer
formulationof existing ones. And either way, one can't help
wonder: does he actually expect enough people to learn this
stuff to make a difference? Does he really think topos theory
can break the Microsoft-like grip that ordinary set theory has
on mathematics? (Note the software analogy raising its ugly
head again. Zermelo-Fraenkel set theory is a bit like the
Windows operating system: once you're locked into it, it's
hard to imagine breaking out. You use it because everyoneelse
does and you're too lazy to do anything about it. Topos theory
is more like the open source movement: you're welcome and even
expec to keep tinkering with the code.)I have some sense of
the answer to these questions. First of all, Lawvere wants to
do math the right way regardless of whether it's popular. But
secondly, he's been hard at work trying to make the subject
accessible to beginners. He's recently written a couple of
textbooks you don't need a degree in math to read:3) F.
William Lawvere and Steve Schanuel, Conceptual Mathematics: A
First Introduction to Categories, Cambridge U. Press,
Cambridge, 1997. 4) F. William Lawvere and Robert Rosebrugh,
Sets for Mathematics,Cambridge U. Press, Cambridge, 2002. And
third, the great thing about topos theory is that you
don'tneed to accept it to profit from it. In math, what really
mattersis not believing the axioms but coming up with good
ideas. Topos theory is full of good ideas, and these are bound
to propagate.I'll finish off with some references to help you
learn more aboutthis stuff.Alas, I believe Lawvere's thesis is
still lurking in the stacks at Columbia University:5) F. W.
Lawvere, Functorial semantics of algebraic theories,
Dissertation, Columbia University, 1963.and so far he's only
gotten around to publishing a brief summary:6) F. William
Lawvere, Functorial semantics of algebraic
theories,Proceedings, National Academy of Sciences, U.S.A. 50
(1963), 869-872.But, you can find expositions of his work on
algebraic theories hereand there. Here's a gentle one geared
towards computer scientists:7) Roy L. Crole, Categories for
Types, Cambridge U. Press, Cambridge,1993.A considerably more
macho one is available free online:8) Michael Barr and Charles
Wells, Toposes, Triples and Theories. Springer-Verlag, New
York, 1983. Available for free electronically at
http://www.cwru.edu/artsci/math/wells/pub/ttt.html This book
also talks about sketches, which are a way of
syntacticallypresenting a category with finite products. It
also serves as an introduction to topoi... umm, or at least
toposes. I used to find itfearsomely difficult and dry. Now I
don't, which is sort of scary.A really beautiful more advanced
treatment of algebraic theories andalso essentially algebraic
theories can be found here:9) Maria Cristina Pedicchio,
Algebraic Theories, in Textos de Matematica:School on Category
Theory and Applications, Coimbra, July 13-17, 1999,pp. 101-159.
Someone should urge her to make this available online - it's
alreadyin TeX, and it deserves to be easier to get!Shortly
after his thesis, Lawvere tackled topoi in this paper:10) F.
William Lawvere, Elementary theory of the category of sets,
Proceedings of the National Academy of Science 52 (1964),
1506-1511.the like:11) F. William Lawvere, Algebraic theories,
algebraic categories, and algebraic functoin Theory of Models,
North-Holland, Amsterdam (1965), 413-418.12) F. William
Lawvere, Functorial semantics of elementary theories, Journal
of Symbolic Logic, Abstract, 31 (1966), 294-295.13) F. William
Lawvere, The category of categories as a foundation for
mathematics, in La Jolla Conference on Categorical Algebra,
Springer, Berlin 1966, pp. 1-20.14) F. William Lawvere, Some
algebraic problems in the context offunctorial semantics of
algebraic theories, in Reports of the MidwestCategory Seminar,
eds. Jean Benabou et al, Springer Lecture Notes inMathematics
No. 61, Springer, Berlin 1968, pp. 41-61.Then came his work on
doctrines, which I vaguely alluded to a whileback:15) F.
William Lawvere, Ordinal sums and equational doctrines,
Springer Lecture Notes in Mathematics No. 80, Springer,
Berlin,1969, pp. 141-155.I think he first published on
synthetic differential geometry inLawvere star publishing his
ideas on mathematical physics in the late 1970s, though he
must have been thinking about them all along:17) F. William
Lawvere, Categorical dynamics, in Proceedings of Aarhus May
1978 Open House on Topos Theoretic Methods in Geometry,
Aarhus/Denmark (1979).18) F. William Lawvere, Toward the
description in a smooth topos of the dynamically possible
motions and deformations of a continuous body, Cahiers de
Topologie et Geometrie Differentielle Categorique 21 (1980),
337-392.In 1981, Anders Kock came out with a textbook on
synthetic differentialgeometry:19) Anders Kock, Synthetic
Differential Geometry, Cambridge U. Press, Cambridge, 1981.
More recently, Lawvere came out with a book on applications of
category theory to physics:19) F. William Lawvere and S.
Schanuel, editoCategories in Continuum Physics, Springer
Lecture Notes in Mathematics No. 1174,Springer, Berlin, 1986.
The quote about Lawvere's teachers is from:20) F. William
Lawvere, Foundations and applications: axiomatization andat
http://www.math.ucla.edu/~asl/bsl/0902/0902-006.psand this
gives a good overview of his ideas, though not easy to
read!Finally, Colin McLarty - whom I was deligh to meet in
Florence - hasa nice quick introduction to synthetic
differential geometry inhis textbook on categories and topos
theory:21) Colin McLarty, Elementary Categories, Elementary
Toposes, Clarendon Press, Oxford, 1995. Along with Lawvere's
books Conceptual Mathematics and Sets forMathematics, this is
the one reference that's really good forbeginners! Okay... now
that everyone is gone except the people who are absolutelynuts
about category theory, let me say a bit more about doctrines
and theory-model duality. The nuts who are still reading are
probably disappoin that I kept everything very gentle and
expository and didn't drop any mind-blowing bombshells of
abstraction, which is what they like about category theory!
So, let's turn up the abstraction afew notches.What's a
doctrine?Well, in week89 I described a monad in an arbitrary
2-category. But most of the time when people talk about monads
they mean monads in Cat, the 2-category of all categories.
These are the most importantmonads - but I've never really
said what they're good for! I need tocome clean and explain
this now, since a doctrine is a categorifiedversion of a
monad. What monads are good for is to describe how objects in
one category can be regarded as objects of some other category
equipped with extra structure. This theme pervades mathematics,
and is of the utmost importance. For example: groups are sets
equipped with extra structure, abelian groups are groups
equipped with extra structure, rings are abelian groups
equipped with extra structure, and so on. We keep building up
fancier gadgets from simpler ones. And pretty much whenever
wedo, there's a monad lurking in the background, running the
show! Suppose we've got two categories C and D, and the
objects of D areobjects of C equipped with extra structure.
Then we get a pair of adjoint functors:R: D -> CL: C -> DThe
right adjoint R sends each D-object to its underlying
C-object, and the left adjoint L sends each C-object to the
free D-object on it. Often R is called a forgetful functor.
For example, ifC = SetandD = Grpthen we can take the
underlying set of any group, and thefree group on any set.We
get a monad on C by lettingT = LR: C -> CThen, we can use
facts about adjoint functors to get natural transformations
called multiplicationm: TT => Tand the uniti: 1_C => TUsing
more facts about adjoint functowe can check that these satisfy
associativity and the left and right unit laws. I didall this
in week92 so I won't do it again here. The upshot isthat T is
a lot like a monoid - which is why Mac Lane dubbed it a monad.
Now, monoids like to *act* on things, and the same is true
formonads. It turns out that a monad T on C can act on any
object of C. When this happens, we call that object an algebra
of T,or a T-algebra for short. And when our monad comes from a
pairof adjoint functors as above, the main way we get
T-algebras isfrom objects of D. And in nice cases, T-algebras
are the *same*as objects of D.So, for example, we can describe
groups as T-algebras where T issome monad on the category of
sets. And we can describe abeliangroups as T-algebras where T
is some monad on the category of groups.And we can describe
rings as T-algebras where T is some monad onthe category of
abelian groups. And so on!To really see how this works, we'd
need to look at a few examples.I remember when James Dolan was
first teaching me this stuff in a little coffeeshop here in
Riverside, which has since gone out ofbusiness. I considered
monads too abstract and dug my heels in like a stubborn mule,
refusing to learn about them - until I went through a bunch of
examples and saw that *yes*, this monad business really *does*
capture the essence of what it means to build up fancy gadgets
from simple ones by adding extra structure! And by now I'm
completely sold on it. One reason is the relation to topology,
which I explained in part N of week118, and also week174.But
alas, I'm too eager to get to the *really* cool stuff to work
through examples right now. So if you're a complete novice at
monads, you'll have to work out some examples yourself. Right
now, I'll just say a bit of fancier stuff to fill in a couple
gaps for the semi-experts.First, when I said in nice cases, I
really meant that the category of T-algebras is equivalent to
D when the forgetful functor R: D -> C is monadic. A bit more
precisely: for any monad T on C there's a categoryof
T-algebras, which is usually called C^T for some silly
reason.And, whenever we have a pair of adjoint functors R: D
-> C and L: C -> D, we get a monad T = LR and a functor from D
to C^T. This is just a careful way of saying that any D-object
gives us a T-algebra. And finally, we say that R is monadic if
this functor from D to C^T is an equivalence of categories.
There's a theorem by Beck that says how to tell when a functor
is monadic, just by looking at it.Second, to make the analogy
between monoids and monads precise,we just need to realize
that a monad on C is a monoid object in the monoidal category
hom(C,C). I already explained this in week92,in even greater
generality than we need here, but we need this nowbecause I'm
about to categorify monads and get doctrines.Okay: so, monads
are good for describing objects equipped with extrastructure
and properties. But now suppose we want to describe
*categories* equipped with extra structure and properties! For
example, the categories with finite products that I was talking
about earlier, or topoi. There are LOTS of different
interestingkinds of categories equipped with extra structure
and properties, andeach of them gives a different kind of
*logic*: the logic that worksinside this kind of category! The
more structure and properties ourcategory has, the more
powerful logic we can use inside it. This iswhat gives the
hierarchy of expressive power I was talking about.So, it pays
to have a good general way to describe categories equippedwith
extra structure and properties. And this is what Lawvere's
doctrines do!I've said how monads on a category C are good for
describing objects of C equipped with extra structure and
properties. But there's a certain category called Cat whose
objects are categories! So, let's take C = Cat! A monad on Cat
will describe categoriesequipped with extra structure and
properties.And this is the simplest definition of doctrine: a
monad on Cat.However, those of you familiar with n-categories
will realize thatit's odd to talk about the category of all
categories. Not because of Russell's paradox - though that's a
problem too, forcing us to talkabout the category of *small*
categories - but because what's reallyimportant is the
2-CATEGORY of all categories. It's best to thinkof Cat as a
2-category. But this suggests that we should work witha
categorified, *weakened* version of monad when defining
doctrines.For this, we need to categorify and weaken the
concept of monad.People have done this, and the result is
sometimes called a pseudomonad, but I prefer to call it a weak
2-monad, since I have dreams of categorifying further, and I
don't want my notation to becomeridiculous. I'd rather talk
about weak 3-monads than pseudopseudomonads,wouldn't you?
Furthermore, if you look up pseudomonad in thedictionary
you'll get this: PSEUDOMONAD: bacterium usually producing
greenish fluorescent water-soluble pigment; some pathogenic
for plants and animals.Yuck! So, let's be very general and
sketch how to define a weak 2-monad in any weak 3-category
(aka tricategory). Given a weak 3-category C and an object c
of C, a weak 2-monad on c is just a weak monoidal category
object in hom(c,c). Huh? Well, hom(c,c) is a weak monoidal
2-category, which is precisely the right environment in which
to define a weak monoidal category object, and that's what
we're doing here. Start with the usual definition of a weak
monoidal category, which is a gadget living in Cat. Cat is an
example of a weak monoidal 2-category, and we can write down
the same definition in *any* weak monoidal 2-category
X,getting the concept of weak monoidal category object in X.
Then,take X = hom(c,c). (Of course I'm lying slightly here:
Cat is more strict than youraverage weak monoidal 2-category,
so it may not be immediately obvioushow to generalize the
concept of weak monoidal category as I'm suggesting. Still, I
claim it's not hard if you know about this stuff.)Now that you
know how to define a weak 2-monad on any object c of a
3-category C, you can take c to be Cat and C to be 2Cat... and
this is what we really should call a doctrine.Unsurprisingly,
people often consider stricter versions of theconcept of
2-monad and doctrine. For example, most people define their
pseudomonads not in a weak 3-category but just a semistrict
one, also known as a Gray-category - since 2Cat is oneof
these. For more details, try these papers:22) R. Blackwell, G.
M. Kelly, and A. J. Power, Two-dimensional monadtheory, Jour.
Pure Appl. Algebra 59 (1989), 1-41.23) Brian Day and Ross
Street, Monoidal bicategories and Hopf algebroids, Adv. Math.
129 (1997) 99-157. 24) F. Marmolejo, Doctrines whose structure
forms a fully faithful adjoint string, Theory and Applications
of Categories 3 (1997), 23-44.Available at
http://www.tac.mta.ca/tac/volumes/1997/n2/3-02abs.html23) S.
Lack, A coherent approach to pseudomonads, Adv. Math. 152
(2000),179-202. Also available at
http://www.maths.usyd.edu.au:8000/u/stevel/papers/
psm.ps.gzAnyway, suppose T is a doctrine. Then we get a
2-category of T-algebras Cat^T, whose objects we should think
of as categories equipped with extra structure of type T. The
classic example would be categories with finite products. Just
as Lawvere thought of these as algebraic theories, we can think
of *any* T-algebra as a theory of type T, and define its
category of models: given T-algebras C and D, the category of
models of C in D is hom(C,D), where the hom is taken in Cat^T.
Depending on what doctrine T we consider, we get many different
forms of logic, and I'll just list a few to whet your appetite:
Cat^T = categories with finite products = algebraic theories
gives what one might call algebraic logic - purely equational
reasoning about n-ary operations. The theory of groups, or
abelian groups, or rings lives here. Cat^T = symmetric
monoidal categories gives a sort of logic that allows for
theories known as operads and PROPs - see week191 for more.
This doctrine is weaker than the previous one, since we can
only use equations where all the same variables appear on both
sides, with no duplications or deletions. If we go further in
this direction we obtain various sorts of quantum logic. Cat^T
= categories with finite limits = essentially algebraic
theories gives what one might call essentially algebraic
logic. This doctrine is strong than that of algebraic
theories, since it allows partially defined operations that
are defined only when some equations hold. The theory of
categories lives here, since composition of morphisms is an
operation of this sort. Cat^T = regular categories gives
regular logic. This doctrine is even stronger, since it allows
for theories that involve relations as well as n-ary
operations. Cat^T = cartesian closed categories gives the
typed lambda-calculus. This allows for operations on
operations on operations... etc. Cat^T = topoi gives topos
logic.The typed lambda-calculus is very popular in theoretical
computerscience, and I recommend Crole's book ci above for more
about howit's rela to cartesian closed categories. A good
introduction totopos logic is McLarty's book ci above. For an
exhaustive study of many other sorts of logic that should be
on this list but aren't, I recommend part D of this book:24)
Peter Johnstone, Sketches of an Elephant: a Topos
TheoryCompendium, Oxford U. Press, Oxford. Volume 1,
comprising Part A:Toposes as Categories, and Part B:
2-categorical Aspects of ToposTheory, 720 pages, 2002. Volume
2, comprising Part C: Toposes as Spaces, and Part D: Toposes
as Theories, 880 pages, 2002.We can do a lot of fun stuff with
all these different forms of logic,and people have indeed done
so... but I think I'll stop here. Mypoint is merely that
higher category theory and logic go hand-in-glove, and there
is plenty of room for exploration here, especially if we keep
categorifying - and also keep trying to craft our logic to
real-worldapplications, both in quantum theory and computer
science.I wish you all a Happy New Year, and good luck on all
your
adventures.---------------------------------------------------
--------------------mathematics and physics, as well as some
of my research papecan beobtained
athttp://math.ucr.edu/home/baez/For a table of contents of all
the issues of This Week's Finds,
tryhttp://math.ucr.edu/home/baez/twf.htmlA simple jumping-off
point to the old issues is available
athttp://math.ucr.edu/home/baez/twfshort.htmlIf you just want
the latest issue, go
tohttp://math.ucr.edu/home/baez/this.week.html===Subject: Re:
Left-invariant metricContent-Length: 561Originator:
rusin@vesuvius> So how about weakening the hypothesis, to
allow for anti-automorphisms? It is easy to see that the
universal covering G of E(2), the group oforien isometries of
the affine euclidean plane, acts simplytransitively on R^3 as
isometries, and hence has a flat metricisometric to the
3-dimensional euclidean space.So, endowed with this metric, G
has a three-dimensional compact groupof isometries, whereas G
has no nontrivial compact subgroups.Of course, G is not simple
of higher rank, so that this does notcontredict the result quo
by L. Kramer.===Subject: Re: Left-invariant metricOriginator:
baez@math-cl-n01.math.ucr.edu (John Baez)Content-Length:
2015Originator: rusin@vesuvius>> Let G be a connec
(unimodular) Lie group, endowed with a left-invariant>>
metric. Let u be an isometry such that u(1)=1. Does it follows
that u is a>> group automorphism?>No.>Consider the group SU(2)
and see it as the set of quaternions>with norm 1. I will
denote by ||q|| the norm of a quaternion>q (i.e. ||q|| =
sqrt(q*q^*), wher q^* stands for the conjugate>quaternion).
Consider in SU(2) the distance d defined by> d(q,r) = ||q -
r||.>Then d is left-invariant (BTW, it is also
right-invariant),>since the quaternionic norm preserves the
product.>Now take u(q) = q^* = q^(-1). Then u is an isometry
and u(1) =>= 1, but u is not a group homomorphism.Indeed, this
counterexample works for every compact simple Lie group.Every
such group has a metric which is both left- and
right-invariant.In fact, the metric with these properties is
unique up to multiplication by a positive constant. It's
defined using the so-called Killing form on the Lie
algebra.Every automorphism is an isometry with respect to this
metric. But,so is the map sending each group element to its
inverse ... and this map is never an automorphism, since
taking inverses is an automorphism only for abelian groups,
and I'm using the usual definition of compact simple Lie
group, which explicitly excludes abelian groups. It is,
however, true that if our compact simple Lie group is
connec,and we use any metric that's both left- and right-
invariant, every isometry mapping the identity to itself is
*either* an automorphism *or* the composite of an automorphism
and taking inverses. I should also warn Cornulier that we can
easily find a compact Liegroup with a metric on it that's both
left- and right- invariant, and an automorphism of this group
that's not an isometry. (Hint:use the torus.) So, the answer
to the converse of his question is also No. ===Subject: Re:
Set Theoretic Statement Equivalent to the Existence of the
Algebraic ClosureContent-Length: 1397Originator:
rusin@vesuvius>> I've been told, that the existence of the
algebraic closure of a>> field doesn't require full AC, but
requires some. Is there a set>> theoretic formulation of the
weaker version of AC, that is equivalent>> to the fact, that
every field has an alegbraic closure?>You might want to look
at the book The Axiom of Chioce by Thomas J.>Jech published in
1973 by North-Holland. It is out of print now, but>you may
be>able to find a copy.>Many of the proofs are brief outlines.
It discusses weaker versions of>the AC.>Hope this helps.Another
place to look is the book by Paul Howard and Jean
Rubin,_Consequences of the Axiom of Choice_. It seems to me
that thePrime Ideal Theorem would be adequate, as the finite
sets generaby the solutions of equations have the property
that any finite number of them can be put together
consistently, and thus theTychonov Compactness Theorem for
Hausdorff spaces would give acommon solution.-- This address
is for information only. I do not claim that these viewsare
those of the Statistics Department or of Purdue
University.Herman Rubin, Department of Statistics, Purdue
Universityhrubin@stat.purdue.edu Phone: (765)494-6054 FAX:
(765)494-0558===Subject: Re: Inequality with complex
numbersIn-reply-to:
<Pine.GSO.4.58.0312300150260.18499@ascc.artsci.wustl.edu>
Epigone-thread: jantorvermOriginator: rusin@vesuvius but this
is not what I have in my mind.Well, let z(k)=exp(2*k*Pi*i/n);
k=0,1, ... ,n-1 ,and f,g functions such that
1<=|f(0)|<=|f(1)|<= ... <=|f(n-1)|, and, 1<=g(0)<=g(1)<= ...
<=g(n-1); A(n):=Sum[z(k)*f(k),{k,0,n-1}]
B(n):=Sum[z(k)*f(k)*g(k),{k,0,n-1}]
C(n):=Sum[z(k)*f(k)*(g(k)^2),{k,0,n-1}]. It seems to me that
|B(n)-A(n)|<=|C(n)-A(n)| and |B(n)-C(n)|<=|C(n)-A(n)| for
every n,f,g. Am I correct? Luka.