A well kept secret?

A well kept secret?

[Ronnie Brown]

In reply to André :


What seems reasonable to do is analysis, namely what is behind the 
success of category theory and how is this success  related to the 
progress of mathematics.
Which implies asking questions of mathematics, some of which have been 
aired in this discussion list. In this way, it should be possible to 
avoid seeming partisan, but to ask serious questions, which should help 
to steer directions, or suggest new ones. Of course lots of great maths 
does not arise in this way, but by following one's nose, but that does 
not mean that such analysis of direction is unhelpful.

I know some argue that this excursion into what might be called the 
theory of knowledge, or into methodology,  seems unnecessary to some. In 
reply I sometimes point to remarks of Einstein on my web site
www.bangor.ac.uk/r.brown/einst.html
or more mundanely retort that normal activities normally require some 
meta discussion: if you want to go on a holiday, you do some planning, 
you don't just rush to the station and buy some tickets. I develop this 
theme in relation to the teaching of mathematics in an article
What should be the output of mathematical education?
on my popularisation and teaching page.

I gave a talk to school children on `How mathematics gets into knots' in 
the 1980s, and a teacher came up to me afterwards and said: `That is the 
first time in my mathematical career that anyone has used the word 
`analogy' in relation to mathematics.' Yet abstraction is about analogy, 
and very powerful it is too. This was part of the motivation behind the 
article
146. (with T. Porter) `Category Theory: an abstract setting for analogy 
and comparison', In: What is Category Theory? Advanced Studies in 
Mathematics and Logic, Polimetrica Publisher, Italy, (2006) 257-274. pdf

There is also interest in the question of how category theory comes to 
be successful, and more successful than, say,  the theory of monoids. 
This seems connected with the underlying geometric structure being a 
directed graph, i.e. allowing a `geography of interaction'. A category 
is also a partial algebraic structure, with domain of definition of the 
operation defined by a geometric condition. Is this enough to explain 
the success?

It is worth noting that the article
Atiyah, Michael, Mathematics in the 20th century, Bull. London Math. 
Soc., {34},  {2002}, 1--15,
suggests that important trends in the 20th century were:
                              higher dimensions, commutative to non 
commutative, local-to-global, and the unification of mathematics,
but does not include the words `category' or `groupoid', let alone 
`higher dimensional algebra'!

This kind of analysis needs to be presented to other scientists, and to 
the public, not only to mathematicians. There is a hunger for knowing 
what mathematics is really up to, in common language as far as possible, 
what new concepts, ideas, etc., and not just `we have solved Fermat's 
last theorem'.

If your analysis of what category theory should do suggests some gaps, 
then that is an opportunity for work!

Good luck

Ronnie Brown


Joyal wrote:

Category theory is a powerful mathematical language.
It is extremely good for organising, unifying and suggesting new directions of research.
It is probably the most important mathematical developpement of the 20th century.

But we cant say that publically.

André Joyal

  



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Re: A well kept secret?

[Andrew Stacey]

This discussion has been very interesting.  I have a couple of comments and
a request, but first a little background.  I've only recently truly
encountered category theory - I describe myself as a differential topologist
and as yet see no reason to change that description, but I've increasingly
needed to use at least the language of category theory to express some of the
things that I come across in algebraic and differential topology and this has
led to me learning some category theory at last.

However, I sometimes feel as though I've stumbled into a party by mistake and
can't find the way out.  I'm quite enjoying being at the party, I ought to
say, but every now and then I sit down in a corner and wonder how I got in,
and also suspect that I missed the Big Announcement at the beginning that said
what the party was for.

All this discussion about a "well kept secret" has gone a bit over my head.
I'm not sure what the secret is!  My forays into the categorical landscape
have been two-fold: understanding operations in cohomology theories and
understanding smooth spaces.  The first, paradoxically, relates to trying to
un-categorify something ("decategorify" now has a mathematical meaning and
I don't intend that); namely, the previous description of what we wanted to
understand was extremely categorical and we wanted a much more "hands on"
description, but that actually just led us from one categorical description to
another (our own journey was quite tortuous, I should say).  The second foray
wouldn't have happened if those I'd been talking to hadn't already been
speaking in categories - I had to learn the language just to join the
conversation.

So when you all talk of a "well kept secret" and something that "went wrong in
the 60s" (didn't everything?), please remember that some of us weren't even
born in the 60s, let alone thinking about mathematics, so haven't a clue
what's going on.  And, as I've tried to say above, I'm an outsider but one
with a favourable view of category theory so if it's hard for me to figure out
what the fuss is about, I'm not surprised that it's hard for anyone further
out.

Let me make these remarks a little more concrete with a request (or
a challenge if you prefer).  In my department, the colloquium is called
"Mathematical Pearls" (gosh, I actually wrote "Perls" first time round; I've
been writing too many scripts lately!).  I'm giving this talk in January.  My
original plan was to say something nice and differential, with lots of fun
pictures of manifolds deforming or knots unknotting, or something like that.
However, the discussion here has set me to thinking about saying something
instead about category theory.  It is a pearl of mathematics, it does have
a certain beauty, there's certainly a lot that can be said, even to a fairly
applied audience as we tend to have here (it is the Norwegian university of
Science and Technology, after all), even without talking about programming
(about which I know nothing).

But for such a talk, I need a story.  I don't mean a historical one (I'm not
much of a mathematical historian anyway), I mean a mathematical one.  I want
some simple problem that category theory solves in an elegant fashion.  It
would be nice if there was one that used category theory in a surprising way;
beyond the idea that categories are places in which things happen (so perhaps
about small categories rather than large ones).

I'm not trying to get anyone to write my talk for me!  It's just that as
someone who only recently engaged with category theory then I'm aware
- painfully aware - that I often miss the point.  But to counter that, then as
  someone who only recently engaged with category theory then I can still
remember fairly vividly why I like it and what convinced me that it was worth
thinking about (and learning about), which will hopefully give the talk
a little more omph.

Thanks in advance for your suggestions,

Andrew Stacey

PS I just remembered something else I was going to mention.  Someone else
mentioned MathOverflow.  Well, there was a question about what was missing
from undergraduate mathematics.  I said "category theory".  It currently lies
5th in the list (out of 28, my other suggestion "how to write with chalk so it
doesn't squeak" is 12th).  More interesting than it's placing is the vast
number of comments that followed, mostly saying that too much "abstract
nonsense" would be off-putting to students.  You can read it at:

http://mathoverflow.net/questions/3973/what-should-be-offered-in-undergraduate-mathematics-thats-currently-not-or-isn



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Re: A well kept secret?

[John Baez]

Andrew Stacey wrote:

> All this discussion about a "well kept secret" has gone
> a bit over my head.  I'm not sure what the secret is!

We could tell you...

... but then it wouldn't be a secret, now, would it?

Seriously, I think the so-called "secret" is the power and glory of
category theory.  And I think some of the older category theorists on
this mailing list have a different attitude than youngsters like you
and me.  They fought to convince the world that category theory was
worthwhile. Some feel they lost that fight.  We came along later and
are a bit puzzled by that attitude: if you look around at the
landscape of mathematics today, categories are everywhere!  From
Grothendieck to Voevodsky to Lurie, etc., much of the most exciting
mathematics of our era would be inconceivable without categories.

I don't think I'll try to tell you the old war stories: others are
better qualified.  But I hope the veterans of those wars take heed of
your comments and realize many young mathematicians naturally find
categories interesting,  exciting, and/or useful.  Certainly there is
much about categories that these youngsters don't understand.  But
they can learn it if you explain it.`

Best,
jb


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Re: A well kept secret

[Joyal, André]

Dear Andrew,

You wrote

>Let me make these remarks a little more concrete with a request (or
>a challenge if you prefer).  In my department, the colloquium is called
>"Mathematical Pearls" (gosh, I actually wrote "Perls" first time round; I've
>been writing too many scripts lately!).  I'm giving this talk in January.  My
>original plan was to say something nice and differential, with lots of fun
>pictures of manifolds deforming or knots unknotting, or something like that.
>However, the discussion here has set me to thinking about saying something
>instead about category theory.  It is a pearl of mathematics, it does have
>a certain beauty, there's certainly a lot that can be said, even to a fairly
>applied audience as we tend to have here (it is the Norwegian university of
>Science and Technology, after all), even without talking about programming
>(about which I know nothing).

>But for such a talk, I need a story.  I don't mean a historical one (I'm not
>much of a mathematical historian anyway), I mean a mathematical one.  I want
>some simple problem that category theory solves in an elegant fashion.  It
>would be nice if there was one that used category theory in a surprising way;
>beyond the idea that categories are places in which things happen (so perhaps
>about small categories rather than large ones).

A colloquium is a good place for expressing wild ideas.
But they must be related to something everyone can understand and touch.
I suggest you talk about "The field with one element" if you think 
the subject can fit your audience.

http://en.wikipedia.org/wiki/Field_with_one_element

Many things in this subject are very speculative
but there are also a few concrete developpements. 
One is the algebraic geometry "under SpecZ" of Toen and Vaquié.
Another due to Borger is using lambda-rings.
What is a lambda-ring?
In their book "Riemann-Roch-Algebra" Fulton and Lang define a lambda-ring
to be a pre-lambda-ring satisfying two complicated identities [(1.4) and (1.5)]
[Beware that F&L are using an old terminology: they call a lambda-ring a "special lambda-ring"
and they call a pre-lambda-ring a "lambda-ring"]
The notion of lambda-ring (ie of "special lambda-ring" in the terminology of F&L)
can be defined in a natural way if we use category theory. 
Let Z[]:CMon ---> CRing be the functor which associates to a commutative monoid M the 
ring Z[M] freely generated by M (it is the left adjoint to the forgetful functor in the
opposite direction). If we compose the functot Z[] with the forgetful functor U:CRing --->Set
we obtain a functor V:CMon --->Set. The algebraic theory of lambda-rings 
can be defined to be the theory of natural operations on the functor V.
The total lambda operation V(M)--->V(M)[[t]] is the group homomorphism Z[M]--->1+tZ[M][[t]] 
which takes an element x\in M to the power series 1+tx.


Best,
André



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RE : categories: Re: A well kept secret

[Joyal, André]

Dear Andrew,

Please disregard my suggestion about a new definition of lambda ring!
My memory may have failed me!
I am not sure the new definition is right!

Best,
André


-------- Message d'origine--------
De: categories@mta.ca de la part de Joyal, André
Date: mar. 15/12/2009 15:14
À: Andrew Stacey; categories@mta.ca
Objet : categories: Re: A well kept secret
 
Dear Andrew,

You wrote

>Let me make these remarks a little more concrete with a request (or
>a challenge if you prefer).  In my department, the colloquium is called
>"Mathematical Pearls" (gosh, I actually wrote "Perls" first time round; I've
>been writing too many scripts lately!).  I'm giving this talk in January.  My
>original plan was to say something nice and differential, with lots of fun
>pictures of manifolds deforming or knots unknotting, or something like that.
>However, the discussion here has set me to thinking about saying something
>instead about category theory.  It is a pearl of mathematics, it does have
>a certain beauty, there's certainly a lot that can be said, even to a fairly
>applied audience as we tend to have here (it is the Norwegian university of
>Science and Technology, after all), even without talking about programming
>(about which I know nothing).

>But for such a talk, I need a story.  I don't mean a historical one (I'm not
>much of a mathematical historian anyway), I mean a mathematical one.  I want
>some simple problem that category theory solves in an elegant fashion.  It
>would be nice if there was one that used category theory in a surprising way;
>beyond the idea that categories are places in which things happen (so perhaps
>about small categories rather than large ones).

A colloquium is a good place for expressing wild ideas.
But they must be related to something everyone can understand and touch.
I suggest you talk about "The field with one element" if you think 
the subject can fit your audience.

http://en.wikipedia.org/wiki/Field_with_one_element

Many things in this subject are very speculative
but there are also a few concrete developpements. 
One is the algebraic geometry "under SpecZ" of Toen and Vaquié.
Another due to Borger is using lambda-rings.
What is a lambda-ring?
In their book "Riemann-Roch-Algebra" Fulton and Lang define a lambda-ring
to be a pre-lambda-ring satisfying two complicated identities [(1.4) and (1.5)]
[Beware that F&L are using an old terminology: they call a lambda-ring a "special lambda-ring"
and they call a pre-lambda-ring a "lambda-ring"]
The notion of lambda-ring (ie of "special lambda-ring" in the terminology of F&L)
can be defined in a natural way if we use category theory. 
Let Z[]:CMon ---> CRing be the functor which associates to a commutative monoid M the 
ring Z[M] freely generated by M (it is the left adjoint to the forgetful functor in the
opposite direction). If we compose the functot Z[] with the forgetful functor U:CRing --->Set
we obtain a functor V:CMon --->Set. The algebraic theory of lambda-rings 
can be defined to be the theory of natural operations on the functor V.
The total lambda operation V(M)--->V(M)[[t]] is the group homomorphism Z[M]--->1+tZ[M][[t]] 
which takes an element x\in M to the power series 1+tx.


Best,
André



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a conjecture

[Joyal, André]

Dear Andrew,

My statement about the theory of lambda ring was wrong.
I would like to correct it.
Let me first recall the original statement.


>What is a lambda-ring?
>In their book "Riemann-Roch-Algebra" Fulton and Lang define a lambda-ring
>to be a pre-lambda-ring satisfying two complicated identities [(1.4) and (1.5)]
>[Beware that F&L are using an old terminology: they call a lambda-ring a "special lambda-ring"
>and they call a pre-lambda-ring a "lambda-ring"]
>The notion of lambda-ring (ie of "special lambda-ring" in the terminology of F&L)
>can be defined in a natural way if we use category theory. 
>Let Z[]:CMon ---> CRing be the functor which associates to a commutative monoid M the 
>ring Z[M] freely generated by M (it is the left adjoint to the forgetful functor in the
>opposite direction). If we compose the functot Z[] with the forgetful functor U:CRing --->Set
>we obtain a functor V:CMon --->Set. The algebraic theory of lambda-rings 
>can be defined to be the theory of natural operations on the functor V.
>The total lambda operation V(M)--->V(M)[[t]] is the group homomorphism Z[M]--->1+tZ[M][[t]] 
>which takes an element x\in M to the power series 1+tx.

I woke up in the middle of the following night
with the strong impression that it could not work.
Here is the problem: the theory of natural operations on the functor V=Z[-] 
is uncountable, whilst the theory of lambda-rings is countable.
To see this, let analyse the set of unary operations on the functor Z[-].
For any monoid M, the algebra Z[M] is equipped with a natural augmentation e:Z[M]--->Z
from which we obtain a natural transformation e:Z[-]--->Z
from the functor Z[-] to the constant functor Z. 
The natural transformation exibits Z as the colimit of the functor Z[-], 
since we have Z[1]=Z and since the monoid 1 is terminal in
 the category of commutative monoids CMon.
Let me denote by Z[M]_n the fiber of the map e:Z[M]--->Z at n\in Z.
The (set valued) functor Z[-]_n is connected, since Z[1]_n={n.1}.
The decomposition
Z[-]=disjoint union_n Z[-]_n 
coincide with the canonical decomposition of the functor Z[-]
as a disjoint union of connected components.
Notices that the functor Z[-]_n is isomorphic to the functor Z[-]_0
since we have n.1+Z[M]_0=Z[M]_n for any monoid M.
Hence the functor Z[-] is isomorphic to the functor Z\times Z[-]_0.
It then follows from the connectedness of the functor Z[-]_0
that we have a bijection

End(Z[-])=End(Z\times Z[-]_0)=Z^Z\times End(Z[-]_0)^Z

Hence the set End(Z[-]) is uncountable, since the set Z^Z is uncountable. 


It seems that the basic idea can be saved by
making a slight modification to the functor V.  
Let me say that an element z in a monoid M is a ZERO ELEMENT 
if we have zx=xz=z for every x\in M. A zero element is unique
when it exists.  I will denote a zero element by 0.
Let me denote by CMonz the category of commutative
monoids with zero element, where a map M--->N
should preserve the zero elements. Then the obvious
forgetful functor CRing--->CMonz has a left
adjoint which associates to M a commutative ring Z[M']. 
The additive group of Z[M'] is the free abelian group on M'=M\setminus{0}. 
If we compose the functot Z['] with the forgetful functor U:CRing --->Set
we obtain a functor V':CMonz --->Set. I conjecture that 
the theory of natural operations on the functor V' is
the algebraic theory of lambda-rings.
The total lambda operation V'(M)--->V'(M)[[t]] 
is the group homomorphism Z[M']--->1+tZ[M'][[t]] 
which takes an element x\in M to the power series 1+tx (notices that 1+t0=1).


Best, 
André




-------- Message d'origine--------
De: Joyal, André
Date: mer. 16/12/2009 08:08
À: Andrew Stacey; categories@mta.ca
Objet : RE : categories: Re: A well kept secret
 
Dear Andrew,

Please disregard my suggestion about a new definition of lambda ring!
My memory may have failed me!
I am not sure the new definition is right!

Best,
André



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Re: A well kept secret?

[Ross Street]

On 15/12/2009, at 5:41 AM, Andrew Stacey wrote:

>  In my department, the colloquium is called
> "Mathematical Pearls"
> I'm giving this talk in January.
>
> But for such a talk, I need a story.

Dear Andrew

Back in the early 90s Todd Trimble gave a beautiful colloquium talk to
our Mathematics Department at Macquarie. It was based on a question in
a book by Halmos which involved finding some group (topological I
think) doing something or other. It was not a categorical problem as
such.

Todd spoke about groups in a category with finite products. The only
categorical theorem he needed was that finite product preserving
functors take groups to groups. I believe he took the definition of
category as known but defined functor, product and internal group.

My vague memory is that he found a group solving the analogous problem
in some fairly combinatorial (presheaf?) category, then found a
product preserving functor to topological spaces to obtain the desired
group.

I hope Todd is reading this and I have jogged his memory enough to
write in more detail. It takes work and ingenuity to design such pearls.

Best wishes,
Ross

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