Re: Help!

Re: Help!

[Mikael Vejdemo Johansson]

On Fri, 5 Oct 2007, Michael Barr wrote:
> What would you say to an undergraduate math club about categories?  I have
> been thinking about it, but I am not sure what to say.  Talk about
> cohomology, which is what motivated E-M?  I don't think so.  Talk about
> dual spaces of finite-dimensional vector spaces?  Maybe, but then what?
>

How about talking about simultaneously existing results in several
categories? The Noetherian isomorphism theorems, while not necessarily the
easiest to nail down exactly when they hold, have always been a strong
motivator at the back of my head for why one might want to look at
algebraic entities codifying things like "All Xs and maps between them".

-- 
Mikael Vejdemo Johansson |  To see the world in a grain of sand
mik@math.su.se           |   And heaven in a wild flower
                          |  To hold infinity in the palm of your hand
                          |   And eternity for an hour

Re: Help!

[Ross Street]

Dear Mike

Some categories that are easily described (even to talented high
school students) are:

the category fun of functions (where objects are natural numbers and
morphisms are functions);
the category mat of matrices (again the objects are natural numbers);
the category brd of braids; and,
the category tang of tangles.

There are enough functors amongst these to be interesting. They are
all monoidal categories.

One can try to discuss other structure the categories have in common
so that strong monoidal
functors (although I probably wouldn't introduce too much such
terminology) preserve it.
For example, duals in mat and tang; trace and braid closure; etc.

One could try to show how the specialized, seemingly ad hoc
Reidemeister moves translate
naturally into the braided monoidal setting.

A hint about how the "new" (mid 1980s)  polynomial link invariants
come from a functor
tang --> mat might be of interest.

Best wishes,
Ross

On 05/10/2007, at 10:52 PM, Michael Barr wrote:

> What would you say to an undergraduate math club about categories?
> I have
> been thinking about it, but I am not sure what to say.  Talk about
> cohomology, which is what motivated E-M?  I don't think so.  Talk
> about
> dual spaces of finite-dimensional vector spaces?  Maybe, but then
> what?

Re: Help!

[Saul Youssef]

Speaking of videos, I think that this one could be great for
motivating students to learn about categories

http://claymath.msri.org/voevodsky2002.mov

Besides making some strong statements about the importance of
categories in the middle of the talk, it's all related to things that
undergraduates know about or are about to learn.

- Saul

On 10/8/07, Toby Bartels <toby+categories@ugcs.caltech.edu> wrote:
> Vaughan Pratt wrote at last part:
>
> >In case You-tube
> >ever has a video on triples you should probably mention any synonyms for
> >"triple" so the students can find the video.
>
> YouTube has a series of 5 video on triples under the name "monads":
> < http://www.youtube.com/results?search_query=monads&search=Search >,
> among others by the Catsters < http://www.youtube.com/user/TheCatsters >.
>
>
> --Toby
>
>
>
>

Re: Help!

[Toby Bartels]

Vaughan Pratt wrote at last part:

>In case You-tube
>ever has a video on triples you should probably mention any synonyms for
>"triple" so the students can find the video.

YouTube has a series of 5 video on triples under the name "monads":
< http://www.youtube.com/results?search_query=monads&search=Search >,
among others by the Catsters < http://www.youtube.com/user/TheCatsters >.


--Toby

Re: Help!

[Vaughan Pratt]

> (It helps to know ahead of time what answers are likely;
>  fortunately there were no pure number theorists at my school.)

At the risk of sounding like a cracked record, how about the division
category in lieu of the division lattice, namely the coproduct
completion of the set P of primes as a discrete category?  For a longer
story use P* instead of P, P with a final object adjoined.  Motivate the
division category by pointing out that only the square-free positive
integers can be recovered as sups of primes in the division lattice.

Vaughan

Re: Help!

[Patrik Eklund]

On Sun, 7 Oct 2007, Michael Barr wrote:

> Hi-tech whiteboards and even video-taping are out.  I don't think we
have
> any of the former and the one case that I know of a lecture that was
> video-taped (a fascinating lecture by Conway in the early '70s in which
he
> showed how the game of Life allowed the simulation of self-reproducing
> Turing-power automata) seems to have disappeared without a trace.  I
will
> probably use a blackboard (or greenboard) and chalk, my favorite medium.

Hi Michael,

Video-taping certainly is out, if it ever was in. Taping is
non-interactive and just silly. Your attitude towards "Hi-tech
whiteboards" sounds too hi-tech as my point indeed was to say what you
say about your favourite medium. That is still also my favourite medium,
but I accept to write or meet virtually in particular if my audience is a
flight distance away. Something is lost when you go virtual, but you also
win some.

Do you resist virtual whiteboards per se, or would you be interested in
trying out a session? Installation is less than 15 minutes, and once we
are online, we could spend another 15 discussing idempotent functors
extendable to monads where E-M and Kleisli coincide. The mouse is your
chalk and your board colour is white.

I've used it so much already over the last years so I cannot work without
it anymore. I can supervise a student from my home or a hotel room in
Tokyo, and nobody knows or even cares who's where.

Cheers,

Patrik

PS And for those who didn't see my mail to Michael, here it is, and
apologies to those who view this purely as spam:

Date: Sun, 7 Oct 2007 07:19:34 +0200 (MEST)
From: Patrik Eklund <peklund@cs.umu.se>
To: Michael Barr <barr@math.mcgill.ca>
Cc: Patrik Eklund <peklund@cs.umu.se>
Subject: Re: categories: Help!

Dear Michael,

No comment (at this point) on content, but let me refer to a previous mail
I sent out on the subject and related to execution.

My idea was to suggest a setup of virtual classrooms so that students and
teacher indeed all over the world can attend a class. Of course, students
and teacher, and in the end content, must be carefully selected.

The reason for my suggestion is that the number of students at many sites
is usually bery low for these courses and we should join forces.

My suggestion is to use "sound-video-whiteboard" techniques as provided
e.g. by Adobe and Marratech. I use the latter.

"Sound-video" is nothing but Skype, but adding whiteboards, that can be
saved and worked with also offline, you have very good possibilities.
The whiteboard mainly accepts non-formatted text, drawings and images. You
can read doc and ppt file which are "pasted" as bitmaps on the whiteboard.
They include desktop sharing if that would be required. Mathematical text
I add through LaTeX, compiling, converting to pdf, and using the snapshot
tool to paste bitmapped formulas on the whiteboard.

Once you get used to it you are actually not (much) slower on the virtual
whiteboard as compared to a real whiteboard. Virtual advantages are e.g.

         - several whiteboards and easy to switch between them
         - more than one can jointly add to whitebooard content
         - can save and open (as mentioned)
         - can prepare whiteboards offline (as mentioned)

If this is inline with your thoughts and you would like to try out
Marratech, let me know.

Best,

Patrik

Re: Help!

[Vaughan Pratt]

I would think the best topics would be those that can be described with
a minimum of jargon.  The problem with category theory is that it is so
steeped in its own jargon as to make it quite an effort to strip it out.

Here are some topics where I would expect that effort to be minimal,
arranged in roughly increasing order of intricacy of definition.  This
should more than fill a one-hour lecture, especially if there are questions.

1.  Thinking of each object T of a category C as both a type and a dual
type, characterize a product AxB in C as an object consisting of all
pairs of T-elements of A and B over all types T, and A+B as dually
consisting of all pairs of T-functionals of A and B over all dual types
T in ob(C).  Section 6 needs pullbacks, they could be done either here
or there, it's neither here nor there.

2.  The category FinBip of finite bipointed sets as the theory of
cubical sets.  The models are arbitrary functors M: Bip --> Set.  You
could look at \Delta for simplicial sets as well or instead, I'm partial
to Bip perhaps because in kindergarten we tended to work more with
cubical than simplicial sets (Western Australian kindergartens had
strong PTOs reflecting epic entanglements).  You could then continue
with FinSet^op as the algebraic theory of Boolean algebras, but that
would entail giving up one of the other segments.

3.  Enriched categories as generalized metric spaces.  People who have a
hard time with abstract objects mixed in with concrete homsets (I
certainly did) will be relieved to know that making the homsets just as
abstract as the objects turns the definition of category into a familiar
object not normally considered part of the categorical basement.

4.  Presheaves on J as colimits of diagrams in J.  If you use Yoneda to
hide the concept of colimit the idea becomes almost trivial.  In the
case J = 1, as C starts from 1 and grows towards Set^1 each new set X as
a new object of C is installed along with an arbitrary choice of C(1,X).
  The composites at X are defined by first installing C(X,Y) for all
existing Y in C, defining fx: 1 --> X --> Y for each x in X and new f in
C(X,Y), and taking C(X,Y) to be maximal subject to Ax[fx=gx] ==> f=g.
These composites then uniquely determine the remaining composites
gf: X --> Y --> Z and fg: W --> X --> Y for W other than 1.  The
completion is complete when every new set is necessarily isomorphic to
one already present.  (Does this have anything to do with Yoneda
structures?  Trying to read about those I discovered I no longer talked
Strine.)

For J the ordinals 1 and 2 as respectively the primitive vertex and the
primitive edge, namely the two reflexive graphs priming the pump for the
rest, there are now two types of element, vertices and edges, with Ax
interpreted as quantifying over all elements of both types; otherwise
everything is as for J = 1.  Point out that whereas all sets are free,
the free graphs are just those with trivial incidences.  If you do
section 6 (triples for matrix multiplication), also point out at some
point that whereas Set^1 is tripleable on Set (the identity), Set^J in
general is tripleable only on Set^{|J|}, important when talking Czech.

5.  Toposes, but *not* the way it is explained on You-tube, which is
completely unmotivated and incomprehensible for anyone who hasn't
already understood them.  The Explanation section
http://en.wikipedia.org/wiki/Topos_theory#Explanation in the Wikipedia
article on elementary toposes touches on the two points that should be
in any explanation of the concept, namely
(i) "subobject" predates "topos," witness CTWM which defines it in
second-order language, and
(ii) monics m: X' --> X are in bijection with pullbacks of the element
(hence monic) t: 1 --> \Omega along morphisms f: X --> \Omega, allowing
one to speak of *the* characteristic function of a monic, thereby
classifying the monics by their characteristic functions, a first-order
notion (whence the "elementary" in "elementary topos") that is in full
agreement with the second-order notion in (i) when applied to a topos.

6.  Matrix multiplication in terms of the Kleisli construction for the
triple for Vct_k.  I just sent out a crib sheet for that which focused
on a difficulty with non-finitary (square summable) linear combinations,
but that difficulty is impossible to absorb in the available time,
better to stick to the finitary operations where matrix multiplication
is tripleable.  You could mention the Haskell programming language and
how they blended the second component of the triple and Kleisli into a
single operator Bind: T(X) --> (X --> T(Y)) --> T(Y), which might get
any programmers in the club interested in Haskell; also point out the
possibility of replacing (X --> T(Y)) by T(Y*X) and its implications for
matrix algebra including Hilbert space.  Stick to finite X in T(X) = k^X
to save the extra step of defining finitary k^X for infinite X (but if
you do decide to do that step it should suffice to point out that 6 of
the 16 binary Boolean operations as 2x2 truth tables have constant rows
or columns or both and then generalize to infinity).  In case You-tube
ever has a video on triples you should probably mention any synonyms for
"triple" so the students can find the video.

Vaughan


Michael Barr wrote:
> What would you say to an undergraduate math club about categories?  I have
> been thinking about it, but I am not sure what to say.  Talk about
> cohomology, which is what motivated E-M?  I don't think so.  Talk about
> dual spaces of finite-dimensional vector spaces?  Maybe, but then what?
>
> Michael
>
>
>

Re: Help!

[Toby Bartels]

Michael Barr wrote:

>What would you say to an undergraduate math club about categories?  I have
>been thinking about it, but I am not sure what to say.  Talk about
>cohomology, which is what motivated E-M?  I don't think so.  Talk about
>dual spaces of finite-dimensional vector spaces?  Maybe, but then what?

When I was a graduate student (recently),
I gave a talk on category theory to other (mostly new) grad students
(as part of a series where advanced students discussed their work).
I began with my definition of category theory for nonmathematicians
("a general theory of how mathematical structures can fit together"),
then gave some basic definitions and an example
(duality in finite-dimensional vector spaces).

Then I asked the audience a very open-ended question:
Tell me what's your favourite branch of mathematics,
and I'll tell you what category theory has to say about it
(to justify the generality in my beginning statement).
What attracted me first to category theory,
and what I think remains impressive about it,
is that you can you can really make good on this challenge.
(It helps to know ahead of time what answers are likely;
 fortunately there were no pure number theorists at my school.)


--Toby Bartels

Re: Help!

[Vaughan Pratt]

George Janelidze wrote:
> (a) a topological space (defined via open sets) has no constant sets, one
> variable set X, and its structure is an element t in PP(X) that is closed
> under finite intersections and arbitrary unions.

The point of categories being (presumably) to shift the burden of
structure from the objects to the morphisms, one would illustrate this
point using your example by pointing out that the topological structure
imputed to a space by the above definition is at least as well imputed
with the definition of a space as the set of continuous functions to the
space from the one-point space together with the set of continuous
functions from it to the Sierpinski space.  It sounded like you were
headed in roughly that direction but then moved on to other points
before getting there.

> (b) a vector space has one constant set A ("the set of scalars"), one
> variable set X ("the set of vectors"), and its structure can be defined as
> element (a,b,c,d) of P(AxAxA)xP(AxAxA)xP(AxXxX)P(XxXxX), where a, b, c, d
> are addition of scalars, multiplication of scalars, scalar (-on-vector)
> multiplication, and addition of vectors respectively; that (a,b,c,d) should
> satisfy familiar conditions of course.

Ditto with the one-dimensional space in place of the one-point and
Sierpinski space (which itself is a kind of one-dimensional space for
topology).

> 4. Linear algebra tells us that instead of working with linear
> transformations of finite-dimensional vector spaces we can work with
> matrices, but one cannot formulate this properly without using the concept
> of equivalence of categories (the category of finite-dimensional K-vector
> spaces is equivalent to the category of natural numbers with matrices with
> entries from K as morphisms).

You may be setting the bar for "proper" higher than necessary to satisfy
us engineers.  I'm currently involved in a computer project where the
question of the proper formulation of matrices came up.  One team had
formulated them in terms of the Kleisli construction for monads as
defined in CTWM, the monad in question being the one that you yourself
would surely come up with for the variety Vct_C, C the complex numbers.

Unfortunately that formulation was giving the computer conniptions.
This could have been construed as bearing out your point were it not for
the fact that another team came along with a reformulation of monads
that overcame the difficulty.

Since I know this list is good at keeping secrets (such as the secret of
categories) I'll be happy to share with you all, in my next message, my
confidential report on the current status of this reformulation.  Our
CEO is not convinced of the correctness of the reformulation, the fact
that it fixed the buggy behavior notwithstanding, and has asked me for a
qualified expert second opinion---where better than this list for a
question about monads?

Vaughan

Re: Help!

[Michael Barr]

I want to thank all who replied and I will take all your comments
seriously.  I would love to talk about Stone duality and such but I don't
think many of our undergrads have ever heard of a topological space.  They
have heard of topology of course, but mostly they think it concerns things
like toruses and Klein bottles.  So they know nothing of the point set
underpinnings of algebraic topology.  My last term before retirement I
taught a course called topology and spent exactly 6 lectures on point-set
theory (taking a beeline to the Tychonoff theorem) before introducing pi_1
and covering spaces.  The students were last year undergrads and a couple
of grad students.

Do they know what a boolean algebra is?  Probably some do, some don't.
Groups and abelian groups they will know about, probably modules, etc.
Vector space duality is a familiar example, for finite dimension at least.

Hi-tech whiteboards and even video-taping are out.  I don't think we have
any of the former and the one case that I know of a lecture that was
video-taped (a fascinating lecture by Conway in the early '70s in which he
showed how the game of Life allowed the simulation of self-reproducing
Turing-power automata) seems to have disappeared without a trace.  I will
probably use a blackboard (or greenboard) and chalk, my favorite medium.

One suggestion that does appeal is to start with universal mapping
properties to explain products and sums.  One thing that always struck me
was Bill Lawvere's observation that the dual of the usual definition of
function as a subset of a product s.t.... namely as a quotient of a sum
s.t.... actually corresponds closely to the usual picture we draw when we
introduce functions for the first time.    I guess I could do worse than
build the whole lecture around universal mapping properties.  I could
mention the somewhat unmotivated definition of (infinite) product of
topological spaces as a perfect example of the universal viewpoint.
Especially as topologists had come up with that definition independent of
category theory.

Michael

RE: Help!

[Marta Bunge]

Hi, Michael,

I am in the same predicament but, since I am speaking at this math club one
week after you (November 6), I do hope to be able to use anything you do in
your own talk!

I also thought a lot about this problem and discarded one topic after
another. Finally, I have decided to speak about the uses of infinitesimals
in the synthetic calculus of variations, aiming at giving an algebraic
(synthetic) proof of the well known fact that, for a paths functional
("energy"), its critical points agree with the geodesics. This requires that
I introduce adjoint functors and cartesian closed categories and the notion
of a ring object of line type. If you will do any of these yourself I could
use it. Informally, I will argue constructively and acually prove things.
Historical considerations may be briefly mentioned at the beggining of the
talk,  and the conceptual advantages of the synthetic method at the end.
This will be an expanded portion of my paper "Synthetic Calculus of
Variations" (with M. Heggie) in Contemporary Mathematics 30, 1983. I hope
that this helps you as well as me.

Best wishes,
Marta


>From: Michael Barr <barr@math.mcgill.ca>
>To: Categories list <categories@mta.ca>
>Subject: categories: Help!
>Date: Fri, 5 Oct 2007 08:52:29 -0400 (EDT)
>
>What would you say to an undergraduate math club about categories?  I have
>been thinking about it, but I am not sure what to say.  Talk about
>cohomology, which is what motivated E-M?  I don't think so.  Talk about
>dual spaces of finite-dimensional vector spaces?  Maybe, but then what?
>
>Michael
>
>
>

_________________________________________________________________
Send a smile, make someone laugh, have some fun! Check out
freemessengeremoticons.ca

Re: Help!

[Ronnie Brown]

The great thing about categories is that they allow analogies between
different mathematical structures: see the paper
R. Brown and  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.
An example of the analogy is between the category of sets and the category
of directed graphs:
``Higher order symmetry of graphs'', {\em Bull. Irish Math.
Soc.} 32 (1994) 46-59.
Here one easily sees non Boolean logics, of course.

The word `analogy' seems to be underused in teaching undergraduates, but
that is what abstraction is about, is it not? A teacher told me after a
lecture on knots that was the first time he had heard the word analogy used
in relation to mathematics! ( I discussed prime knots.)

The other possibility is to advertise categorical structures: I advertised
higher dimensional algebra to an international  conference of
neuroscientists in Delhi in 2003, pointing out the unlikelihood of the brain
working entirely serially, and also the concept of colimit with an email
analogy. A senior Indian neuroscientist came up to me afterwards and said
that was the first time he had heard a seminar by a mathematician which made
any sense! This is written up in
(R. Brown and  T. Porter), `Category theory and higher dimensional
algebra: potential descriptive tools in neuroscience', Proceedings
of the International Conference on Theoretical Neurobiology,
Delhi, February 2003, edited by Nandini Singh, National Brain
Research Centre, Conference Proceedings 1 (2003) 80-92.

These are all downloadable from
http://www.bangor.ac.uk/r.brown/publicfull.htm
or my home page.

See also
http://www.bangor.ac.uk/r.brown/outofline/out-home.html
for a general talk.

As said before, I see higher dimensional algebra as the study of
mathematical structures with operations defined under geometrical
conditions, thus allowing a combination of algebra and geometry, in a way
which even Atiyah might like (see his paper on `20th century mathematics'
Bull LMS 44  (2002) 1-15, in which the words `category' and `groupoid' do
not appear).

I have found giving general talks makes one think hard about the underlying
ideas and motivation.

Ronnie





----- Original Message -----
From: "Michael Barr" <barr@math.mcgill.ca>
To: "Categories list" <categories@mta.ca>
Sent: Friday, October 05, 2007 1:52 PM
Subject: categories: Help!


> What would you say to an undergraduate math club about categories?  I have
> been thinking about it, but I am not sure what to say.  Talk about
> cohomology, which is what motivated E-M?  I don't think so.  Talk about
> dual spaces of finite-dimensional vector spaces?  Maybe, but then what?
>
> Michael

Re: Help!

[George Janelidze]

Dear Michael,

The question is so impossibly big, and it was asked and answered in various
form so many times, and no responsibility for the originality/completeness
of the answer can be assumed... So, I am not afraid to begin with a few
obvious remarks, looking forward to seeing many other remarks from others:

1. Many people believe that mathematics is about mathematical structures,
but what is a mathematical structure in general? According to Bourbaki, one
should begin with two finite sequences of sets, say, A, B, C,... and X, Y,
Z,...; let us call them constant sets and variable sets respectively. Then
build any scale, which is a finite sequence of sets obtained by taking
finite products and power sets of the sets above. Then, (briefly and
roughly!) call a structure (of a fixed type) an element of one of the sets
in the scale satisfying certain conditions. For example:

(a) a topological space (defined via open sets) has no constant sets, one
variable set X, and its structure is an element t in PP(X) that is closed
under finite intersections and arbitrary unions.

(b) a vector space has one constant set A ("the set of scalars"), one
variable set X ("the set of vectors"), and its structure can be defined as
element (a,b,c,d) of P(AxAxA)xP(AxAxA)xP(AxXxX)P(XxXxX), where a, b, c, d
are addition of scalars, multiplication of scalars, scalar (-on-vector)
multiplication, and addition of vectors respectively; that (a,b,c,d) should
satisfy familiar conditions of course.

Then, according to Bourbaki again, structures are useless without
morphisms - but what is a morphism? It turns out that only isomorphisms can
be defined, and the class of morphisms should in each case be CHOSEN
depending on the "experience" of working with a given class of structures in
such a way, that it is closed under composition and contains all
isomorphisms (or, better, also determine isomorphisms as invertible
morphisms). Is it possible that the most fundamental concept of mathematics
is described as such a monster? Is not it better to study abstract
categories?

And what is the problem of defining morphisms? To answer this question one
should learn about functors, covariant and contravariant ones, and what do
they preserve and what not.

2. Set theory is wonderful: it gives precise mathematical definitions to
concepts that were only intuitively understood before. But... it often makes
definitions complicated. And category theory very often solves this problem
by using universal properties. For example the set N - better to say, the
structure (N,0,s) - of natural numbers is "designed to count"; therefore N
should have the first element 0 (unfortunately 0 is better than 1) and the
successor function s from N to N - and be "the best such", i.e. initial
such. Moreover, developing basic properties of this structure out of
initiality is much easier than out of Peano axioms. In fact all classical
number systems have simple elegant definitions via universal properties.
Moreover, the axioms of set theory itself are much less elegant than their
elementary-topos-theoretic counterparts.

3. We can say that set theory is more fundamental than arithmetics: e.g.
children learn addition by counting the number of elements in the disjoint
union. But category theory is more fundamental than set theory: e.g. it
makes the disjoint union a more natural operation... but, more importantly

(a) we all know that, say, a+b=b+a, ab=ba, PvQ<=>QvP,... for all natural
numbers a and b and all logical formulas P and Q - but one needs category
theory to see these as the same result (and we can add cartesian products,
free product, intersection, union, and many other operations to it).

(b) or, say, we all know that composites of injections are injections and
composites of surjections are surjections - but again, one needs category
theory to see these as the same result;

(c) and even exponentiation and implication are the same... and categorical
logic follows...

4. Linear algebra tells us that instead of working with linear
transformations of finite-dimensional vector spaces we can work with
matrices, but one cannot formulate this properly without using the concept
of equivalence of categories (the category of finite-dimensional K-vector
spaces is equivalent to the category of natural numbers with matrices with
entries from K as morphisms). And there are so many other equivalences and
dualities (that are not isomorphisms) playing fundamental roles in various
branches of mathematics. Not to mention that the aforementioned matrices
themselves arise from a categorical observation (finite products = finite
coproducts).

5. Proper understanding of Eilenberg - Mac Lane work, and the work of their
followers, friends, and not-quite-friends in category theory and proper
understanding of what the 21st century mathematics would be without it
obviously requires far better knowledge of mathematics than the 21st century
students have. But may be we should at least say that our Fields Medal
(Grothendieck) is not less than any other Fields Medal...

George Janelidze

----- Original Message -----
From: "Michael Barr" <barr@math.mcgill.ca>
To: "Categories list" <categories@mta.ca>
Sent: Friday, October 05, 2007 2:52 PM
Subject: categories: Help!


> What would you say to an undergraduate math club about categories?  I have
> been thinking about it, but I am not sure what to say.  Talk about
> cohomology, which is what motivated E-M?  I don't think so.  Talk about
> dual spaces of finite-dimensional vector spaces?  Maybe, but then what?
>
> Michael

Help!

[Michael Barr]

What would you say to an undergraduate math club about categories?  I have
been thinking about it, but I am not sure what to say.  Talk about
cohomology, which is what motivated E-M?  I don't think so.  Talk about
dual spaces of finite-dimensional vector spaces?  Maybe, but then what?

Michael