Re: Help!

[Vaughan Pratt <pratt@cs.stanford.edu>]

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
>
>
>