Subject: categories: Re: cracks and pots

From: David Yetter <[email-masked]>
Date: Fri, 24 Mar 2006 22:22:27 -0500

<fontfamily><param>Times</param>Fellow categorists,

Jim Stasheff has been appealing to me to comment on the role of
category theory in knot theory in the context of the ‘cracks and pots’
thread.

In that regard, let me begin with the story I tell in the introduction
to my book:  

At a Joint Summer Research conference a number of years back, Moishe
Flato at some point offered the usual dismissal of category
theory--‘it is a mere language’.  Kolya Reshetikhin and I undertook
that evening to disabuse him of the notion by explaining Shum’s
coherence theorem (the one-object version, being ‘the braided monoidal
category with two-sided duals compatible with the braiding is
monoidally equivalent to the category of framed tangles’).  This is a
remarkable theorem--a structure absolutely natural from the internal
structure of category theory is essentially identical to the key
geometric sturctures in 3- and 4-manifold topology, framed tangles
being simply ‘relative versions of’ the framed links on which the
Kirby calculi for 3- and 4-manifolds depend.  It is one of several
theorems relating category theory, and with it  a great deal of
algebra, to geometric topology, all of which had a “who ordered that?”
feel about them.  It is also the only basis on which the connection
between knot theory and quantum groups can be explained:  the category
of representations of a quantum group has the algebraic structure for
which framed tangles are a free model!

I retired before the point had sunk in, leaving Kolya to continue the
discussion.  The next morning as I sat in the back of the main lecture
hall, Flato came in, tapped me on the shoulder, and with a thumbs up,
said “Hey! Viva les categories!. . .these new ones, the braided
monoidal ones.”

Now, Shum’s theorem is merely the first of several, all of which give
one the “who ordered that?” impression, at least once on starts
thinking of TQFT’s.  The other two that come to mind require a bit of
set up to state fully, but I will spare you all now:  they are Abrams
theorem that a 2-dimensional TQFT is equivalent to a Frobenius
algebra, and a theorem due to myself and Crane, and Kerler, that in a
certain category of cobordisms between surfaces with boundary, the
handle (a torus with a hole cut in it) has the structure of a Hopf
algebra (CY, K) which is self-dual (CY) and admits a right integral
(K), and that every surface with a circle boundary is a
Yetter-Drinfel’d module over the handle.  (For an ordinary finite
dimensional Hopf algebra, YD-modules are modules over the Drinfel’d
double, but they exist more generally, for infinite dimensional Hopf
algebras or Hopf algebra objects in arbitrary monoidal categories,
where Drinfel’d doubles don’t exist.)

All of these are part and parcel of a different face of category
theory than one saw in the old days:  category theory as algebra,
rather than category theory as foundations.

Flato’s dismissal was directed at category theory as foundations.  It
is easily ignored if one is interested in foundations of mathematics,
since most mathematicians really don’t care about foundations.  For
example, many mathematicians pay lip service to the attitude ‘set
theory is ‘the’ foundation of mathematics’, but then turn around and
talk about ‘the real numbers’.  Which ‘real numbers’?  Dedekind cuts? 
equivalence classes of Cauchy sequences? a complete Archimedian field
constructed from surreal numbers?  Now we, as categorists, know the
question is silly:  one doesn’t bother asking which of a family of
isomorphic structures one means, because they are isomorphic.  It is
in the practical sense of describing the basic structure of what
mathematicians actually do, that category theory is a superior
‘foundation’ to set theory.  (Was there ever a time when the epsilon
tree defining an element of a smooth manifold ever mattered to
anyone?)  The dim view of category theory in many mathematical circles
is surely due to mathematicians’ boredom with foundations--and
attitude which might be summed up as “Set theory was bad enough.  Why
open up all those questions again? Just let me do my geometry,
algebra, or whatever.” 

I reading this thread, I wonder how much of the concern about public
perceptions of category theory is really concern that ‘categories as
algebra’ has become the public face of category theory, concern on the
part of those who are fond of ‘categories as foundations’.  

‘Categories as foundations’ served the subject poorly in relations to
most mathematicians, but well in relation to computer science:  only
categorists were willing to take up the challenge of polymorphic type
theory.  If you thought ‘set theory is *the* foundation’, you bashed
your head against Russell’s paradox and were no help the the folk in
CS. We (really those of *you* who took up the challenge) were the only
mathematicians who had any hope of being helpful.

On the other hand, those of us who set our sights on ‘core
mathematics’ have been better served by ‘categories as algebra’:  the
applications to knot theory (and geometric topology more generally),
homotopy theory, deformation theory, and physics all flow from this
‘face’ of category theory.  

Even if those of us whose love is ‘categories as foundations’ can be a
little uneasy with the other face of the subject getting applied to
physics and drawing fire from outside mathematics, those of us whose
love is ‘categories as algebra’ can be uneasy about applications of
the other face to philosophy (as pointed to in Peter Arndt’s last
post), which are sure to be vilified (philosophers and humanists
always vilify their rivals, as I am learning from my daughter who is
studying philosophy).  ‘Categories as algebra’ at least got a ‘Viva!’
from one of the fathers of deformation quantization.

Best Thoughts,

David Yetter</fontfamily>

On 23 Mar 2006, at 14:45, Peter Arndt wrote:

<excerpt>Dear category theorists,

I would like to support Krzysztof Worytkiewicz's remark that "cat
theory

needs to be demystified in first place rather than to be sold" from a

different side: I have recently come across several publications and

research projects of philosophers who have become over-enthusiastic
with

category theory. In certain circles category theory seems to have
gained a

nimbus of an all-encompassing theory of everything, be it part of

mathematics or not, see for example

http://lists.debian.org/debian-devel/2000/10/msg02048.html for an
expression

of such opinions or http://ru.philosophy.kiev.ua/rodin/Endurance.htm
for a

crude offspring of them. Such exaggerated propaganda is very likely to
cause

railings like the one of Lubos Motl. Has anyone observed the same
phenomenon

or does it only exist among the people I have to do with?

All the best,

Peter

</excerpt>