Jim Stasheff has been appealing to me to comment on the role of=20 category theory in knot theory in the context of the =91cracks and pots=92= =20 thread. In that regard, let me begin with the story I tell in the introduction=20= to my book: At a Joint Summer Research conference a number of years back, Moishe=20 Flato at some point offered the usual dismissal of category theory--=91it=20= is a mere language=92. Kolya Reshetikhin and I undertook that evening = to=20 disabuse him of the notion by explaining Shum=92s coherence theorem (the=20= one-object version, being =91the braided monoidal category with = two-sided=20 duals compatible with the braiding is monoidally equivalent to the=20 category of framed tangles=92). This is a remarkable theorem--a=20 structure absolutely natural from the internal structure of category=20 theory is essentially identical to the key geometric sturctures in 3-=20 and 4-manifold topology, framed tangles being simply =91relative = versions=20 of=92 the framed links on which the Kirby calculi for 3- and 4-manifolds=20= depend. It is one of several theorems relating category theory, and=20 with it a great deal of algebra, to geometric topology, all of which=20 had a =93who ordered that?=94 feel about them. It is also the only = basis=20 on which the connection between knot theory and quantum groups can be=20 explained: the category of representations of a quantum group has the=20= algebraic structure for which framed tangles are a free model! I retired before the point had sunk in, leaving Kolya to continue the=20 discussion. The next morning as I sat in the back of the main lecture=20= hall, Flato came in, tapped me on the shoulder, and with a thumbs up,=20 said =93Hey! Viva les categories!. . .these new ones, the braided=20 monoidal ones.=94 Now, Shum=92s theorem is merely the first of several, all of which give=20= one the =93who ordered that?=94 impression, at least once on starts=20 thinking of TQFT=92s. The other two that come to mind require a bit of=20= set up to state fully, but I will spare you all now: they are Abrams=20 theorem that a 2-dimensional TQFT is equivalent to a Frobenius algebra,=20= and a theorem due to myself and Crane, and Kerler, that in a certain=20 category of cobordisms between surfaces with boundary, the handle (a=20 torus with a hole cut in it) has the structure of a Hopf algebra (CY,=20 K) which is self-dual (CY) and admits a right integral (K), and that=20 every surface with a circle boundary is a Yetter-Drinfel=92d module over=20= the handle. (For an ordinary finite dimensional Hopf algebra,=20 YD-modules are modules over the Drinfel=92d double, but they exist more=20= generally, for infinite dimensional Hopf algebras or Hopf algebra=20 objects in arbitrary monoidal categories, where Drinfel=92d doubles = don=92t=20 exist.) All of these are part and parcel of a different face of category theory=20= than one saw in the old days: category theory as algebra, rather than=20= category theory as foundations. Flato=92s dismissal was directed at category theory as foundations. It=20= is easily ignored if one is interested in foundations of mathematics,=20 since most mathematicians really don=92t care about foundations. For=20 example, many mathematicians pay lip service to the attitude =91set=20 theory is =91the=92 foundation of mathematics=92, but then turn around = and=20 talk about =91the real numbers=92. Which =91real numbers=92? Dedekind = cuts? =20 equivalence classes of Cauchy sequences? a complete Archimedian field=20 constructed from surreal numbers? Now we, as categorists, know the=20 question is silly: one doesn=92t bother asking which of a family of=20 isomorphic structures one means, because they are isomorphic. It is in=20= the practical sense of describing the basic structure of what=20 mathematicians actually do, that category theory is a superior=20 =91foundation=92 to set theory. (Was there ever a time when the epsilon=20= tree defining an element of a smooth manifold ever mattered to anyone?)=20= The dim view of category theory in many mathematical circles is surely=20= due to mathematicians=92 boredom with foundations--and attitude which=20 might be summed up as =93Set theory was bad enough. Why open up all=20 those questions again? Just let me do my geometry, algebra, or=20 whatever.=94 I reading this thread, I wonder how much of the concern about public=20 perceptions of category theory is really concern that =91categories as=20= algebra=92 has become the public face of category theory, concern on the=20= part of those who are fond of =91categories as foundations=92. =91Categories as foundations=92 served the subject poorly in relations = to=20 most mathematicians, but well in relation to computer science: only=20 categorists were willing to take up the challenge of polymorphic type=20 theory. If you thought =91set theory is *the* foundation=92, you bashed=20= your head against Russell=92s paradox and were no help the the folk in=20= CS. We (really those of *you* who took up the challenge) were the only=20= mathematicians who had any hope of being helpful. On the other hand, those of us who set our sights on =91core = mathematics=92=20 have been better served by =91categories as algebra=92: the = applications=20 to knot theory (and geometric topology more generally), homotopy=20 theory, deformation theory, and physics all flow from this =91face=92 of=20= category theory. Even if those of us whose love is =91categories as foundations=92 can be = a=20 little uneasy with the other face of the subject getting applied to=20 physics and drawing fire from outside mathematics, those of us whose=20 love is =91categories as algebra=92 can be uneasy about applications of = the=20 other face to philosophy (as pointed to in Peter Arndt=92s last post),=20= which are sure to be vilified (philosophers and humanists always vilify=20= their rivals, as I am learning from my daughter who is studying=20 philosophy). =91Categories as algebra=92 at least got a =91Viva!=92 = from one=20 of the fathers of deformation quantization. Best Thoughts, David Yetter On 23 Mar 2006, at 14:45, Peter Arndt wrote: > Dear category theorists, > I would like to support Krzysztof Worytkiewicz's remark that "cat=20 > 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=20 > with > category theory. In certain circles category theory seems to have=20 > 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=20 > expression > of such opinions or http://ru.philosophy.kiev.ua/rodin/Endurance.htm=20= > for a > crude offspring of them. Such exaggerated propaganda is very likely to=20= > cause > railings like the one of Lubos Motl. Has anyone observed the same=20 > phenomenon > or does it only exist among the people I have to do with? > > All the best, > > Peter >