For the last two half-days of the conference, I managed to take a break
in skiing precisely when the conditions were at their very worst; with
sight down to a few meters and angry winds. Miles Gould and Arne Weiner,
however, managed to sit in a chair lift that kept stopping every 5
meters - AND they managed to break a T-bar lift. Suddenly the rope
broke, they told me, and they had to ski down to the warden with the
T-bar in the hand.
First out in this mathematical expose, though, is André Henriques,
talking about
An operad coming from representation theory
There is a way to connect to a finite Lie algebra [tex]\mathfrac
g[/tex] first it's universal enveloping algebra [tex]U\mathfrac g[/tex]
and quantum groups [tex]U_q\mathfrac g[/tex]. From representations of
[tex]U_q\mathfrac g[/tex], one path leads on over braided tensor
products to braided tensor categories. Such categories are described by
[tex]E_2[/tex] operads, which occur in the study of Gerstenhaber
algebras and their homology.
By instead of studying representation, studying crystals, Henriques
finds an interesting operad as the result of an analogous chain of
associations. A crystal is built up in analogy to a representation; as
follows:
| Representation |
Crystal |
|---|
| V a vector space with direct sum decomposition to weight spaces |
B a finite set with disjoint union decomposition |
- Chevalley operators:
- [tex]e_i:V(\lambda)\to V(\lambda+\alpha_i)[/tex]
[tex]f_i:V(\lambda)\to V(\lambda-\alpha_i)[/tex]
|
- Raising and lowering operators:
- [tex]e_i:B(\lambda)\to B(\lambda+\alpha_i)[/tex]
[tex]f_i:B(\lambda)\to B(\lambda-\alpha_i)[/tex]
|
Would it be possible to find bases for the representation that gets
mapped to "itself" under the operators, then the work would be done
here, and crystals would be the same as representations. This is,
however, not possible.
Which isn't to say that they don't have anything to do with each other.
There is a bijection of isomorphism classes between representations and
crystals over a Lie algebra [tex]\mathfrac g[/tex]
The analogy to the braid group, when studying the categorical properties
of the crystals, is the cactus group - which also is the fundamental
group of the manifold whose points correspond to isomorphism classes of
real algebraic curves such that
- each component is homeomorphic to [tex]\mathbb{RP}^1[/tex]
- the components are glued together along a tree
- all singularities are at gluepoints between components
- each component has at least three points - either crossing points or
marked points
These end up being governed by an operad; which in turn has
2-Gerstenhaber algebras as their homology.
Scherer, later, held a talk on
Relative homotopy cyclic homology
in which he seems to want to recast the +-construction of Quillen in
operadic terms.
Next morning, the funky stuff starts. First out is Eugenia Cheng
Operads and multicategories
The talk was expository, early, brilliant and very lucid. In my humble
opinion the best of the whole conference.
Cheng set out to illustrate the theory and current state of research of
multicategories; and did this by displaying multicategories as a
simultaneous generalisation of operads and categories. Operads describe
operations with several inputs and one input. This will be generalized
to encompass both many different objects and inputs in some interesting
configuration. (This requires nifty pictures which I don't really have
any decent way of reproducing here. Poke me if you want the pictures
that go with the text...)
Categorically, we're motivated by the possibility of getting composition
of higher cells (0-cells = objects, 1-cells = morphisms/arrows, 2-cells
= natural transformations, .... an arrow between two n-1-cells is an
n-cell)
Topologically, this gives us a way to deal with composition in loop
spaces; or to even deal with "path spaces" with many possible objects,
but compatibility requirements on compositions of paths.
So. To get her idea through, Cheng starts by giving the definition of
category as she sees it:
A category C is a collection ob C of objects, for any pair of
objects, a set of arrows [tex]a\to b[/tex], for any object a
canonical arrow [tex]a\to a[/tex] and a composition [tex]a\to b\to
c[/tex]; with axioms that make this work as expected.
A multicategory C, as defined by Lambek in 1969, is a collection of
objects ob C, for any sequence of inputs [tex]a_1,\dots,a_n[/tex] and
an output, b, a set [tex]C(a_1,\dots,a_n;b)[/tex] of arrows, with a
canonical arrow in C(a;a) and a composition of arrows.
A non-symmetric operad in the category of sets is a multicategory with
one object. The hom-sets can be indexed by their number of inputs, and
compotision needs no source/target matching.
This way of looking at categories with a single object is quite useful:
| Many |
One |
|---|
| category |
monoid |
| groupoid |
group |
| bicategory |
monoidal category |
| multicategory |
operad |
| abelian category |
ring |
| topological category |
topological monoid |
| topologial multicategory |
topological operad |
The topological varieties go on beyond the pair listed here, but the
idea is clear.
In some terminology, multicategories are called "coloured operads".
Cheng points out that in that case, it would be consistent to call
categories "coloured monoids" and groupoids "coloured groups" et.c.;
which would be ... poetic.
Some examples
A category is a multicategory where every arrow is unary.
A monoidal category has an underlying multicategory; with arrows in
[tex]C(a_1,\dots,a_n;b)[/tex] given by the arrows
[tex]a_1\otimes\dots\otimes a_n\to b[/tex] for [tex]\otimes[/tex]
the monoidal operation.
For many monoidal thingies, the monoidality is not needed, but merely a
multicategory conditions. As the categorists like to hunt down minimal
required conditions, this is a very relevant observation.
Now, this multicategory game could just as well be seen as using a
monad to capture the input. More specifically, the "free monoid"-monad
[tex]\mathcal F[/tex] (i.e. the composition of the adjoint pair of a
forgetful and a free functor) gives rise to these strings of inputs that
occured up previously. By replacing [tex]\mathcal F[/tex] by any other
monad T, and expanding the obvious (for some value of obvious) axioms
and propositions, you get a sane theory for T-multicategories.
By expanding on this idea, various combinations of base categories and
monads in these give rise to a new, cool and exciting sequence of new
categories.
Tom Leinster in Glasgow has expanded on the idea of PROPs by introducing
monads in a similar manner.
After this talk, Mark Weber gave a highly technical talk on
Applications of 2-categorical algebra to the theory of operads
This talk tried, it seems, to generalize operads to higher category
levels. I didn't really understand much. At all.
Finally, Peter May gave
An oldfashioned elementary talk
Peter May started by lamenting that:
I had good results in topology to talk about - but not all here
are topologists.
I had nice category theory to talk about - but not all here are
categorists.
I knew some nice things about operads in algebraic geometry - but
definitely not all here are algebraic geometers
So he ended up talking about constructions of Steenrod operations in
various situations, based on a construction through the Eilenberg-Zilber
operad. Among other things, it seems that for good algebras A,
[tex]\operator{Ext}_A^{*,*}(\mathbb F_p,\mathbb F_p)[/tex] has
Steenrod operations. What this means, and whether it's of any use
outside of topology, is unknown to me and quite interesting. Or so it
seems.