Report from Villars (5 in a series)

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

  1. each component is homeomorphic to [tex]\mathbb{RP}^1[/tex]
  2. the components are glued together along a tree
  3. all singularities are at gluepoints between components
  4. 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.