Back to the Future
The format was informal, the idea being to speak about some big problem in one's own area of interest that should have wider appeal. I have never been to a workshop where this worked so well. Tuesday kicked off with Steve Lack giving an introductory talk on Topos Theory. For this week the term introductory assumes that one is either really into Category Theory or really, really good at Algebraic Geometry or Homotopy Theory.
James Borger followed with some revision on Algebraic Spaces and a question about related, weaker topos like structure. To begin with, an old problem with Rng, the category of rings, is that it doesn't have nice limits for doing Algebraic Geometry. The traditional solution is to use schemes. But then one ends up relying on the Axiom of Choice even though it isn't really needed, and the whole theory is very complicated. So instead of a category of schemes one defines a category AlgSp of Algebraic Spaces. This is the opposite category to Rng (= affine schemes) equipped with an etale topology. So things start to look more topos theoretic. James wants to think of a subfunctor from AlgSp into etale sheaves as a kind of half topos.
Boris Chorny modestly launched into some pretty technical ideas on small presheaves. A motivation for his questions was the Motivic Homotopy of Voevodsky, something I would dearly like to understand if I lived another 100 years. This means looking at functors from (wait for it) the opposite category of finite smooth schemes over S with the Nisnevich topology into the category of simplicial sets. Boris says that the new understanding of Motivic Spaces means that we need to redo the classical theory. For instance, is there a way of doing Motivic Homotopy in bigger categories?
Mark Weber spoke about 2-toposes. He has been doing some very interesting work on this. A nice example should be a category of categories CAT. So what is the analogue of the subobject classifier and truth arrow? One thing that works is the forgetful functor from pointed sets into Set. So the suboject classifier becomes the whole category Set! Mark's notion of 2-topos has the advantage of being able to axiomatise the notion of size. Other examples are based on internal categories in globular sets.
On Tuesday afternoon we had a departmental seminar by Ross Street on his recent work (with Craig Pastro) on quantum categories and weak Hopf algebras in braided monoidal categories. I will talk about this at some later date.
Wednesday morning was Operad Time. Yummy! After a short talk by a certain disreputable physicist, Michael Batanin asked the participants about homotopy types and the search for an ideal theorem about their characterisation by higher groupoids. Fortunately this involved some introductory words on higher operads. Things get particularly interesting when one gets to dimension three. 3-homotopy types require Gray groupoids. These arise as algebras of the Gray 3-operad G. Let's briefly describe this. For the 0-tree and for any 1-tree, G is the singleton. For a 2-tree which is labelled by ordinals (m1,m2,...,mk), G is the shuffles on (m1,m2,...,mk). And finally, for a 3-tree G is given by a Cartesian product of G for the boundary 2-trees. This structure is due to the weakened interchange law of Gray categories. The question is: what are the higher dimensional analogues of this operad? Batanin also gave the Colloquium talk on Thurdsday afternoon, about the relation of his work to Deligne's conjecture.
Simona Paoli was wondering about a model structure for internal categories. She discussed a 2005 paper by Everaert et al and open questions related to the Tamsamani approach to higher categories. This involves some intriguing looking cubical structures, which I don't understand at all. Alexei Davydov then spoke about autoequivalences, and Amnon Neeman about equivalences for derived categories.
On Thursday morning we were visited by Peter Bouwknegt, who posed the question of a good definition for C* algebra objects in monoidal categories, with motivation from his work on T-duality. At this point there was some laughter on the invasion of physics into such a pure mathematics workshop. Later on we actually found some time for more introductory talks. Boris Chorny told us about the Calculus of Functors. He focused on a dictionary between the traditional calculus of functions and the Homotopy Calculus. To begin with, instead of a function from a manifold M into R one has a functor from pointed topological spaces into either a category of topological spaces or a category of spectra. The notion of |x - y| small becomes f: X --> Y is highly connected. The replacement of derivatives defined using h-->0 is quite abstract: the derivative of a functor F with respect to a pointed space X is the homotopy colimit (as n --> oo) of the n-fold loop space of the homotopy fibrations of F(X v S^n) --> F(X). Er, yeah, OK. Anyway, the point is that spheres S^n for large n are like highly contractible spaces. So we had a real h-->0 and now we have a discrete n --> oo.
My favourite talk for the week was the last: James Borger speaking about Lambda Rings and related goodies. But enough blathering from me on all this!