The n-Category Café

I’m giving two talks — you can see the slides here.

The first is at the special session on Homotopy Theory and Higher Categories, run by Tom Fiore, Mark Johnson, Jim Turner, Steve Wilson and Donald Yau. When I last checked, it was scheduled for Wednesday January 7th, 1–1:20 pm in Virginia Suite C, Lobby Level in the Marriott:

• Classifying Spaces for Topological 2-Groups — joint work with Danny Stevenson.
  
  Abstract: Categorifying the concept of topological group, one obtains the notion of a topological 2-group. This in turn allows a theory of ‘principal 2-bundles’ generalizing the usual theory of principal bundles. It is well-known that under mild conditions on a topological group $G$ and a space $M$, principal $G$-bundles over $M$ are classified by either the Cech cohomology $H^1(M,G)$ or the set of homotopy classes $[M,B G]$, where $B G$ is the classifying space of $G$. Here we review work by Bartels, Jurco, Baas-Bökstedt-Kro, Stevenson and myself generalizing this result to topological 2-groups. We explain various viewpoints on topological 2-groups and the Cech cohomology $H^1(M,\mathbf{G})$ with coefficients in a topological 2-group $\mathbf{G}$, also known as ‘nonabelian cohomology’. Then we sketch a proof that under mild conditions on $M$ and $\mathbf{G}$ there is a bijection between $H^1(M,\mathbf{G})$ and $[M,B|N\mathbf{G}|]$, where $B|N\mathbf{G}|]$ is the classifying space of the geometric realization of the nerve of $\mathbf{G}$.

Here are some talks by friends of mine at the same session:

• Peter May, Permutative and bipermutative categories revisited.
• Mike Shulman, Limits, derived functors, and homotopical category theory.
• Julie Bergner, Homotopical versions of Hall algebras.
• Tom Fiore, The homotopy theory of n-fold categories.

I’ll also give talk at the special session on Categorification and Link Homology, run by Aaron Lauda and Mikhail Khovanov. When I last checked, it was scheduled for Wednesday January 7th, 5:10 - 5:30 pm, in the Harding Room, Mezzanine Level, Marriott:

• Groupoidification — joint work with James Dolan, Todd Trimble, Alex Hoffnung and Christopher Walker.
  
  Abstract: There is a systematic process that turns groupoids into vector spaces and spans of groupoids into linear operators. ‘Groupoidification’ is the attempt to reverse this process, taking familiar structures from linear algebra and enhancing them to obtain structures involving groupoids. Like quantization, groupoidification is not entirely systematic. However, examples show that it is a good thing to try! For example, groupoidifying the quantum harmonic oscillator yields combinatorial structures associated to the groupoid of finite sets, while groupoidifying the $q$-deformed oscillator yields structures associated to finite-dimensional vector spaces over the field with $q$ elements. Starting with flag varieties defined over the field with $q$ elements, we can also groupoidify Hecke and Hall algebras.

Here are some talks by friends and fellow math bloggers at the same session:

• Scott Morrison, The 2-point and 4-point Khovanov categories.
• Ben Webster, A categorification of quantum tangle invariants via quiver varieties.
• Hendryk Pfeiffer, Every modular category is the category of modules over an algebra.
• Alex Hoffnung, A categorification of Hecke algebras.

Alex’s talk will pick up where mine leaves off.

My student Alissa Crans will also be there; she and Sam Nelson are running the special session on Algebraic Structures in Knot Theory. Charles Frohman and Louis Kauffman will be talking at that — I used to see them fairly often, back when I was into knots and quantum gravity.

During the conference I’ll stay with my friend the combinatorialist Bill Schmitt, who lives in Bethesda, conveniently located next to the subway into DC. We went to grad school at MIT together. He’s a student of Rota, and he’s the one who got me interested in Joyal’s work on ‘species’.

Given this profusion of friends attending the conference, I expect there’ll be some serious socializing going on. Mathematically speaking, that’s actually more productive than the ridiculously short 20-minute talks. And, it’s more fun. I really like it when two people I know and admire, who haven’t ever met, finally meet. There’ll be a lot of that going on here.

Who else reading this will be there?