The n-Category Café

Whoops! Weird.

Fixed. Thanks!

David wrote:

Could either or both of Derek and Jeffrey be persuaded to give a brief summary of the Traces conference as a guest post?

No. They are both having thesis defenses at the end of this month, and they’re desperately writing stuff up. No blogging for them!

Jeff’s thesis is on extended TQFTs — rigorously constructing the Dijkgraaf-Witten model as a 2-functor

$$Z:{\mathrm{nCob}}_2\to 2\mathrm{Vect}$$

for all $n$. Lots of nice spans of groupoids show up here!

Derek’s thesis is on topological gauge theory, Cartan geometry and gravity. Right now he’s busily classifying conjugacy classes in $\mathrm{SO}(\mathrm{4,1})$ and showing how they match up to various kinds of particles.

What I should really do is get them to write guest blogs about their theses after they’re done!

The conference on traces sounded fun, but I have not been able to extract any specific new insights from their descriptions of what went on.

There has been an ominous silence surrounding my attempt to view 2-geometry as being about bundles (perhaps also orbifolds).

I thought I said something concenring the path groupoid, regarded as a “finite” version of the tangent bundle.

Notice that every smooth category for which both source and target maps are surjective submersions may be regarded as a bundle (of morphisms) over the space of objects in two ways.

Regarding a vector bundle as a smooth category where composition of morphisms is addition of vectors in a given fibre is just one special degenrate case of this:

the category is the skeletal Lie groupoid whose vertex groups are the abelian groups given by addition in the fibers.

I am just saying this to emphasize that – indeed – (certain kinds of) bundles naturally fit into the realm of 2-geometry, if the latter is the study of categories internal to suitable categories of “spaces” (smooth spaces in the above examples).