The n-Category Café

Wow. Big thanks for clock info!!! :-) I could miss my plane as well (accidentally I also go to Belgium, but to Louvain-la-Neuve, for Ulrike Tillmann lectures).

Thanks once more and have a good philosophising :-).

These notes are very interesting.

I enjoyed seeing that part of the discussion of two things and 2-things has made it into your talk (slide 6).

Another good class of examples for complicated-looking conglomerates of data that can be realized as mere components of a single $n$-categorical gadget are a plethora of various “cocycle relations” that people usually write out as long lists of symbols with, usually, confusingly many signs scattered all over the place.

Often these can be seen to be nothing but coherence conditions for structure morphisms in higher categories.

Well, I guess the encapsulation of structure is actually a recursive phenomenon:

highly complicated-looking structures may be realized as algebras for operads.

Defining an operad itself in the standard way also takes a couple of lines.

But then we follow Tom Leinster and realize that all this data going into the definition of an operad in turn may be re-assembled into the mere statement of a monad in generalized spans.

And a monad, in turn, is just a component way of saying “lax functor on the trivial 2-category”.

:-)

David talked about

Mathematics organised by stories or dramatic narratives

By the way, probably you have mentioned this elsewhere, but I’d think that one reason why physics, and in particular what might be called “modern formal high energy physics” has been such a rich source of mathematical insight, at least at the level of conjectures (Witten, Kontsevich,…) is that whatever its relation to the physical world really is (which is an issue of heated – and overheated – debate, as we know #) it does a great job of identifying lots of mathematical entities as secretly being actors in one grand story: it allows to “visualize” all kinds of sophisticated math in terms of something like physical processes.