The n-Category Café

Perhaps it’ll be time soon for an update to Tom Leinster’s survey of definitions of $n$-category. There he looked at ten definitions. In his talk yesterday he mentioned that there are at least twelve. But, as he pointed out to me, counting definitions as different is not so straightforward.

Tom promised to bore all the experts present with his introductory talk. So we all strove to be bored. Not, as he noted, that boredom would establish expertise.

Eugenia Cheng gave us a talk on the periodic table, and the work she has done with Nick Gurski on degeneracy. Seeing the old table again, I was wondering what happens in the limit down that diagonal where $n$-categories stabilise. John and Jim call them stable $\omega$-categories (p. 25). The free such one on one generator they call $Braid_{\infty}$. Presumably one would call the analogous one with duals $Tang_{\infty}$, but it would turn out to be equivalent to $\Pi(\Omega^{\infty} \S^{\infty})$ in view of Eugenia’s result.

These are not to be confused with Lurie’s stable infinity categories, which are stable ($\infty, 1)$-categories.

An idle thought occurred to me on ($n, r$)-categories as to whether you might have an additional interval where $j$-morphisms have duals up to a certain point, then are equivalences, before becoming trivial.