Higher Segal spaces

A Segal space is a simplicial topological space $X = (X_n)$ with a certain condition expressing $X_n$ as an $n$-fold homotopy fiber product of $X_1$ over $X_0$. This concept can be seen as encoding a weak (higher) categorical structure.

The talk, based on joint work in progress with T. Dyckerhoff, will discuss a "higher" (level 2) generalization of this concept where the fiber products are taken with respect to a triangulation of a 2-dimensional polygon (the product in the usual Segal condition can be thought of as corresponding to a subdivision of an interval). It turns out that such "2-Segal spaces" give rise to associative algebras, of which the Hall algebra of an abelian category is a particular instance. It corresponds to the Waldhausen space used in algebraic K-theory, which is an example of a 2-Segal space. Other examples include cyclic nerves of categories and Bruhat-Tits complexes. Further, a 2-Segal space which is cyclic (in the sense of Connes) gives rise to a kind of 2-dimensional TQFT: a system of homotopy types with actions of the Teichmueller groups.