Incidence algebras and Moebius inversion in Rezk categories
   and decomposition spaces

   I'll explain how the classical theory of incidence algebras
   of locally finite posets (Rota) and Moebius categories
   (Leroux) can be generalised to higher categories, leading to
   the new notion of decomposition space: it is a simplicial
   (infinity) groupoid satisfying an exactness condition weaker
   than the Segal condition, expressed in terms of generic and
   free maps in Delta.  Just as the Segal condition expresses
   up-to-homotopy composition, the new condition expresses
   decomposition (and as everybody knows from watch repairs, it
   is much easier to decompose than to compose).  New examples
   covered by the theory include the Faa di Bruno and
   Connes-Kreimer bialgebras, the Lawvere-Menni category of
   Moebius intervals which contains the universal Moebius
   function (but is not itself a Moebius category), and Hall
   algebras: the Waldhausen S-construction of abelian (or stable
   infinity) categories are decomposition spaces, and their
   incidence algebras are Hall algebras.  This is joint work
   with Imma Galvez and Andy Tonks.