Joachim Kock (UAB): "Joyal theorems for homotopical species"
  (Joint work with David Gepner)

  Joyal, in the early 1980s, introduced combinatorial species and
  the accompanying theory of analytic functors in order to explain
  the use of generating functions in combinatorics.  He proved
  that an endofunctor of Set is analytic iff it preserves filtered
  colimits, cofiltered limits and weak pullbacks.  Work in
  progress with David Gepner extends this theory from sets to
  infinity groupoids, introducing (various flavours of)
  homotopical species, supposed to be thought of as data
  structures holding homotopy cardinalities, i.e. 'counting'
  problems with values in homotopy types; as an example
  Grassmannians 'count' vector subspaces, and form the
  coefficients for a homotopical species in two variables.
  Baez-Dolan 'stuff types' (a categorification of the harmonic
  oscilator) is a special case of homotopical species.  For each
  of the flavours of homotopical species we prove a 'Joyal
  theorem'.  Assuming the theory of quasi-categories, the proofs
  are actually easier in the homotopical setting than in the
  classical case, essentially because the technically inconvenient
  notion of weak pullback evaporates, and the notion of 'analytic'
  merges with 'polynomial', allowing exploitation of the
  representability features of polynomial functors.  For the
  simplest of the notions of homotopical species, the exactness
  conditions are: preservation of sifted colimits and connected
  limits.  The classical Joyal theorem results as a corollary by
  taking connected components.  In my talk I will first review the
  classical theory of species, then recall some notions from the
  theory of quasi-categories and outline the proof of our 'Joyal
  theorem' for homotopical species.  If time permits I will
  speculate about future directions of the theory.