Infinity-operads as polynomial monads

Polynomial functors play an important role in logic and computer
science, for example as semantics for inductive and coinductive
types, polymorphic functions, interaction systems, etc.  Many
monads in functional programming are polynomial.  In
combinatorics and algebraic topology, polynomial functors have
been less successful, due to symmetries and higher homotopies:
polynomial functors can model only flat species, not all
species, and polynomial monads correspond to sigma-cofibrant
operads, not all operads.

In this talk I will explain how the homotopy version of the
theory of polynomial functors remedies this, upgrading from sets
to groupoids to infinity-groupoids.  This involves a Joyal
theorem for homotopical species, an initial-algebra theorem for
accessible endofunctors, a description of the free infinity
monad on a polynomial endofunctor in terms of trees, and a nerve
theorem implying that finitary polynomial monads are a model for
infinity operads.

This is joint work with David Gepner and Rune Haugseng.