# Infinity-operads as polynomial monads

- 14:00 5th September 2017 ( Trinity Term 2017 )Lecture Theatre B

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.