Infinity-operads as polynomial monads
Classical operads can fruitfully be regarded as monoids in
the monoidal category of species/analytic functors under the
substitution product. We establish an infinity version of
this interpretation by developing the theory of polynomial
functors over infinity categories. In the infinity world,
analytic functors enjoy a representability feature not
shared by classical analytic functors and operads: they are
polynomial. We give a description of the free monad on an
analytic endofunctor in terms of trees, and prove a nerve
theorem implying that the infinity category of analytic
monads is equivalent to the infinity category of dendroidal
Segal spaces of Cisinski and Moerdijk, one of the known
equivalent models for infinity operads. A byproduct of the
development is a Joyal theorem for homotopical species.
This is joint work with David Gepner and Rune Haugseng.