Algebra/Topology Seminar

Speaker: Rune Haugseng

Title: Yet another description of ∞-operads

Abstract: A common description of (one-object) operads in Set is as algebras in symmetric sequences under the composition product. Joyal realized that symmetric sequences are equivalent (via left Kan extension) to the so-called analytic endofunctors of Set, with the composition product corresponding to composition of functors, so that operads can be described as analytic monads on Set. (Under this correspondence an operad O corresponds to the monad for free O-algebras in Set, which is analytic.) In joint work with Gepner and Kock, we introduced analytic monads on slices of the ∞-category of spaces, and proved that these are equivalent to the dendroidal Segal spaces of Cisinski and Moerdijk through an embedding of the dendroidal category into analytic monads. In this talk I will explain how analytic monads are related to Lurie's model of ∞-operads, via an equivalence that assigns to an ∞-operad O the monad for free O-algebras in spaces.