From analytic monads to $\infty$-operads through Lawvere theories

Abstract

We show that a variant of Lurie’s $\infty$-operads, where the category of finite pointed sets is replaced by that of spans of finite sets, can be identified as precisely the Lawvere theories of the analytic monads previously studied by Gepner, Kock, and the author. This gives an equivalence between $\infty$-operads (or more precisely a “flagged” or “pinned” version thereof) and analytic monads, with an $\infty$-operad $\mathcal{O}$ corresponding to the monad for $\mathcal{O}$-algebras in spaces. In particular, the $\infty$-operad $\mathcal{O}$ is completely determined by this monad. To prove this we extend the equivalence between $\infty$-categorical Lawvere theories and algebraic monads, due to Gepner, Groth, and Nikolaus, to the many-object (or many-sorted) case.