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

TL;DR

This paper establishes an equivalence between Lurie's model of $ abla$-operads and analytic monads, showing that $ abla$-operads are fully determined by their associated monads through Lawvere theories.

Contribution

It demonstrates that Lurie's $ abla$-operads correspond exactly to pinned $ abla$-operads derived from Lawvere theories of analytic monads, providing a new perspective on their structure.

Findings

Equivalence between Lurie's $ abla$-operads and analytic monads.

Lawvere theories characterize $ abla$-operads via spans in finite sets.

$ abla$-operads are determined by their monads for $ abla$-algebras.

Abstract

We show that Lurie's model for $\infty$-operads (or more precisely a "flagged" or "pinned" version thereof) is equivalent to the analytic monads previously studied by Gepner, Kock, and the author, 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 study the Lawvere theories of analytic monads, and show that these are precisely pinned $\infty$-operads in a slight (equivalent) variant of Lurie's definition, where finite pointed sets are replaced by spans in finite sets.