The root functor

TL;DR

This paper demonstrates that any ∞-operad can be represented as a localization of a discrete Σ-free operad using dendroidal sets, extending known results from ∞-categories to operads.

Contribution

It introduces the root functor for dendroidal sets and proves its operadic weak equivalence, generalizing Joyal's last vertex map result to ∞-operads.

Findings

Any ∞-operad is equivalent to the localization of a discrete Σ-free operad.

The root functor is an operadic weak equivalence after localization.

The ∞-category of algebras over an ∞-operad matches that of locally constant algebras over its discrete resolution.

Abstract

In this paper we show that any $\infty$-operad is equivalent to the localization of a discrete $\Sigma$-free operad, working in the formalism of dendroidal sets. The key point is defining the root functor of a dendroidal set $X$, a functor from the dendroidal nerve of a discrete operad $\mathbf{\Omega}/X$ into $X$, which we show to be an operadic weak equivalence after localizing $\mathbf{\Omega}/X$. This extends an analogous result for $\infty$-categories due to Joyal: when $X$ is a simplicial set, $\mathbf{\Omega}/X$ is its category of elements, and the root functor is the last vertex map. As an application, we deduce that the $\infty$-category of algebras over an $\infty$-operad is equivalent to that of locally constant algebras over its discrete resolution.