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The root functor

Francesca Pratali

TL;DR

The paper proves that any $\infty$-operad can be realized as the localization of the nerve of a discrete operad via the root functor in the dendroidal setting, generalizing the simplicial last-vertex localization of Joyal and Stevenson. It constructs the operad of elements $\mathbf{\Omega}/X$ and proves that $\mathcal{N}_d(\mathbf{\Omega}/X)$ localizes to $X$ under a natural class of root-preserving maps, yielding an operadic weak equivalence after localization. This yields a concrete description of the $\infty$-category of algebras over an $\infty$-operad as locally constant algebras over its discrete resolution, together with an operadic un/straightening framework compatible with the covariant model structure. The results extend classical localization phenomena from simplicial sets to dendroidal sets and establish a robust bridge between dendroidal $\infty$-operads and their discrete resolutions, with implications for understanding algebras via locally constant data. The paper also develops an operadic décalage perspective, offering a conceptual approach to the root construction that may generalize beyond the dendroidal setting.

Abstract

In this paper we show that any $\infty$-operad is equivalent to the localization of a discrete $Σ$-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 $\mathbfΩ/X$ into $X$, which we show to be an operadic weak equivalence after localizing $\mathbfΩ/X$. This extends an analogous result for $\infty$-categories due to Joyal: when $X$ is a simplicial set, $\mathbfΩ/X$ 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.

The root functor

TL;DR

The paper proves that any -operad can be realized as the localization of the nerve of a discrete operad via the root functor in the dendroidal setting, generalizing the simplicial last-vertex localization of Joyal and Stevenson. It constructs the operad of elements and proves that localizes to under a natural class of root-preserving maps, yielding an operadic weak equivalence after localization. This yields a concrete description of the -category of algebras over an -operad as locally constant algebras over its discrete resolution, together with an operadic un/straightening framework compatible with the covariant model structure. The results extend classical localization phenomena from simplicial sets to dendroidal sets and establish a robust bridge between dendroidal -operads and their discrete resolutions, with implications for understanding algebras via locally constant data. The paper also develops an operadic décalage perspective, offering a conceptual approach to the root construction that may generalize beyond the dendroidal setting.

Abstract

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

Paper Structure

This paper contains 22 sections, 19 theorems, 82 equations, 3 figures.

Table of Contents

  1. Dendroidal recollections
  2. Discrete operads
  3. The dendroidal category
  4. Some terminology
  5. Tensor product of dendroidal sets
  6. Dendroidal $\infty$-operads
  7. Dendroidal boundary and horn inclusions
  8. The homotopy theory
  9. Notation and conventions
  10. Localization of dendroidal $\infty$-operads
  11. The definition of localization
  12. The root functor
  13. The category of forests
  14. Wide maps of forests
  15. The operad of elements and the root functor
  16. ...and 7 more sections

Key Result

Theorem 1

Let $X$ be a dendroidal set, and let ${\mathcal{R}}$ be the set of morphisms of $\mathbf{\Omega}/X$ sent to identities by $\mathcalligra{r}\,_X$. The root functor $\mathcalligra{r}\,_X$ induces an operadic weak equivalence of dendroidal sets between the localization of ${\mathcal{N}}_d(\mathbf{\Omega}/X)$ at ${\mathcal{R}}$ and $X$.

Figures (3)

  • Figure 1: Some typical trees in $\mathbf{\Omega}$.
  • Figure 2: The embedding $\Delta\to \mathbf{\Omega}$
  • Figure 3: The Boardman-Vogt interchange relation for $\Omega(C_3)\otimes\Omega(C_1)$.

Theorems & Definitions (63)

  • Theorem : \ref{['main']}
  • Corollary : \ref{['westable']}
  • Corollary : Joyal, Stevenson
  • Proposition : \ref{['localg']}
  • Corollary : \ref{['corofip']}
  • Remark 1.1
  • Definition 1.2
  • Remark 1.3
  • Definition 1.4
  • Example 1.5
  • ...and 53 more